ABSTRACT
Title of dissertation: PREDICTING THE TRANSPORT
PROPERTIES OF AEROSOL PARTICLES
IN CREEPING FLOW
FROM THE CONTINUUM
TO THE FREE MOLECULE REGIME
James Corson, Doctor of Philosophy, 2018
Dissertation directed by: Professor Michael R. Zachariah
Department of Chemical
and Biomolecular Engineering
The transport of nanoscale aerosol particles plays an important role in many
natural and industrial processes. Despite its importance, the transport behavior
of aerosol aggregates is poorly understood, largely due its complex dependence on
particle size, shape, and orientation. Often, these particles are in the transition
regime, where neither the continuum approximation for large particles nor the free
molecule approximation for small particles is valid.
At present, methods for calculating the aerodynamic force on and diffusive
behavior of fractal aggregates in the transition regime either rely upon scaling laws
fitted to experimental data or computationally-intensive direct simulation Monte
Carlo or molecular dynamics approaches. Thus, there is a pressing need for a new
method for determining aerosol transport properties.
This dissertation introduces such a method for calculating the drag and torque
on an aerosol aggregate as a function of the primary sphere size and the aggregate
size and shape. This method is an extension of Kirkwood-Riseman theory to the
transition regime, using an appropriate model for interactions between the individual
spheres in an aggregate.
This dissertation also describes the application of this extended Kirkwood-
Riseman (EKR) method to a number of problems related to aerosol transport, in-
cluding computation of the scalar translational and rotational friction coefficients of
aggregates formed by diffusion-limited processes, analysis of the effects of alignment
on particle migration in an electric field, and the strength of interactions between
particles due to their effects on the surrounding fluid flow field.
In each of these applications, results from the EKR method are in good agree-
ment with published experimental data and computational results. EKR results
also demonstrate that particle translational and rotational behavior becomes more
continuum-like as both primary particle size and the number of spheres in the ag-
gregate increase.
Using these results, new correlations have been developed for the translational
and rotational friction coefficients of aggregates formed by diffusion-limited processes
(e.g. soot); these correlations are more accurate than the empirical models currently
available in the literature.
PREDICTING THE TRANSPORT PROPERTIES
OF AEROSOL PARTICLES IN CREEPING FLOW
FROM THE CONTINUUM TO THE FREE MOLECULE REGIME
by
James Corson
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2018
Advisory Committee:
Professor Michael R. Zachariah, Chair
Dr. George W. Mulholland
Professor Richard V. Calabrese
Professor Panagiotis Dimitrakopoulos
Professor Elaine S. Oran
©c Copyright by
James Corson
2018
Dedication
To my wife, Holly, who convinced me to go back to school to earn my Ph.D.
Were it not for her encouragement, I would still be Mr. Corson.
ii
Acknowledgments
First and foremost, I would like to thank my adviser, Prof. Michael Zachariah,
as well as Dr. George Mulholland. They provided me with valuable guidance
throughout my time at the University of Maryland, from suggesting that I take
on a more focused and manageable project than the mess I had in my mind when
I started at Maryland, to introducing me to the Kirkwood-Riseman method that
plays such a prominent role in this dissertation, to mentioning ways to improve my
papers and presentations. I could not ask for two better advisers.
I would also like to thank Prof. Howard Baum in the Department of Fire
Protection Engineering at the University of Maryland for introducing me to the
BGK model and explaining how one might go about solving the Boltzmann transport
equation for flow around a sphere in the transition regime.
Thank you to Dr. Walid Keyrouz and Dr. Derek Juba at the National Institute
of Standards and Technology, who provided me with the ZENO code. I have used
the Deepthought2 cluster at Maryland to run Zeno and some of my own codes; thank
you to the Division of Information Technology for maintaining the cluster and to
Prof. Jeffery Klauda in the Department of Chemical and Biomolecular Engineering
for helping me get an initial allocation to the cluster.
Finally, thank you to my wonderful family and friends, especially my parents
and my wife. My parents always encouraged me to challenge myself and instilled in
me the work ethic needed to complete all 21 years (!) of my formal education. My
iii
wife encouraged me to go back to school for my Ph.D. and supported me throughout
my time at Maryland, for which I am forever grateful.
This research was performed while under appointment to the U.S. Nuclear
Regulatory Commission Graduate Fellowship Program.
iv
Table of Contents
List of Tables x
List of Figures xii
List of Abbreviations xv
1 Introduction 1
1.1 Aerosol Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Creeping Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Aerosol Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 Continuum Regime . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.2 Free Molecule Regime . . . . . . . . . . . . . . . . . . . . . . 14
1.3.3 Transition Regime . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4 Experimental Techniques for Obtaining Particle Size . . . . . . . . . 20
1.4.1 Mobility Measurements . . . . . . . . . . . . . . . . . . . . . . 21
1.4.2 Optical Measurements . . . . . . . . . . . . . . . . . . . . . . 24
1.5 Scope of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 The BGK Model Equation 31
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Solution of the Krook Equation for Isothermal Mass Transfer to a
Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3 Solution of the Krook Equation for Uniform Flow Around a Sphere . 43
2.3.1 Derivation of the Governing Equations . . . . . . . . . . . . . 43
2.3.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . 62
2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3 Friction Coefficient for Translating Particles 69
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 Velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3 Kirkwood-Riseman theory . . . . . . . . . . . . . . . . . . . . . . . . 76
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
v
4 Analytical Expression for the Friction Coefficient of DLCA Aggregates based
on Extended Kirkwood-Riseman Theory 85
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.2 Theoretical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2.1 Kirkwood-Riseman Theory . . . . . . . . . . . . . . . . . . . . 89
4.2.2 Flow around a Sphere . . . . . . . . . . . . . . . . . . . . . . 92
4.2.3 Application of BGK Results to Kirkwood-Riseman Theory . . 96
4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3.1 Comparison to Experimental Data and Power-Law Models . . 98
4.3.2 Uncertainty in the Calculated Friction Coefficients . . . . . . . 105
4.3.3 Analytical Expression for Friction Coefficients of Aggregates . 108
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5 Calculating the Rotational Friction Coefficient of Fractal Aerosol Particles
in the Transition Regime using Extended Kirkwood-Riseman Theory 115
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2 Drag and torque on a rigid particle . . . . . . . . . . . . . . . . . . . 117
5.2.1 Kirkwood-Riseman Theory . . . . . . . . . . . . . . . . . . . . 119
5.2.2 Extension to the Transition Regime . . . . . . . . . . . . . . . 123
5.2.3 Monte Carlo Calculations for Free Molecule Drag and Torque 125
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.3.1 Continuum regime . . . . . . . . . . . . . . . . . . . . . . . . 129
5.3.2 Free Molecule Regime . . . . . . . . . . . . . . . . . . . . . . 132
5.3.3 Transition Regime . . . . . . . . . . . . . . . . . . . . . . . . 134
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6 Analytical Expression for the Rotational Friction Coefficient of DLCA Ag-
gregates over the Entire Knudsen Regime 142
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.2 Theoretical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.2.1 Extended Kirkwood-Riseman Method for the Rotational Fric-
tion Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.2.2 Adjusted Sphere Method for the Rotational Friction Coefficient150
6.2.3 Scaling Laws for the Rotational Friction Coefficient in the
Continuum and Free Molecule Regimes . . . . . . . . . . . . . 152
6.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.3.1 Comparison to Experimental Data . . . . . . . . . . . . . . . 155
6.3.2 Comparison to Results in the Continuum and Free Molecule
Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.3.3 Relative Importance of Translational and Rotational Diffusion 161
6.3.4 Uncertainty in the Calculated Rotational Friction Coefficients 163
6.3.5 Analytical Expression for Rotational Friction Coefficients of
DLCA Aggregates . . . . . . . . . . . . . . . . . . . . . . . . 167
6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
vi
7 The Effect of Electric Field Induced Alignment on the Electrical Mobility of
Fractal Aggregates 172
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
7.2 Theoretical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.2.1 Particle Orientation in an Electric Field . . . . . . . . . . . . 175
7.2.2 Average Drift Velocity of a Particle in an Electric Field . . . . 179
7.2.3 Friction Tensor for an Aggregate . . . . . . . . . . . . . . . . 181
7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.3.1 Comparison to Experimental Data . . . . . . . . . . . . . . . 183
7.3.2 Effects of Aggregate Size and Field Strength on Mobility . . . 185
7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.4.1 General Observations . . . . . . . . . . . . . . . . . . . . . . . 192
7.4.2 Validity of the Slow Rotation Assumption . . . . . . . . . . . 193
7.4.3 Polarizability Versus Friction . . . . . . . . . . . . . . . . . . 194
7.4.4 Using Field-dependent Mobility to Evaluate Particle Shape . . 196
7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
8 Hydrodynamic Interactions between Particles 200
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
8.2 Theoretical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
8.2.1 Two spheres in continuum flow . . . . . . . . . . . . . . . . . 203
8.2.2 Aggregates in continuum flow . . . . . . . . . . . . . . . . . . 206
8.2.3 Extended Kirkwood-Riseman theory . . . . . . . . . . . . . . 209
8.2.4 Point force approach . . . . . . . . . . . . . . . . . . . . . . . 211
8.3 Two particle results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
8.3.1 Sphere results . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
8.3.2 Aggregate results . . . . . . . . . . . . . . . . . . . . . . . . . 217
8.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8.4.1 Aerosol clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.4.2 Additional considerations . . . . . . . . . . . . . . . . . . . . 225
8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
9 NGDE: A MATLAB-based Code for Solving the Aerosol General Dynamic
Equation 229
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
9.2 Overview of Numerical Methods for Solving the GDE . . . . . . . . . 232
9.3 NGDE Code Description . . . . . . . . . . . . . . . . . . . . . . . . . 235
9.3.1 Coagulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
9.3.2 Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
9.3.3 Surface Growth . . . . . . . . . . . . . . . . . . . . . . . . . . 240
9.3.4 Solution Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 242
9.4 Sample Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
9.4.1 Pure Coagulation . . . . . . . . . . . . . . . . . . . . . . . . . 245
9.4.2 Pure Surface Growth . . . . . . . . . . . . . . . . . . . . . . . 248
9.4.3 Nucleation and Coagulation . . . . . . . . . . . . . . . . . . . 251
vii
9.4.4 Full GDE for Condensation of Aluminum . . . . . . . . . . . . 252
9.5 Limitations of NGDE . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
9.6 NGDEplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
9.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
10 Conclusions and Recommendations for Future Work 263
10.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
10.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . 268
10.2.1 Friction Coefficient Expressions for non-DLCA Aggregates . . 268
10.2.2 Aggregates with Polydisperse Primary Spheres . . . . . . . . . 269
10.2.3 Rotational and Coupling Interactions . . . . . . . . . . . . . . 272
10.2.4 Brownian Dynamics . . . . . . . . . . . . . . . . . . . . . . . 273
A Derivation of Expressions in Chapter 2 278
A.1 Derivation of the Expression for g [Eq. (2.42)] . . . . . . . . . . . . . 278
A.2 Derivation of the Source Term Expressions (Eqns. 2.40–2.41) . . . . . 284
A.3 Derivation of H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
A.4 Derivation of the Drag Expression (Eq. (2.54)) . . . . . . . . . . . . . 296
B BGK Results 301
C Monte Carlo Drag and Torque Results 327
C.1 Drag on a Translating Sphere . . . . . . . . . . . . . . . . . . . . . . 327
C.2 Drag on an Aggregate . . . . . . . . . . . . . . . . . . . . . . . . . . 327
C.3 Torque on a Rotating Sphere . . . . . . . . . . . . . . . . . . . . . . . 328
D Relationship between the Rotation and Coupling Interaction Tensors and the
Flow around a Sphere 329
E Supplemental Material for Chapter 7 333
E.1 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
E.2 Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 333
E.3 Sample Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
E.4 Effects of Knudsen Number and the Number of Primary Spheres on
Fully-Aligned Particle Mobility . . . . . . . . . . . . . . . . . . . . . 341
F NGDE User Manual 351
F.1 Running NGDE and NGDEplot . . . . . . . . . . . . . . . . . . . . . 351
F.2 Description of Input and Output . . . . . . . . . . . . . . . . . . . . 356
F.2.1 NGDE Input (ngdein) . . . . . . . . . . . . . . . . . . . . . . 356
F.2.2 NGDEplot Input (plotoptions) . . . . . . . . . . . . . . . . 358
F.2.3 NGDE and NGDEplot Output . . . . . . . . . . . . . . . . . 359
F.3 Code Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
F.3.1 NGDE Subroutines . . . . . . . . . . . . . . . . . . . . . . . . 361
F.3.2 NGDE Main Program . . . . . . . . . . . . . . . . . . . . . . 364
F.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
viii
G MATLAB Codes Referenced in this Disseration 367
G.1 Code for Calculating the Velocity around a Sphere . . . . . . . . . . . 367
G.1.1 Code Listing for bgk sphere par . . . . . . . . . . . . . . . . 369
G.2 Codes for Calculating the Friction and Diffusion Tensors . . . . . . . 382
G.2.1 Code Listing for continuum tensors . . . . . . . . . . . . . . 382
G.2.2 Code Listing for continuum tensors 3rd . . . . . . . . . . . . 384
G.2.3 Code Listing for bgk tensors . . . . . . . . . . . . . . . . . . 387
G.3 Codes for Calculating the Average Friction Coefficient of a Particle
in an Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
G.3.1 Code Listing for avg bgk velocity . . . . . . . . . . . . . . . 391
G.3.2 Code Listing for avg bgk drag . . . . . . . . . . . . . . . . . . 394
G.4 Codes for Hydrodynamic Interactions between Particles . . . . . . . . 398
G.4.1 Code Listing for bgk two particles . . . . . . . . . . . . . . 398
G.4.2 Code Listing for bgk cloud . . . . . . . . . . . . . . . . . . . 402
Bibliography 407
Publications and Presentations 420
ix
List of Tables
3.1 Comparison of my results for F/FFM to Millikan’s data and to results
from previous computational studies . . . . . . . . . . . . . . . . . . 75
5.1 Continuum friction coefficient for fractal aggregates, normalized by
the monomer friction results . . . . . . . . . . . . . . . . . . . . . . . 132
5.2 Free molecule results for fractal aggregates, normalized by the mon-
omer friction results . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.1 Comparison of EKR results to experimental data from the literature . 156
8.1 Speed of two spheres moving parallel to their line of centers, relative
to the speed of isolated spheres subjected to the same external force . 206
8.2 Speed of two spheres moving anti-parallel to their line of centers,
relative to the speed of isolated spheres subjected to the same external
force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
8.3 Speed of two spheres moving perpendicular to their line of centers,
relative to the speed of isolated spheres subjected to the same external
force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
9.1 Moments, Mi, of the particle size distribution for the pure coagulation
calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
B.1 Results for a = 0.01 (Kn = 88.8) . . . . . . . . . . . . . . . . . . . . . 302
B.2 Results for a = 0.025 (Kn = 35.5) . . . . . . . . . . . . . . . . . . . . 303
B.3 Results for a = 0.05 (Kn = 17.8) . . . . . . . . . . . . . . . . . . . . . 304
B.4 Results for a = 0.075 (Kn = 11.8) . . . . . . . . . . . . . . . . . . . . 305
B.5 Results for a = 0.1 (Kn = 8.88) . . . . . . . . . . . . . . . . . . . . . 306
B.6 Results for a = 0.25 (Kn = 3.55) . . . . . . . . . . . . . . . . . . . . . 307
B.7 Results for a = 0.5 (Kn = 1.78) . . . . . . . . . . . . . . . . . . . . . 308
B.8 Results for a = 0.75 (Kn = 1.18) . . . . . . . . . . . . . . . . . . . . . 309
B.9 Results for a = 1.0 (Kn = 0.888) . . . . . . . . . . . . . . . . . . . . . 310
B.10 Results for a = 1.25 (Kn = 0.710) . . . . . . . . . . . . . . . . . . . . 311
B.11 Results for a = 1.5 (Kn = 0.592) . . . . . . . . . . . . . . . . . . . . . 312
x
B.12 Results for a = 1.75 (Kn = 0.5074) . . . . . . . . . . . . . . . . . . . 313
B.13 Results for a = 2.0 (Kn = 0.444) . . . . . . . . . . . . . . . . . . . . . 314
B.14 Results for a = 2.5 (Kn = 0.355) . . . . . . . . . . . . . . . . . . . . . 315
B.15 Results for a = 3.0 (Kn = 0.296) . . . . . . . . . . . . . . . . . . . . . 316
B.16 Results for a = 4.0 (Kn = 0.222) . . . . . . . . . . . . . . . . . . . . . 317
B.17 Results for a = 5.0 (Kn = 0.178) . . . . . . . . . . . . . . . . . . . . . 318
B.18 Results for a = 6.0 (Kn = 0.148) . . . . . . . . . . . . . . . . . . . . . 319
B.19 Results for a = 7.0 (Kn = 0.1269) . . . . . . . . . . . . . . . . . . . . 320
B.20 Results for a = 8.0 (Kn = 0.111) . . . . . . . . . . . . . . . . . . . . . 321
B.21 Results for a = 9.0 (Kn = 0.0987) . . . . . . . . . . . . . . . . . . . . 322
B.22 Results for a = 10.0 (Kn = 0.0888) . . . . . . . . . . . . . . . . . . . 323
B.23 Results for a = 50 (Kn = 0.0178) . . . . . . . . . . . . . . . . . . . . 324
B.24 Results for a = 100 (Kn = 0.00888) . . . . . . . . . . . . . . . . . . . 325
B.25 Results for c1 and c2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
E.1 Coordinates of the center of each sphere in my sample aggregate . . . 337
xi
List of Figures
1.1 Differential mobility analyzer . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Geometry for the mass transfer problem . . . . . . . . . . . . . . . . 40
2.2 Geometry for Eq. 2.25 . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.3 Geometry for Eq. 2.26 . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4 Geometry for determining A(r0) . . . . . . . . . . . . . . . . . . . . . 51
3.1 Ratio of the calculated drag from the Krook equation to the free
molecule drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.2 Open and dense 20 particle aggregates . . . . . . . . . . . . . . . . . 80
3.3 Comparison of my results for the slip correction factor to DSMC
results for a dimer and open and dense 20-particle aggregates . . . . . 81
3.4 Calculated slip correction factors for a range of aggregate morpholo-
gies, plotted versus the aggregate Knudsen number . . . . . . . . . . 83
4.1 Friction factor results for fractal aggregates with primary sphere di-
ameter 19.5 nm in ambient air (Kn = 7) . . . . . . . . . . . . . . . . 99
4.2 Comparison of self-consistent field results to other models for the
scalar friction factor for several Knudsen numbers . . . . . . . . . . . 101
4.3 Ratio of friction coefficients from other models to my results . . . . . 103
4.4 Normalized friction coefficient results for a range of aggregate sizes . . 104
4.5 Relationship between the mobility radius and the radius of gyration
for several Knudsen numbers . . . . . . . . . . . . . . . . . . . . . . . 105
4.6 Normalized friction coefficient as a function of the primary sphere
Knudsen number and the number of primary spheres, N , calculated
using Eq. (4.38) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.7 Error of my harmonic sum model for the friction coefficient relative
to my EKR results for a range of Knudsen numbers . . . . . . . . . . 112
5.1 Representations of the fractal aggregates used in this study . . . . . . 130
5.2 Calculated rotational slip correction factor for a dimer, linear hex-
amer, and octahedral hexamer . . . . . . . . . . . . . . . . . . . . . . 136
5.3 Calculated rotational slip correction factor for four fractal aggregates 137
xii
5.4 Rotational slip correction factor plotted versus an aggregate Knudsen
number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.1 Rotational friction coefficient results for Kn = 0.1, 1, 2, and 10 . . . . 159
6.2 Ratio of rotational friction coefficients for N = 2000 . . . . . . . . . 160
6.3 Ratio of the characteristic translational diffusion time to the charac-
teristic rotational diffusion time for DLCA aggregates . . . . . . . . . 162
6.4 Torque on each sphere of two rotating 20-particle aggregates . . . . . 166
6.5 Ratio of the drag on each sphere in a rotating 20-particle aggregate
to the drag on an isolated sphere . . . . . . . . . . . . . . . . . . . . 167
6.6 Error in the analytical expression for the rotational friction coefficient
relative to my EKR results . . . . . . . . . . . . . . . . . . . . . . . . 169
7.1 Comparison of my calculated orientation-averaged mobilities to pub-
lished experimental data . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.2 Normalized mobility as a function of electric field strength for 100-
sphere and 1000-sphere aggregates . . . . . . . . . . . . . . . . . . . . 188
7.3 Normalized mobility as a function of electric field strength for aggre-
gates with primary sphere radii of 25 nm and 5 nm . . . . . . . . . . 189
7.4 Maximum electric field strength at which particles are randomly ori-
ented . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
7.5 Ratio of fully-aligned to random electric mobilities . . . . . . . . . . . 191
7.6 Reduced rotation velocity for a range of primary sphere sizes and
Knudsen numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.1 Two spheres in the parallel, anti-parallel, and perpendicular flow con-
figurations, and two 10-sphere aggregates with random orientations
in parallel flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
8.2 Speed of two spheres moving parallel, anti-parallel, and perpendicular
to their line of centers . . . . . . . . . . . . . . . . . . . . . . . . . . 216
8.3 Hydrodynamic force on an aggregate as a function of the distance
between its center of mass and the center of mass of an identical
aggregate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
8.4 Effects of orientation on the hydrodynamic force on one of two 500-
sphere aggregates with primary sphere Kn = 2.7 . . . . . . . . . . . . 219
8.5 Average velocity for a cloud of spheres . . . . . . . . . . . . . . . . . 224
9.1 NGDE volume nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
9.2 Illustration of the NGDE algorithm . . . . . . . . . . . . . . . . . . . 243
9.3 Non-dimensional size distribution for the pure coagulation problem . 247
9.4 Size distribution calculated by NGDE for the pure surface growth
sample problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
9.5 Volume-mean particle diameter calculated by NGDE for the pure
surface growth problem . . . . . . . . . . . . . . . . . . . . . . . . . . 251
9.6 Particle size distribution at select times for nucleation and coagula-
tion of aluminum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
xiii
9.7 Critical volume and nucleation rate for nucleation and coagulation of
aluminum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
9.8 Particle size distribution at select times for nucleation, coagulation,
and surface growth of aluminum . . . . . . . . . . . . . . . . . . . . . 254
9.9 Nucleation rate and saturation ratio early in the simulation for nu-
cleation, coagulation, and surface growth of aluminum by NGDE . . . 254
9.10 Monomer and total particle concentration for nucleation, coagulation,
and surface growth of aluminum . . . . . . . . . . . . . . . . . . . . . 255
9.11 Screenshot of the particle size distribution movie from NGDEplot . . 260
9.12 Screenshot of the light scattering movie from NGDEplot . . . . . . . 261
E.1 Representation of the Euler angles that relate the body-fixed coordi-
nates to the space-fixed coordinates . . . . . . . . . . . . . . . . . . . 334
E.2 Probability distributions for particles with 652 primary spheres with
5 nm radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
E.3 Ratio of fully-aligned to random electric mobilities for wide range of
primary sphere Knudsen numbers and the number of primaries . . . . 342
E.4 Comparison of the continuum mobility and radius of gyration ratios . 345
E.5 Comparison between the average continuum mobility ratio and the
average radius of gyration ratio as a function of N . . . . . . . . . . . 346
E.6 Comparison of the free molecule mobility and projected area ratios . 347
E.7 Comparison between the average free molecule mobility ratio and the
average projected area ratio as a function of N . . . . . . . . . . . . . 348
xiv
List of Abbreviations
APM Aerosol Particle Mass [analyzer]
ASM Adjusted Sphere Method
BGK Bhatnagar-Gross-Krook [model]
DLCA Diffusion-Limited Cluster Aggregation
DMA Differential Mobility Analyzer
DSMC Direct-Simulation Monte Carlo
EKR Extended Kirkwood-Riseman [method]
GDE General Dynamic Equation
KR Kirkwood-Riseman [theory]
MC Monte Carlo
NGDE Nodal General Dynamic Equation [solver]
ODE Ordinary Differential Equation
PA Projected Area
PFDMA Pulsed Field Differential Mobility Analyzer
PSD Particle Size Distribution
RPY Rotne-Prager-Yamakawa [tensor]
SPD Self-Preserving Distribution
TDMA Tandem Differential Mobility Analyzer
xv
Chapter 1: Introduction
Nanoscale aerosol particles formed at high temperature are found in many
natural and engineered environments [1–3]. A particle’s size and shape significantly
affect its transport properties, most notably the aerodynamic force it experiences as
it moves through an external force field [4–6]. Research on this topic is motivated
by the widespread use of aerosol reactors for the manufacturing of carbon black,
ceramics (e.g. SiO2 and TiO2), catalysts, and optical fibers present in numerous
consumer products [2, 7–10]; the impacts of aerosols generated in the combustion of
fossil fuels or from volcanic eruptions on climate, both through direct absorption or
scattering of incident solar radiation and through its influence on cloud formation
[11–16]; and the adverse human health effects of particle uptake by the body [17–20]
and exposure to radioactive particles from nuclear reactor accidents [21–25].
In many practical situations, these aerosol particles move very slowly with
respect to the surrounding gas. As a result, one can neglect inertial effects in the
fluid and treat the particle as if it is in the creeping flow regime. This significantly
simplifies the fluid dynamics and makes problems of aerosol transport more tractible.
A further simplification is to treat particles as if they are spherical. The
equivalent sphere size may be based on the particle mass – as if often done in aerosol
1
dynamics codes [26, 27] – or on its aerodynamic behavior, such as its experimentally-
measured mobility. Using this equivalent sphere size, one can estimate any number
of transport properties (e.g. diffusion and friction coefficients, coagulation rates,
phoretic velocities, etc.) using various theoretical or experimental relations that
have been developed for spheres [2].
Unfortunately, particles are often non-spherical. In fact, particles formed by
random processes are often fractal aggregates of N spheres (or monomers) with
radius a. The number of primary spheres in a fractal aggregate is related to the
radius of gyration Rg of the particle by
( )d
R fg
N = k0 (1.1)
a
Here, df and k0 are the fractal dimension and prefactor. One important class of
particles, those formed by diffusion-limited cluster-to-cluster aggregation (DLCA),
have a fractal dimension of approximately 1.78 and a prefactor around 1.3 [4].
Treating a fractal aggregate as a sphere with an equivalent mass or equivalent
mobility leads to an erroneous estimate of particle migration, coagulation, and depo-
sition rates. There are existing methods for calculating the drag on a non-spherical
particle, but most of these methods are only applicable in the continuum [28–31]
or free molecule [32–36] regimes corresponding to particles much larger or much
smaller than the mean free path of gas molecules. However, nano-scale aerosols
typically have characteristic sizes that place them in the transition regime between
the continuum and free molecule limits. There is also some ambiguity as to how one
2
should approach a situation where the aerosol is an aggregate of very many small
spheres (a λ, where a is the sphere diameter and λ the mean free path) while the
characteristic size of the aggregate (such as Rg) is comparable to or larger than λ.
Methods for calculating the drag on a particle in the transition regime are largely
based on fits to empirical data [37–39], or rely on expensive computational methods
such as the direct simulation Monte Carlo (DSMC) method [40, 41].
This dissertation describes a new method for calculating the drag and torque
on an aggregate in creeping flow when continuum approaches are invalid. Before
describing my method and presenting my results, I will first provide brief intro-
ductions to concepts in aerosol physics that are relevant to my dissertation. This
introduction includes basic definitions on topics such as aerosol particle size distri-
butions, an overview of creeping flow and its characteristics, review of the existing
literature on drag and torque on particles in creeping flow, and a discussion of per-
tinent experimental equipment and methods used in aerosol transport studies that
play some role in validating my theoretical methods. To conclude the introduction,
I will outline the remaining scope of my dissertation.
1.1 Aerosol Basics
Aerosols are two-phase systems consisting of a dispersed phase of solid particles
or liquid droplets in a continuous gas phase [1–3]. The particles may form from gas-
phase processes (such as condensation of super-saturated vapor) or from breakup of
solids or liquids. Gas-phase processes typically produce smaller particles (i.e. less
3
than 1 µm in diameter), while disintegration processes yield larger particles [2].
Particle size is often described in terms of the non-dimensional Knudsen number,
which is defined as the ratio of the mean free path of molecules in the gas to the
characteristic size of the particle, Kn = λ/L. The mean free path is the average
distance gas molecules travel between collisions, which is a function of the size of
the gas molecules and their number density and is equal to approximately 65 nm
for air at standard temperature and pressure [2]. The mean free path can be related
to the gas viscosity through relations that depend on the choice of molecular model
(e.g. hard sphere, Lennard-Jones) for the gas [40]. The characteristic size of a
particle depends on its shape; for spheres, the radius is the characteristic size, while
for more general shapes one can use the mobility radius or radius of gyration. The
radius of gyration is a purely geometric quantity, while the mobility radius depends
on the interaction between the particle and the fluid [2]. This topic will be discussed
later in this introduction.
As mentioned previously, many aerosol particles are fractal-like aggregates of
many smaller, primary spheres, that form when the primary spheres coagulate and
stick together. Generally speaking, aggregates contain spheres with a distribution
of sizes; however, the standard deviation in the primary sphere diameter is often
small when compared to the mean diameter [38, 42–44]. As a result, most studies of
aerosol fractal aggregate transport assume that the primary spheres are all the same
size [2, 4]. Primary sphere sizes range from a few nanometers up to about 0.1 µm [2],
while aggregates may include tens, hundreds or even thousands of primary spheres
[2, 4, 7, 42–44].
4
Aerosol systems typically consist of particles with a range of sizes; this range
is described by the particle size distribution n(v, t), where n(v, t)dv is the number of
particles per unit gas volume at time t with particle volume between v and v + dv.
The size distribution can represent either spherical particles or aggregates [2]. The
evolution of this size distribution with time is described by the general dynamic
equation [2, 3, 45]. This equation accounts for changes in the distribution due to
coagulation, condensation/evaporation, and particle formation and removal. Many
of the terms in the general dynamic equation depend on the transport properties of
the system. Thus, one must be able to accurately describe the transport properties
(e.g. drag and torque on the particles as a function of the gas properties and the
particle size, shape, and orientation) in order to predict the dynamic behavior of
the aerosol. Again, this is the primary motivation for the research described in this
dissertation.
1.2 Creeping Flow
The statistical behavior of a dilute gas is described by the Boltzmann transport
equation, ∣
∂f ∣
+ c · ∇ FE · ∂f δff + = ∣∣ (1.2)∂t m ∂c δt coll
where f(r, c, t)drdc is the number of gas molecules in differential volume dr with
velocity c + dc at time t, ∇f is the spatial gradient of f(r, c, t), ∂f/∂c is its
gradient with respect to the molecular velocity, and FE/m is the external force
(e.g. electrical, magnetic) per unit mass on the molecules [40, 46]. The right-hand
5
side of the above equation is the collision integral. Thus, the Boltzmann equation
tracks the probability distribution of molecular velocities as a function of position
and time, accounting for convection (the second term on the left-hand side), external
forces on the molecules (the third term on the left-hand side), and collisions between
gas molecules that alter their velocities (the term on the right-hand side) [40]. It
is a complicated integro-differential equation that can only be solved for a very
small number of cases [46]. (See Chapter 2 for more details about the Boltzmann
equation.)
For near-equilibrium situations where the smallest length scale of the prob-
lem is much greater than the mean free path of the gas (i.e. Kn  1), one can
use Chapman-Enskog theory to derive the mass, momentum, and energy balance
equations that govern continuum transport [47]. Conservation of momentum for
near-equilibrium, continuum flow in an incompressible, Newtownian fluid is given
by the Navier-Stokes equation,
( )
∂u
ρ + u · ∇u = −∇p+ µ∇2u+ ρg (1.3)
∂t
where u(x, t) is the bulk gas velocity at position x at time t, g is gravitational
acceleration, and ρ, p, and µ are the gas density, pressure, and viscosity. The
coefficients of viscosity, diffusion, and heat conduction that appear in the continuum
transport equations can be related to the molecular velocity distribution function
through the Chapman-Enskog expansion. (See Refs. [40, 46] for further discussion
on this topic.)
6
The Navier-Stokes equation is less complicated than the Boltzmann equation,
but it is still a non-linear differential equation, making it difficult to solve analytically
except in special cases. However, we can significantly simplify the equation for cases
where the inertial terms (i.e. the left-hand side of the Navier-Stokes equation) are
negligible, leading to the Stokes equation governing creeping flow in the continuum:
0 = −∇P + µ∇2u (1.4)
Here, ∇P ≡ ∇p+ρg combines the pressure term with the gravitational term, which
can be done because gravity is a conservative vector field [48].
The Stokes equation is valid for Re  1, where the Reynolds number repre-
sents the ratio of inertial to viscous forces and is defined as Re ≡ ρUL/µ. Here, U
and L are the characteristic speed and length scale in the problem. (For a sphere
with radius a moving with velocity U0, U = |U0| and L = a.) The Mach number
(Ma ≡ U/cs, where cs is the speed of sound in the gas) must also be very small for
the creeping flow approximation to be valid, though in the continuum the Reynolds
number condition is typically more restrictive. (Note that the Mach number must
be small in order for a gas flow to be considered incompressible.) Unlike the Boltz-
mann and Navier-Stokes equations, the Stokes equation is linear, which gives it a
number of interesting mathematical properties, some of which I will discuss later.
From a practical standpoint, it makes the equation much easier to solve.
For non-continuum flow, Eq. (1.4) no longer applies; however, one can still be
in the creeping flow regime, provided the usual conditions of very low Reynolds and
7
Mach numbers are satisfied. This allows us to simplify the Boltzmann equation to
determine the flow field around and drag on an aerosol particle, as I will explain in
more detail in this dissertation. (See, especially, Chapters 2 and 3.)
Before describing the methods one might use to calculate the drag on a particle,
I must first introduce two important features resulting from the linearity of the
creeping flow equations (whether the Stokes equation for continuum flow or the
BGK equation – the subject of Chapter 2 – for non-continuum flow).
First, there is a linear relationship between the translational and rotational
velocities UO and ω of a particle and the drag and torque F and TO exerted by the
fluid on the particle [49]:
F = −Ξt ·UO −Ξ†O,c ·ω (1.5a)
T = −ΞO,c ·U −Ξ†O O,r ·ω (1.5b)
Here, the translational, rotational, and translation-rotation coupling friction ten-
sors Ξt, ΞO,r, and ΞO,c are functions of the particle size, shape, and orientation. The
coupling tensor reflects the fact that in general, a translating particle can experience
a net torque, which can induce particle rotation. Likewise, a rotating particle can
experience a net force that induces particle translation. The dagger symbol repre-
sents the transpose of the tensor, while the subscript O signifies that the variable is
defined with respect to the center of mass of the particle (i.e. UO is the translational
velocity of particle center of mass). Note that Ξt and ΞO,r are symmetric. For
8
isotropic particles such as spheres, the coupling tensor is zero, while the transla-
tional and rotational tensors are Ξt = ζtI and ΞO,r = ζO,rI. Here, I is the identity
tensor and ζt and ζO,r are the (scalar) translational and rotational friction coeffi-
cients for the sphere, which are given in the following section. Thus, for an isotropic
particle one need only determine two coefficients to describe the force and torque on
a particle with specified velocity. For an arbitrary particle, one must determine 21
parameters: the 6 independent components of Ξt, the 6 independent components of
ΞO,r, and all 9 components of ΞO,c.
The second important consequence of the linearity of the creeping flow equa-
tions is that one may solve the equations using superposition. This means that we
may add up the velocity results for problems where the solution is known (e.g. Stokes
flow around a sphere moving through an infinite fluid) to get results for a different
problem where the solution is more difficult to determine (e.g. two spheres in Stokes
flow), provided the superposed solution satisfies the equation and boundary condi-
tions of the more difficult problem [48]. This property of linear equations forms the
basis for the Kirkwood-Riseman approach that I will discuss shortly.
1.3 Aerosol Transport
While the work described in this dissertation primarily concerns transport of
particles in the transition regime, it is necessary to first review the theoretical devel-
opment of the drag on a particle in the continuum and free molecule regimes. There
are two main reasons for doing so: first, the extended Kirkwood-Riseman (EKR)
9
method introduced in this dissertation incorporates elements from the continuum
and free molecule regimes; second, the drag computed using the EKR method should
approach the continuum and free molecule expressions in the limits of very small
and very large Knudsen numbers. Thus, the review of aerosol particle transport
is divided into sections relevant to the continuum, free molecule, and transition
regimes.
1.3.1 Continuum Regime
The creeping (or Stokes) flow equation forms the basis for any study of the
behavior of particles in low-Reynolds-number flow in the continuum. Stokes [50] was
the first to solve the creeping motion equation [Eq. (1.4)] for a sphere with radius
a, resulting in the expression now know as Stokes’ law,
F = −6πµaU ≡ −ζct,0U (1.6)
where ζct,0 is the translational friction coefficient for a sphere in continuum flow. One
can also solve Eq. (1.4) for a sphere rotating with angular velocity ω; the torque on
the rotating sphere is given by
T = −8πµa3ω ≡ −ζcr,0ω (1.7)
where ζcr,0 is the rotational friction coefficient for a sphere in continuum flow about
its center of mass.
10
In principal, one can solve the Stokes equation for the velocity and pressure
fields around a particle of arbitrary shape moving with translational velocity U and
angular velocity ω, then integrate the resulting stress profile at the particle surface
to determine the lift and drag forces and the torque on the particle. In this way,
one can obtain the friction tensors relating the translational and angular velocities
of the particle to the drag and torque exerted on the particle by the fluid [Eq. (1.5)].
Brenner [29] describes this process in detail, as well as the relationship between
the force and torque on a particle of arbitrary shape and its diffusive properties.
In practice, the Stokes equation can only be solved analytically for simple shapes,
so that alternative methods are need to determine the forces on an aggregate of
spheres.
Kirkwood and Riseman [28] proposed that the force on each spherical element
of an N -sphere aggregate can be obtained by considering the effects of all of the
elements on the fluid flow pattern. The strength of those effects is given by an appro-
priate hydrodynamic interaction tensor, Tij, so that the force on the ith spherical
element becomes ∑N
F = −ζc ci t,0Ui − ζt,0 Tij ·Fj (1.8)
i=6 j
where Ui is the velocity of the ith sphere. For a rigid, non-rotating particle, all
spheres move with the same velocity, so Ui = U . The total force on the par-
ticle is simply the sum of the forces on the N spherical elements. By repeating
the calculation for flow in three mutually-orthogonal directions, one can obtain the
translational friction tensor Ξt. Ignoring the effects of coupling between transla-
11
tional and rotational motion, the scalar translational friction coefficient ζt is the
harmonic mean of the eigenvalues of Ξt [29, 49].
The Kirkwood-Riseman (KR) framework can also be used to calculate the
torque on a rotating particle and the coupling between translational and rota-
tional motions, as described in the works of Garcia de la Torre and colleagues
(e.g. Refs. [51–53]). This procedure accounts for rotational and coupling hydro-
dynamic interactions and yields the rotational and coupling friction tensors, Ξr and
Ξc. From the three friction tensors, one obtains the translation, rotation, and cou-
pling diffusion tensors through a generalization of the Stokes-Einstein law derived
by Brenner [29].
Kirkwood and Riseman originally applied their theory to flexible macromole-
cules; Bernal et al. [51] and Chen et al. [30] later applied the theory to rigid macro-
molecules and fractal aggregates, respectively. KR theory has been used extensively
to compute the transport properties of macromolecules, colloids, and fractal aggre-
gates [30, 51–56].
In its original form, KR theory used the Oseen tensor for Tij. Subsequent
applications of the theory for pure translational motion have used the Rotne-Prager-
Yamakawa (RPY) tensor [57, 58]. More complicated translational, rotational, and
coupling hydrodynamic interaction tensors are also available in the literature [59–
61]. Carrasco and Garcıa de la Torre [53] have shown that KR theory with the RPY
tensor yields translational friction coefficients within a few percent of the friction
coefficients calculated with more sophisticated methods for the simple particles they
studied.
12
Hubbard and Douglas [31] developed a different approach for calculating the
translational friction coefficient of an arbitrarily-shaped Brownian particle by noting
the approximate relationship between the friction coefficient and the electrostatic
capacitance C,
ζct ≈ 6πµC (1.9)
The Zeno algorithm [62] uses a random walk approach to calculate the electrostatic
capacitance – and thus the translational friction coefficient. The accuracy of the
Hubbard-Douglas approximation is within 1% for shapes where ζct is known and
within a few percent for an arbitarily-shaped particle [63]. These results suggest
that the Hubbard-Douglas method is more accurate than KR for relatively simple
shapes. For larger fractal aggregates, the differences between the two methods is
less significant. (See Chapter 4.)
For fractal aggregates, research suggests that the friction coefficient in the
continuum regime follows a power-law relationship,
ζct = AN
η (1.10)
Sorensen [4] analyzed the results of various experimental and computational studies
and found that η ≈ 0.46 for N < 100 and η ≈ 0.56 for N > 100 for clusters formed
by diffusion-limited cluster aggregation (k0 ≈ 1.3, df ≈ 1.78).
13
1.3.2 Free Molecule Regime
The drag on a particle in the free molecule regime can be calculated using
kinetic theory. However, because the particle is much smaller than the gas mean
free path, it has very little impact on the distribution of molecular velocities in
the gas. As a result, the aerodynamic force on the particle can be calculating by
assuming that the gas molecules impinging on the surface have a Maxwell-Boltzmann
distribution of velocities. This assumption obviates the need to solve the Boltzmann
equation to obtain the drag on a particle in free molecule flow.
Epstein [32] first calculated the drag on a sphere in creeping flow in the free
molecule regime as
√ ( )( )1/2
F = − 2π πα kBT1 + ρa2U ≡ −ζFMt,0 U (1.11)3 8 m
where kB is the Boltzmann constant, T and ρ are the gas temperature and density,
m is the mass of a gas molecule, and α is the fraction of gas molecules that are
in thermal equilibrium after reflecting from the particle surface (i.e. the fraction of
molecules reflected diffusely). Thus, the drag on a sphere in free molecule flow is
proportional to a2, whereas the drag is proportional to a in the continuum. Epstein
also calculated the torque on a rotating sphere,
√ ( )1/2
− 32π kBTT = α ρa4ω ≡ −ζFMω (1.12)
3 m r,0
14
showing that the torque is proportional to a4 for free molecule flow, compared to a3
in the continuum regime [32].
Dahneke [33] extended Epstein’s analysis to develop analytic expressions for
the drag on various convex bodies. The analysis is more complicated for concave
bodies – such as fractal aggregates – due to shielding of incoming molecules by parts
of the surface and the possibility of multiple collisions between a molecule and the
particle. Thus, numerical techniques are required for concave bodies. These tech-
niques track the trajectories of gas molecules near the particle to determine whether
or not the molecules hit the particles. For those molecules that hit the surface,
the momentum transfer is computed using an appropriate reflection law. Chan and
Dahneke [34] used this ballistic approach to compute the drag on straight chain
aggregates for flow parallel and perpendicular to the long axis of the chain. Meakin
and Deutch [55] applied the approach to determine the drag on fractal aggregates
and found that the drag is proportional to the projected area of the aggregate.
Mackowski [36] performed a similar analysis and developed an empirical correlation
for the translational friction coefficient as a function of the fractal dimension and
prefactor and the number of spheres for the range of parameters studied.
One could also apply the ballistic approach to compute the torque on a rotating
particle, though it does not appear anyone has done so based on the dearth of
information in the literature. Instead, researchers have used simplified techniques
to estimate the rotational friction or diffusion coefficient of aggregates in the free
molecule regime [6].
For fractal aggregates, results suggest that there is a power-law relationship
15
between the free molecule translational friction coefficient and the number of spheres
in the aggregate; this is similar to the observed behavior in the continuum. Sum-
marizing the available experimental and computational results in the literature,
Sorensen [4] recommended a power-law exponent of η ≈ 0.92 for DLCA clusters of
all sizes in the free molecule regime.
1.3.3 Transition Regime
As the particle size increases, the assumption that the particle has no impact
on the molecular velocity distribution around the particle no longer holds, and a
more rigorous application of kinetic theory is required. In the transition regime, the
drag can be obtained by solving the Boltzmann equation and integrating the stress
on the surface of the profile. However, the Boltzmann equation is exceeding difficult
to solve even for the simple case of a sphere, so significant simplifications are needed
to make the problem tractable. These simplifications will be described shortly, but
first I will focus on empirical models based on experimental data.
Millikan [64] laid the groundwork for determining the drag on a sphere in
the transition regime. Data from his famous oil drop experiments demonstrates
the transition between the continuum and free molecule regimes, where the drag is
proportional to a and a2, respectively. To cover the entire Knudsen number range,
one can apply a slip correction factor Cc(Kn) to Stokes’ law,
6πµa
ζt,0 = (1.13)
Cc(Kn)
16
where the slip correction factor has the form
Cc(Kn) = 1 + Kn[A+B exp(−C/Kn)] (1.14)
The coefficients A, B, and C are selected to fit the experimental data; the coefficients
of Davies [65] and Allen and Raabe [66] are commonly used in aerosol applications.
Eq. (1.13) approaches the continuum and free molecule friction coefficients defined
in Eqs. (1.6) and (1.11) for Kn 1 and Kn 1, respectively.
Other researchers have developed empirical models for the drag on fractal
aggregates in the transition √regime. Rogak et al. [37] proposed substituting the
projected area radius (aPA = PA/π) for a in Eq. (1.13). Lall and Friedlander [38]
suggested that the drag on an aggregate with fractal dimension less than 2 can be
approximated as a straight chain. Their correlation applies Chan and Dahneke’s
results for chain elements in the free molecule regime [34]. Finally, Eggersdorfer
et al. [39] relate the mobility radius (i.e. the radius of a sphere with the same drag
as the particles) to the number of primary spheres and the fractal dimension and
prefactor of the aggregate. The resulting model is similar to Rogak’s model for
particles formed by DLCA. These three models provide simple relationships for the
drag on fractal aggregates, though they are only valid near the free molecule regime.
(See Chapter 4.)
Dahneke [67] proposed an adjusted sphere method (ASM) for particles of ar-
bitrary shape, where the drag on the particle is given by an expression analogous
to Eq. (1.13). The difference is that a and Kn should be replaced by an appropri-
17
ate characteristic length and aggregate Knudsen number. Through scaling analysis,
Zhang et al. [41] demonstrated that the appropriate characteristic length is the hy-
drodynamic radius RH (i.e. the radius of a sphere that has the same drag as the
particle in continuum flow), and the aggregate Knudsen number is
Knagg = πλRH/PA (1.15)
where PA is the projected area of the particle. RH can be computed using the KR
or Hubbard-Douglas methods, and PA can be computed using ballistic methods.
Thus, the aggregate Knudsen number is proportional to the ratio of continuum and
free molecule measures of the drag. The drag calculated using the ASM is in good
agreement with experimental and computational results [41, 68, 69].
A number of researchers have managed to solve simplified forms of the Boltz-
mann equation for spheres and other axisymmetric bodies. Cercignani and Pagani
[70] described the general approach for solving the Boltzmann equation with the
Bhatnagar-Gross-Krook (BGK) model [71] in place of the Boltzmann collision opera-
tor using a variational technique. The BGK model assumes that the non-equilibrium
distribution of molecular velocities in the gas relaxes to an equilibrium distribution
after one collision. Kogan [72] has shown that this approximation is valid for most
physical situations. The variational approach of Cercignani and Pagani is valid for
any axisymmetric body.
Cercignani et al. [73] applied that technique to determine the drag on a sphere
as a function of Knudsen number. Their results are within a few percent of a fit
18
to Millikan’s data over a wide range of Knudsen numbers. Loyalka and colleagues
[74–76] obtained the velocity profile around the sphere as well as the drag using
methods similar to Cercignani et al. [73]. Later, Loyalka [77] and Takata et al.
[78] solved the problem using a linearized form of the Boltzmann collision operator.
The velocity and drag results from these studies are similar to the previous BGK
model results, which were obtained at a significantly lower computational cost than
the linearized Boltzmann results. Loyalka [77] also solved the linearized Boltzmann
equation to determine the velocity around and torque on a rotating sphere in the
transition flow regime.
In principal, one could solve the Boltzmann or BGK equation numerically to
obtain the drag on and flow field around a particle with arbitrary shape, but this is
exceedingly difficult in practice, especially for concave particles. One approach for
doing so is the direct simulation Monte Carlo method [40], which tracks a number
of test molecules and reconstructs the velocity distribution in the gas from the
behavior of these test molecules. This method is computationally expensive and is
less accurate near the continuum regime due to the finite size of the test domain
[41].
Melas et al. [79] determined the friction coefficient for aggregates in the near-
continuum regime by solving the Laplace equation with a slip boundary condition.
This approach is an extension of the Hubbard-Douglas approach (which uses the
stick boundary condition at the particle surface). Melas et al. [80] estimated that
this approach is valid for Kn < 2, meaning some other approach is needed for
particles closer to the free molecule regime.
19
Tandon and Rosner [81] developed a method for calculating the friction co-
efficient for fractal aggregates using a porous sphere approach. The porosity is a
function of radial position in the sphere and is obtained from the pair distribution
function for the orientation-averaged coordinates of monomers in the aggregate. The
velocity around the porous sphere is governed by the Stokes equations, while the
flow within the sphere is obtained by solving the Brinkman equation [4, 81]. Rosner
and Tandon [82] have shown that the porous sphere method gives friction coefficient
results in good agreement with the Adjusted Sphere Method of Dahneke [67] and
Zhang et al. [41] for any primary sphere size, provided the aggregate is large enough
that one can accurately treat the outer flow using the Stokes equation. In other
words, the aggregate size (e.g. the radius of gyration) must be very large compared
to the gas mean free path. Again, this means that a different approach is needed
for aggregates closer to the free molecule regime.
1.4 Experimental Techniques for Obtaining Particle Size
The work contained in this dissertation focuses on the theory of aerosol physics;
that is, I have not performed an experiments to support this work. Fortunately,
there is some data in the published literature to validate – or at least qualitatively
support – my theoretical results. To aid the reader in understanding comparisons
between the results in this dissertation and available experimental data, I will pro-
vide an overview of the experimental techniques used to size aerosol particles that
are pertinent to my own work.
20
Figure 1.1: Differential mobility analyzer (DMA) for selecting charged particles with
a specified mobility. The mobility size is selected by controlling the DMA voltage
and air flow rate.
Broadly speaking, we can divide the relevant experimental techniques into
two categories: those techniques that measure the mobility of a particle, and those
that measure its light scattering behavior. I will address each of these categories
separately in the following subsections.
1.4.1 Mobility Measurements
The first class of instruments that I will review measure how a particle moves
in an applied force field. Perhaps the most widely-used such instrument for aerosol
studies is the cylindrical differential mobility analyzer (DMA), shown in Fig. 1.1.
A DMA system works as follows. First, an aerosol stream passes through a
21
neutralizer to obtain a known equilibrium charge distribution [2, 83]. The aerosol
stream enters the the cylindrical DMA at flow rate Qa along with a stream of clean
air (i.e. the sheath flow, Qsh). Particles are advected with the sheath flow; at the
same time, positively charged particles drift from the outer cylinder wall to the inner
wall, which has a negative potential. This drift velocity is the velocity required to
balance the electrical and aerodynamic drag forces on the particle:
ζtVd = qE (1.16)
Here, ζt is the particle translational friction coefficient, Vd is the drift velocity, q
is the charge on the particle, and E is the electric field strength. The field and
drift velocity are in the same direction. Particles that travel a radial distance R
in the time they travel an axial distance L pass through the slit in the DMA. The
remaining particles either deposit on the outer (negatively charged particles) or
inner (positively charged particles with higher mobility than the sampled particles)
wall of the DMA or pass out of the DMA with the excess flow. One selects the
voltage (and thus the field strength) and the sheath flow rate to obtain particles
with a desired electrical mobility Z, where
Vd q
Z = = (1.17)
E ζ
By scanning through a series of voltages and counting the number of particles in the
sample flow at each voltage (e.g. with a condensation particle counter, as explained
22
in Chapter 6 of Friedlander [2]), one can determine the particle size distribution for
the aerosol that enters the DMA [2].
Often, researchers present the mobility as an equivalent sphere size by solving
Eq. (1.13) implicitly to find the mobility diameter dm [37–39, 43, 44, 68, 84]. Of
course, for a sphere the mobility diameter is equivalent to its geometric diameter.
The situation is much more complicated for fractal aggregates, so it is difficult to
obtain information about particle mass from a DMA. Furthermore, the DMA selects
some particles that have multiple charges in addition to those with a single charge,
which results in some error in the size distribution (since the larger, multiply-charged
particles have the same electrical mobility as the smaller, singly-charged particles).
To address these difficulty, systems for characterizing non-spherical particles
often involve both mobility measurements in a DMA and mass measurements in an
aerosol particle mass (APM) analyzer [43, 44, 85, 86]. Just as a DMA relies on a
balance between the electric force and the aerodynamic drag on a particle, an APM
sizes particles by balancing the electric force with the centrifugal force. When the
two forces are equal, the particle passes through the APM; otherwise, the particle
deposits on the inner or outer wall of the APM. In this way, one can size-select
particles based on their mass-to-charge ratio.
Researchers have developed other systems using combined mass and/or mo-
bility measurements to obtain additional size and shape information about aerosol
particles [44, 87, 88]. In many cases, mass and mobility measurements are sup-
plemented by information about primary particle size from transmission electron
microscopy (TEM) images; this information can be used with basic assumptions
23
about the fractal dimension of the aggregate to estimate the number of primary
spheres it contains [6, 44].
1.4.2 Optical Measurements
The second class of aerosol instruments that are relevant to my research in-
volve measuring the intensity of light scattered by the particles. (See Bohren and
Huffman [89] for the detailed discussion on the theory of light scattering by small
particles.) These optical instruments are used for a variety of purposes, from de-
termining the number density of particles in a gas stream (e.g. in a condensation
particle counter), to obtaining information about particle shape, to determining the
rotational diffusion coefficient of a particle. I will focus my attention on the latter
application.
In general, the angular distribution of the light scattered by a particle is a
function of that particle’s orientation. In the absence of a strong external force
field, the light scattered by a nano-scale aerosol particle is an average over all orien-
tations, where all orientations are equally probable due to the randomizing effects
of Brownian motion. However, particles can become aligned in a strong electric field
if the interaction energy between the induced dipole in the particle and the field is
much larger in magnitude than the Brownian energy kBT , where kB is the Boltz-
mann constant and T is the temperature. By measuring the change in scattered
light intensity when the field is on and off, one can obtain some information about
the shape of particles. One can also obtain the rotational diffusion coefficient of a
particle by turning off the field and measuring the time required for the scattered
24
light intensity to relax to a value corresponding to the random particle orientation.
Such measurements have been reported in the literature for soot particles from var-
ious sources [90, 91]; I will later compare my results for the rotational diffusion
coefficients of soot-like aggregates to the experimental data of Colbeck et al. [91].
This experimental technique also offers an alternative approach to the method I
describe in Chapter 7 for obtaining particle shape information.
1.5 Scope of the Dissertation
This dissertation describes a method [92] for calculating the drag on an aggre-
gate of spheres in point contact in the transition flow regime, based on Kirkwood-
Riseman theory originally developed for the continuum regime [28]. Generally speak-
ing, Chapters 2, 3, and 5 introduce the method, while Chapters 4 and 6-8 focus on
applications of the method.
In Chapter 2, I discuss the Bhatnagar-Gross-Krook model equation and its
solution for flow around a sphere as a function of the Knudsen number. This chapter
follows the earlier work of Loyalka and colleagues [74–76]. I include this discussion
because my EKR method uses the flow around an isolated sphere to determine the
drag and torque on an N -sphere aggregate.
In Chapter 3, I develop a new approach for computing the hydrodynamic
friction tensor and scalar friction coefficient for an aerosol fractal aggregate in the
transition regime [92]. My approach involves solving the BGK equation for the ve-
locity field around a sphere and using the velocity field to calculate the force on
25
each primary sphere in the aggregate due to the presence of the other spheres. It
is essentially an extension of Kirkwood-Riseman theory from the continuum flow
regime to the entire Knudsen range (Knudsen number from 0.01 to 100 based on
the primary sphere radius). Results compare well to published Direct Simulation
Monte Carlo results and converge to the correct continuum and free molecule limits.
My calculations for clusters with up to 100 spheres support the theory that aggre-
gate slip correction factors collapse to a single curve when plotted as a function of
an appropriate aggregate Knudsen number. This self-consistent field approach cal-
culates the friction coefficient very quickly, so the approach is well-suited for testing
existing scaling laws in the field of aerosol science and technology, as I demonstrate
for the adjusted sphere scaling method.
In Chapter 4, I use the self-consistent field method described in Chapter 3 to
calculate the translational friction coefficient of fractal aerosol particles formed by
diffusion-limited cluster aggregation (DLCA) [93]. The method involves solving the
Bhatnagar-Gross-Krook model for the velocity around a sphere in the transition flow
regime. The velocity and drag results are then used in an extension of Kirkwood-
Riseman theory to obtain the drag on the aggregrate. Results span a range of
primary sphere Knudsen numbers from 0.01 to 100 for clusters with up to N =
2000 primary spheres. Calculated friction coefficients are in good agreement with
experimental data and approach the correct continuum and free molecule limits
for small and large Knudsen numbers, respectively. Results show that particles
exhibit more continuum-like behavior as the number of primary spheres increase,
even when the primary particle is in the free molecule regime; as an illustrative
26
example, the friction coefficient for aggregates with primary sphere Kn = 1 are
approximately equal to the continuum friction coefficient for N > 500. I estimate
that the calculations are within 10% of the true values of the friction coefficients
for the range of Kn and N presented here. Finally, I use my results to develop an
analytical expression (Equation 4.38) for the friction coefficient over a wide range
of aggregate and primary particle sizes.
In Chapter 5, I apply extended Kirkwood-Riseman theory to compute the
translation, rotation, and coupling friction tensors and the scalar rotational friction
coefficient for an aerosol fractal aggregate in the transition flow regime [94]. The
method can be used for particles consisting of spheres in contact. The approach
considers only the linear velocity of the primary spheres in a rotating aggregate
and ignores rotational and coupling interactions between spheres. I show that this
simplified approach is within approximately 40% of the true value for any particle
for Knudsen numbers between 0.01 and 100. The method is especially accurate (i.e.
within about 5%) near the free molecule regime, where there is little interaction
between the particle and the flow field, and for particles with low fractal dimension
(less than ≈ 2) consisting of many spheres, where the average distance between
spheres is large and translational interaction effects dominate. Results suggest that
there is a universal relationship between the rotational friction coefficient and an
aggregate Knudsen number, defined as the ratio of continuum to free molecule ro-
tational friction coefficients.
In Chapter 6, I apply the EKR method to calculate the rotational friction co-
efficient for fractal aerosol particles in the transition flow regime [95]. The method
27
considers hydrodynamic interactions between spheres in a rotating aggregate due
to the linear velocities of the spheres. Results are consistent with electro-optical
measurements of soot alignment. Calculated rotational friction coefficients are also
in good agreement with continuum and free molecule results in the limits of small
(Kn = 0.01) and large (Kn = 100) primary sphere Knudsen numbers. As demon-
strated for the translational friction coefficient (Chapter 4), the rotational friction
coefficient approaches the continuum limit as either the primary sphere size or the
number of primary spheres increases. I apply my results to develop an analytical
expression (Equation 6.26) for the rotational friction coefficient as a function of the
primary sphere size and number of primary spheres. One important finding is that
the ratio of the translation to rotational diffusion times is nearly independent of clus-
ter size. I include an extension of previous scaling analysis for aerosol aggregates to
include rotational motion.
In Chapter 7, I study the effects of electric field strength on the mobility of
soot-like fractal aggregates (fractal dimension of 1.78) [96]. The probability dis-
tribution for the particle orientation is governed by the ratio of the interaction
energy between the electric field and the induced dipole in the particle to the energy
associated with Brownian forces in the surrounding medium. I use the extended
Kirkwood-Riseman method to calculate the friction tensor for aggregates of up to
2000 spheres, with primary sphere sizes in the transition and near-free-molecule
regimes. My results for electrical mobility versus field strength are in good agree-
ment with published experimental data for soot, which show an increase in mobility
on the order of 8% from random to aligned orientations. My calculations show
28
that particles become aligned at decreasing field strength as particle size increases
because particle polarizability increases with volume. Large aggregates are at least
partially aligned at field strengths below 1000 V/cm, though the small change in mo-
bility means that alignment is not an issue in many practical applications. However,
improved DMAs would be required to take advantage of small changes in mobility
to provide shape characterization.
In Chapter 8, I present a method for calculating the hydrodynamic interac-
tions between particles in the kinetic (or transition) regime, characterized by non-
negligible particle Knudsen numbers. Such particles are often present in aerosol
systems. The method is based on my extended Kirkwood-Riseman theory [92],
which accounts for interactions between spheres using the velocity field around a
translating sphere as a function of Knudsen number. Results for the two-sphere
problem at small Knudsen numbers are in good agreement with those obtained us-
ing Felderhof’s interaction tensors for mixed slip-stick boundary conditions, which
are accurate to order r−7 [97]. The strength of interactions decreases with increasing
Knudsen number. Results for two fractal aggregates demonstrate that one can apply
a point force approach for interactions between particles in the transition regime;
the interaction tensor is similar to the Oseen tensor for continuum flow. Using this
point force approach, I present an analysis for the settling of an unbounded cloud of
particles. The analysis shows that for sufficiently high volume fractions and cloud
radii, the cloud behaves as a gas droplet in continuum flow even when the individual
particles are small relative to the mean free path of the gas. The method presented
here can be applied in a Brownian dynamics simulation analogous to Stokesian
29
dynamics to study the behavior of a dense aerosol system.
Chapter 9 is the Reference Manual for the NGDE code. The NGDE code uses
a nodal method to solve the general dynamic equation for an aerosol undergoing
coagulation, nucleation, and surface growth. This method is similar to widely-
used sectional methods for solving the general dynamic equation, but by dividing
the particle size distribution into discrete volume nodes, it eliminates many of the
mathematical complexities of sectional methods. I have converted the original C
version of NGDE to the MATLAB language, added a dynamic time step algorithm
that reduces the code execution time by orders of magnitude, and created a new
post-processing tool for viewing the evolution of the particle size distribution and the
light scattering, extinction, and absorption coefficients. Results of sample problems
compare well to results obtained from other methods. Because of NGDE’s simplicity
and accuracy, it is well-suited for use as part of courses in aerosol dynamics.
The main body of the dissertation closes with a summary of the important
conclusions of my research in Chapter 10.
30
Chapter 2: The BGK Model Equation
2.1 Introduction
Solution of the density, velocity, and temperature fields around a sphere in
the transition regime (0.01 < Kn < 100) requires consideration of the Boltzmann
equation or its derivatives. For steady flow, the Boltzmann equation can be written
in dimensional form as ∣
δf ∣
c̃ · ∇̃f = ∣ (2.1)
δt ∣coll
where c̃ is the molecular velocity, f = f(r̃, c̃) is the molecular velocity distribution
function, and the term on the right-hand side of the equation is the collision op-
erator.1 The distribution function is defined such that f(r̃, c̃)dr̃dc̃ represents the
number of gas molecules in differential volume dr̃ centered at location r̃ that have
a velocity between c̃ and c̃+ dc̃.
The collision operator can be written as the difference between two terms. The
1Throughout this chapter, I am using bold symbols to denote vectors, bold symbols with hats
to denote unit vectors, the subscript ∞ to denote properties far from the perturbation, and a
tilde over dimensional quantities that also appear in non-dimensional form without the tilde. To
expand on the latter point, I do not add a tilde to all dimensional quantities; I only add it to
explicitly differentiate between dimensional and non-dimensional variants of the same quantity.
For example, temperature always appears as T because I only use it as a dimensional quantity,
whereas the molecular velocity c appears with and without the tilde to signify dimensional and
non-dimensional variants.
31
first (positive) term represents the rate at which molecules are scattered into the
interval [c̃, c̃+dc̃] as a result of collisions, while the second (negative) term represents
the molecules in this velocity interval that are scattered out of the interval due to
collisions with other molecules. The collision operator is the chief source of difficulty
in the Boltzmann equation; it makes solving the equation analytically impossible
for all but the simplest cases.
To address this difficulty, Bhatnagar, Gross, and Krook [71] proposed a sim-
plified form of the collision operator,
∣
δf ∣∣∣ f0(r̃, c̃)− f(r̃, c̃)= (2.2)δt coll τ(c̃)
where τ(c̃) is the (velocity-dependent) mean collision time in the gas and
( ) [ ]3/2
m 2
f0 = n exp −
m|c̃− Ũ |
(2.3)
2πkBT 2kBT
is the Maxwell-Boltzmann distribution at the local number density n, bulk velocity
Ũ , and temperature T . The BGK model for the collision operator expresses the fact
that any distribution f decays to the Maxwellian distribution, where the relaxation
time can be approximated as the time between collisions [71, 72]. The second term
on the right-hand side of Eqn. (2.2) has the same general form as the depletion term
in the Boltzmann collision operator: in both cases, depletion is directly proportional
to f(r̃, c̃) and to the frequency of collisions between molecules with velocity c̃ and
all other molecules. Strictly speaking, the collision frequency is a function of the
32
molecular velocity, though it is possible to use an average collision time [46] (hence
the appearance of the mean collision time τ̄ in later equations). The f0 term in
Eq. (2.2) does not have a direct mathematical relationship to the replenishing term
in the Boltzmann equation. Instead, this term in the BGK model assumes that the
distribution relaxes to the local Maxwellian distribution after one collision. Kogan
[72] has demonstrated that in most physical situations, the velocity distribution after
one collision is fairly close to the equilibrium distribution. Thus, the BGK model
is a reasonable approximation to the Boltzmann collision operator, especially for
near-equilibrium situations like creeping flow of a sphere at the same temperature
as the surrounding gas.
Based on the preceding discussion, we can simplify our equation for the dis-
tribution function by substituting the BGK model for the collision operator in the
Boltzmann equation. The resulting equation is known as the Krook equation. For
steady flow, we have
· ∇̃ f0(r̃, c̃)− f(r̃, c̃)c̃ f = (2.4)
τ(c̃)
We can non-dimensionalize the above equation by defining the variables
c̃
c = (2.5a)
(2kBT∞/m)1/2
Ũ
U = (2.5b)
(2k T∞/m)1/2B
r̃
r = (2.5c)
τ̄(2kBT∞/m)1/2
33
where τ̄ is the collision time averaged over all molecular velocities. Setting the mean
collision time as τ̄ = µ/p (where p is pressure) and the viscosity as µ = 0.499λρc̄, the
non-dimensional sphere radius is related to the Knudsen number by the following
expression:2
√
r̃0 π
r0 = = Kn
−1 (2.5d)
τ̄(2k 1/2BT∞/m) 1.996
Applying the length and velocities scales in Eq. (2.5), the non-dimensional
Krook equation for steady flow becomes
c · ∇f = f0(r, c)− f(r, c) (2.6)
While the Krook equation is much simpler than the Boltzmann equation, the
Krook equation is still difficult to solve because the number density, bulk velocity,
and temperature that appear in f0 are all functions of the local conditions in the
gas. In other words, these variables each depend on f .
To simplify the equation further, we can treat any disturbance in the distribu-
tion function as a small perturbation to the linearized far-away distribution function
f 3∞(1 + 2c ·U∞),
f ≈ f∞(1 + 2c ·U∞ + h) (2.7)
2The definition of the mean collision time comes from the Chapman-Enskog solution of the
Krook equation, which shows that µ from the Chapman-Enskog solution is identical to µ for
continuum flow if we set µ = pτ̄ [46]. The expression relating the viscosity to the mean free path
applies to monatomic gases [46], but it is a reasonable approximation for air and is used in many
aerosol studies in the literature (e.g. [41, 73])
3Here, we have linearized the Maxwellian distribution defined in Eq. (2.3) by expanding the
distribution in powers of U∞. This approach assumes that the stream velocity is very small
compared to the thermal speed, which is certainly true for creeping flow.
34
where ( )3/2 ( )
m mc̃2
f∞ = n∞ exp − (2.8)
2πkBT∞ 2kBT∞
We can also linearize the Maxwellian distribution to get
∣
≈ · ∂[f(1 + 2c ·U)]
∣
f0 f∞(1 + 2c U∞) + ∣ ∣∣ (n− n∞)∂n ∞
∂[f(1 + 2c ·U)]∣
+ [ ∣∂U (∣
∣
· − ∂[f(1 + 2c ·U)]
∣
(U )U∞) + ( ∣∂T ∣ (T∞ ∞ )−(T∞) )]
n 3 T
=f∞ 1 + 2c ·U∞ + − 1 + 2c · (U −U∞) + c2 − − 1
n∞ 2 T∞
[ ( ) ]
f0 ≈ f∞ 1 + 2c ·
3
U∞ + ε1 + c · ε 22 + c − ε3 (2.9)
2
where
n(r) = n∞(1 + ε1) (2.10a)
1
U (r) = U∞ + ε2 (2.10b)
2
T (r) = T∞(1 + ε3) (2.10c)
and ε1, ε2, and ε3 are perturbations to the far-field number density, velocity, and
temperature, respectively. These perturbations will be defined shortly.
From kinetic theory, the number density, bulk velocity, and temperature are
defined in terms of moments of the distribution function f :
∫
n = fdc (2.11a)
35
∫
1
U = cfdc (2.11b)
n ∫
3 1
T = c2fdc (2.11c)
2 n
We define the pertrubation ε1 and ε2 by substituting Eq. (2.7) into Eqns. (2.11a)
and (2.11b), respectively:
∫ ∫
n = f∞∫ (1 + 2c ·U∞ + h)dc = n + n π
−3/2
∞ ∞ ∫h exp(−c
2)dc = n∞(1 + ε1)
1 1
U = f∞(1 + 2c ·U∞ + h)cdc = U + π−3/2∞ hc exp(−c2)dc = U∞ + ε2
n∞ 2
Similarly, we define the temperature perturbation by substituting Eq. (2.7) into
Eq. (2.11c):
∫
3 1
T = f∞(1 +∫2c ·U∞ + h)c
2dc
2 n
3 1
T∞(1 + ε3) = f∞(1 + 2∫c ·U∞ + h)c
2dc
2 n∞(1 + ε1)
3 3
[ T∞(∫1 + ε1 + ε3 + ε ε ) = T + π
−3/2 2
1 3 ]∞ ∫ c h exp(−c
2
()dc2 2 )
2 3 2 3
ε3 = π
−3/2 c2h exp(−c2)dc− ε1 = h exp(−c2) c2 − dc
3 2 3π3/2 2
Note that we have ignored the ε 21ε3 term because it is of order ε (and thus very small
compared to the other terms in the equation). To summarize, the number density,
bulk velocity, and temperature perturbations are moments of the perturbation to
the distribution function h and are defined as
ε1(r) = (h, 1) (2.12a)
36
ε2(r) = 2(h, c) (2.12b)
( )
2 3
ε3(r) = h, c
2 − (2.12c)
3 2
where the inner product (h(r, c), g(c)) is defined as
∫ ∞
(h, g) = π−3/2 hg exp(−c2)dc (2.13)
−∞
We can apply the linearized forms of f and f0 given by Eqns. (2.7) and (2.9)
to Eq. (2.6):
c · ∇f =f0 −[ f ( ) ]
c · ∇[f∞(1 + 2c ·U∞ + h)] =f∞ 1 + 2c ·
3
U 2∞ + ε1 + c · ε2 + c − ε3
2
− f∞(1 + 2c(·U∞ +)h)
c · ∇h =ε1 + c · 2 −
3
ε2 + c ε3 − h
2
This equation can be written as
c · ∇h = Lh (2.14)
where the operator L is defined as
( )
L · 2 2 − 3h = ε1 + c ε2 + c ε3 − h (2.15)
3 2
It is in this form [i.e. Eqns. (2.14) and (2.15)] that the Krook equation appears in
37
papers by Cercignani and Pagani [70] and Lea and Loyalka [75].
Now that I have completed the derivation of the BGK model and Krook equa-
tion, I can apply the equation to practical problems of mass transfer to and flow
around a sphere. I will first discuss briefly the isothermal mass transfer problem
because it is simpler mathematically than the flow problem, and thus it provides
a good introduction for solving the Krook equation. Once I have discussed solving
the condensation problem, I will turn my attention to the problem most relevant to
this dissertation, that of uniform flow around a sphere. Note that in solving these
problems, I am following the derivation of Lea [74].
2.2 Solution of the Krook Equation for Isothermal Mass Transfer to
a Sphere
Let us start by considering mass transfer of a dilute vapor to a sphere. There is
no bulk flow, so ε2 = 0. We will also assume that the region of interest is isothermal,
so ε3 = 0. This leaves us with the following equation for the perturbation to the
distribution function:
c · ∇h = (h, 1)− h (2.16)
We can write this equation in terms of the characteristic path, s:
dh h (h, 1)
+ = (2.17)
ds c c
38
Here, the derivative along the characteristic path is related to the diffusion term on
the left-hand side of Eq. (2.14) by
· ∇ ∂h ∂h ∂hΩ̂ h =Ωx + Ωy + Ωz
∂x ∂y ∂z
∂x ∂h ∂y ∂h ∂z ∂h
= + +
∂s ∂x ∂s ∂y ∂s ∂z
dh
=
ds
where Ω̂ = c/c is the direction vector of the molecular velocity. The geometry of
this problem is shown in Fig. 2.1.
We can solve Eq. (2.17) using an integration factor, giving
( ∫ ( )
−s
) s s′′ h′′
h(r, c) = exp exp ds′′ (2.18)
c −∞ c c
where h = h(rp + s Ω̂, c Ω̂) and h
′′ = h(rp + s
′′Ω̂, c Ω̂). Substituting s′ = s − s′′ =
|r − r′| (see Fig. 2.1), our integral equation becomes
∫ ∞ ( s′) ′
h(r, c) = exp − (h )ds′ (2.19)
0 c c
where h′ = h(r−s′Ω̂, c Ω̂). The above equation applies to all paths that are outside
of the solid angle ω. Within the solid angle, we must account for the boundary
condition at the surface, which in this case is
h(r, c) = 0, c · n̂ > 0, r = r0êr (2.20)
39
Figure 2.1: Geometry for the mass transfer problem (adapted from Fig. 1 of Lea
[74])
40
This BC represents the fact that all vapor molecules that reach the sphere are
absorbed and none are reflected. With this condition, the contribution to h from
points within solid angle ω are
∫ |r−r0| ( )s′ (h′, 1)
h(r, c) = exp − ds′ (2.21)
0 c c
where the upper limit of integration is the distance between point r and the surface
of the sphere.
For this problem, we are most interested in the number density moment ε1 =
(h, 1). Taking the moment of Eq. (2.19), we get
∫ ∞ ∫ ∞ ( )2 ′
ε =π−3/2
exp(−c ) s
1 ∫ ∫ ∫ ∫ e(xp − ε
′ ′
)1(s )ds dc0 0 c c∞ ∞ s′
=π−3/2 ∫ ∫ ∫ c exp −c
2 − ε ′ ′1(s ) sin θdφdθds dc
0 0 θ φ c
∞
=π−3/2 T (s′∫ 1 )ε(s
′) sin θdφdθds′
0 θ φ
′
−3/2 T1 (|r − r |)=π ′ ε1(r
′)dr′
|r − r |2
where ∫ ∞ ( x)
Tn(x) = c
n exp −c2 − dc (2.22)
0 c
and the integration includes the region outside of solid angle ω.4 More information
about Tn(x) can be found in Section 27.5 of Abramowitz and Stegun [98]. We get
4In deriving the equation for ε1, we first write the integral over molecular velocities in terms of
polar coordinates, i.e. dc = c2dcdΩ̂ = c2 sin θdφdθdc. Because the unit vector Ω̂ appears in both
the spatial and velocity components of h, and because s′ and c are independent variables, we can
switch the order of integration over ds′ and dc and write sin θdφdθds′ = dsdΩ̂/s′ 2 = dr′/s′ 2.
41
the same result if we take the moment of Eq. (2.21), except the integration bounds
the region inside solid angle ω between point r and the surface of the sphere. Thus,
our equation for the number density moment is
∫
−3/2 T1 (|r − r′|)ε1(r) = π ′′ ε1(r )dr
′ (2.23)
V |r − r |2
where V includes all points in space that can be reached from r without first passing
through the sphere.
It is possible to solve Eq. (2.23) numerically by providing an initial guess for
ε1 (either as a function or as a set of values are specified values of r), performing
the integration for a set of radii, comparing the resulting ε1(r) to the initial guess,
and iterating (i.e. using Newton’s method or some other suitable technique). Note
Eq. (2.23) only gives ε1(r) up to some constant multiplier, since if we multiply ε (r
′
1 )
by a constant in the integral, we get ε1(r) multiplied by that same constant. One
obtains the correct numerical values by noting that far from the sphere, the vapor
concentration is given by the Chapman-Enskog solution for diffusion [21, 99]. It is
possible to simplify Eq. (2.23) further by integrating over the angles θ and φ, leaving
only the integral over the radius. The procedure is similar to the procedure that I
will describe in the next section for flow around a sphere. For further information
on solving the mass transfer problem, refer to Lea [74].
42
2.3 Solution of the Krook Equation for Uniform Flow Around a
Sphere
I will now consider the problem that is more pertinent to this dissertation,
that of uniform flow around a sphere in the transition regime. I will start with the
derivation of the equations, then I will consider the numerical strategies that I will
use to solve the problem.
2.3.1 Derivation of the Governing Equations
For this problem, let us consider the situation where the sphere is at the same
temperature as the surrounding gas. Note that this does not mean the flow field is
isothermal [76, 78], so we must consider all of the terms in the Krook equation.5
We start by writing Krook equation with the diffusion term as a derivative
along the characteristic path, as we did for the problem of mass transfer to the
sphere: [ ( )( )]
dh h 1
+ = (h, 1) + 2c · 2 3(h, c) + c2 − h, c2 − 3 (2.24)
ds c c 3 2 2
Again, we can apply an integrating factor to convert the integro-differential
5In continuum flow, the flow field is isothermal, but this is not the case outside of the continuum
regime. However, Law and Loyalka [76] have shown that accounting for temperature fluctuations
has only a minor impact on the calculated velocity profile and drag force. Their published results
compare favorably to the earlier results of Lea and Loyalka [75] that assume an isothermal flow
field. Nevertheless, I will account for ε3 6= 0 in my calculations because it better represents the
physics of the problem.
43
equation to an integral equation, which gives us
∫ ∞ [ ] [ ( )( )]1 s′ 2 3 3
h(r, c) = exp − (h′, 1) + 2c · (h′, c) + c2 − h′, c2 − ds′
0 c c 3 2 2
(2.25)
for the region of space outside of the solid angle ω and
[ ] ∫ s [ ] [s 1 (s− s′)
h(r, c) = h ′ ′0 exp − + exp − ((h , 1))+(2c · (h , c)c 0 c c )]
2 3 3
+ c2 − h′, c2 − ds′ (2.26)
3 2 2
for the region of space in the solid angle ω between r and the surface of the sphere.
In Eq. (2.25), s′ = |r − r′| and h′ = h(r − s′Ω̂, c). In Eq. (2.26), s − s′ = |r − r′|,
h′ = h(r0+s
′Ω̂, c), h0 = h(r0, c) (i.e. the value of h at the surface of the sphere), and
s = |r−r0|. (See Fig. 2.2 for the geometry described by Eq. 2.25 and Fig. 2.3 for the
geometry described by Eq. 2.26.) The integral terms in these two equations represent
the contribution to h from molecules whose last collision was with another gas
molecule, while the non-integral term on the right-hand side of Eq. (2.26) describes
the contribution to h from molecules whose previous collision was with the sphere.
Next, we must take the moments of h(r, c) to find ε1(r), ε2(r), and ε3. We
44
Figure 2.2: Geometry for Eq. 2.25, which describes the contribution to h(r, c) from
points outside of the solid angle ω
45
Figure 2.3: Geometry for Eq. 2.26, which describes the contribution to h(r, c) from
points inside of the solid angle ω, including points on the surface r0
46
start by taking the moment (h, 1) using Eq. (2.25):
∫ ∞ ∫ ∞ [ ] [1 s′
ε1(r) =π
−3/2 exp − ((h
′, 1) +)2(c · (h
′, c)
0 0 c c )]
2 3 3
∫ ∫ ∫ ∫ + c
2
[ − h
′
] [ , c
2 − ds′dc
3 2 2
∞ ∞ s′
=π−3/2 c sin θ exp − (h′, 1) + 2c · (h′, c)
0 θ φ 0 ( c )( )]
2 2 − 3 ′ 2 − 3∫ ∫ ∫ [ + c h , c dcdφdθds
′
3 2 2
∞
=π−3/2 sin θ T1(s
′)(h′( , 1) + 2 T (s
′
2 )Ω̂ ·)(h
′
0 θ φ (, c) )]
2 ′ 3 ′ ′ 2 3
∫ [ + T3(s )− T(1(s ) h ,)c − ] dφdθds
′
3 2 2
−3/2 dr
′ 3
=π ′ T1 ε
′
1(r ) + T2 Ω̂ · ε ′| − |2 2
+ T3− T1 ε3(r )
r r 2
The argument of the Tn functions in the last equality is |r − r′|.
Taking the moment (h, 1) using Eq. (2.26) gives the same result for the integral
over ds′ (i.e. the second term on the right-hand side of the equation), so now we
∫must determine the contribution from the surface of the sphere. This is simply
h0 exp(−c2 − s)dc, where again s = |r − r0|. Thus, the perturbation to thec
number density is
{∫ ( )
−3/2 − 2 − |r − r0|ε1(r) = π ∫ h(r0, c) ex[p c dcC c
dr′
( ) ]}
+ ′′ T1 ε1(r ) + T2 Ω̂ ·
3
ε (r′) + T − T ε (r′)
|r − r |2 2 3 1 3V 2
(2.27)
The domain C accounts for all molecules reflected from the sphere whose trajectory
47
passes through r, while the domain V includes all points in physical space that can
be reached from r without first passing through the sphere (including points within
the solid angle ω between r and the surface of the sphere). Note that in the above
equations, Ω̂ = c/c = (r − r′)/|r − r′|.
Taking the moment 2(h, c) of Eqns. (2.25) and (2.26) gives us the following
equation for ε2(r):
{∫ ( )
|r − r0|
ε2(r) = 2π
−3/2
∫ ch(r0, c) e[xp −c
2 − dc
C c ( ) ] }
dr′ 3
+ ′ T2 ε
′
1(r ) + T3 Ω̂ · ε ′2(r ) + T4− T2 ε ′3(r ) Ω̂
V |r − r |2 2
(2.28)
Likewise, our equation for ε3(r) is
{∫ ( ) ( )
2 −3/2 3 |r − r0|ε3(r) = π ∫ c
2
[−( h(r0, c)) exp −c
2 − dc
3 C 2 c
dr′
( )
3 ′ 3+ ′ T3− T1 ε1(r ) +( T4− T2 Ω̂)· ε2(r
′)
V |r − r |2 2 2 ]}
9
+ T5−3 T3 + T1 ε ′3(r ) (2.29)
4
These results are very similar to our result for ε1, as are the intermediate mathe-
matical steps required to obtain equations for the density, velocity, and temperature
perturbations.
Before we can solve for ε1, ε2, and ε3, we must determine the perturbation at
the surface of the sphere, h(r0, c). To do so, we must apply the boundary condition
48
at the surface of the sphere,
h(r0, c) = A(r0)− 2c ·U∞, c · n̂ > 0 (2.30)
Note that our other boundary condition,
lim h(r, c) = 0, (2.31)
|r|→∞
states that the perturbation decays to zero far from the sphere.
The form of Eq. (2.30) assumes diffuse reflection from the surface. (n̂ is the
normal vector on the surface of the sphere at r0.) Thus, the reflected molecules
have a Maxwellian distribution, with U = 0 (since the sphere is stationary for
this problem), T = T∞, and unknown number density. If we plug Eq. (2.30) into
Eq. (2.7) for f , c · n̂ > 0, we get
f(r0, c) = f∞[1 + A(r0)], c · n̂ > 0
This shows that the function A(r0) is effectively the perturbation to the number
density of an equilibrium gas with n = n∞, U = 0, and T = T∞, so the form of our
surface boundary condition for h is correct.
We can determine A(r0) by applying mass conservation at the surface:
∫
c · n̂fdc = 0
49
Physically, this integral states that the net mass flux at the surface of the sphere
is zero. Breaking our integral into two parts – the molecules moving towards the
surface and those moving from the surface – and substituting for f , we get
∫ ∫
c · n̂f∞(1 + 2c ·U∞ + h)dc = − c · n̂f∞(1 + 2c ·U∞ + h)dc
c · n̂>0 c · n̂<0
The constant terms cancel in the integrals due to the symmetry of the Maxwellian
distribution, so our mass balance becomes
∫ ∫
c · n̂ exp(−c2)(2c ·U∞ + h0)dc = − c · n̂ exp(−c2)(2c ·U∞ + h0)dc
c · n̂>0 c · n̂<0
(2.32)
For the term on the left-hand side, we can substitute Eq. (2.30) for h and
perform the integration,
∫ ∫
c · n̂ exp(−c2∫ )(2c ·U∞ + h0)dc = c · n̂ exp(−c
2)A(r0)dc
c · n̂>0 c · n̂>0
=A ∫ c∫Ω̂ · n̂ ex∫p(−c
2)dc
c · n̂>0
π/2 2π ∞
=A dθ dφ dc c3 cos θ sin θ exp(−c2)
0 0 0
π
= A(r0)
2
We can also integrate the first term on the right-hand side (i.e. the term
involving c ·U∞). Before performing the integration, we will define α as the angle
between n̂ and U∞, θ as the polar angle between n̂ and Ω̂, and φ as the angle
between U sinα and sin θ. (See Fig. 2.4.) These definitions will be used to evaluate
50
the dot products in the integral. With these conventions established, the integral of
the first term on the right-hand side of Eq. (2.32) is
∫ ∫
−2 2 2 2
c · n̂<∫ c · n̂ ex∫p(−c )c ·U∞dc = −2 c Ω̂ · n̂ exp(−c )Ω̂ ·U∞dc0 c · n̂<0π/2 2π ∫ ∞
=− 2U dθ dφ dc c4 exp(−c2) cos θ sin θ(sinα sin θ cosφ+ cosα cos θ)
0 0 0
π3/2
=− U cosα
2
Figure 2.4: Geometry for determining A(r0)
Finally, we can substitute Eq. (2.25) for h0 in the second term on the right-
51
hand side of Eq. (2.32) and perform the integration:
∫
hc · n̂ exp(−c2)dc
c · n̂<0 ∫ ∫ ∞ [ ] [c s′
= dc ( ds
′
)·(n̂ exp −c
2
)−] (h
′, 1) + 2c · (h′, c)
c · n̂<0 0 c c
2 2 − 3 3∫ +
2
∫ c h, c −3 2∞ ∫ 2∞ [ ′] [s
= ds′ dΩ̂ dc c2 exp −c2 − (h′, 1) + 2c · (h′( , c)0 Ω̂ · n̂<0 )0( )] c
2 3 3
∫ + c
2 − h, c2 − (Ω̂ · n̂)
3 [ 2 2
dr
= T (|r − r |)ε (r) + T (|r
V |r − r(0|2 2 0 1 3 )−]r0|)Ω̂ · ε2(r)
+ T4(|
3
r − r0|)− T2(|r − r0|) (Ω̂ · n̂)
2
Combining these results, we get an equation for A(r0):
√
A(r0) =− π∫U cosα [ ( ) ]
− 2 dr 3T2 ε1(r) + T3 Ω̂ · ε2(r) + T4− T2 ε3(r) (Ω̂ · n̂)
π V |r − r 20| 2
(2.33)
The argument of the Tn functions in the last equality is |r−r0|. The integral in the
equation above sums up the contribution of all molecules that reach the point on
the sphere r0 and are reflected from the surface. The integration domain is every
point in space with a direct line of sight to the point r0 on the surface of the sphere.
We now substitute our boundary condition Eq. (2.30) for h0 in Eqns. (2.27–
52
2.29) to obtain expressions for ε1(r), ε2(r), and ε3(r) that we can solve numerically:
{∫ [ ]
ε (r) = π−3/21 ∫ A(r0) T2[(|r − r0|)− 2(Ω̂ ·U∞) T3(|r − r0|) dΩ̂ω
dr′
+ T (|r − r′|)ε (r′) + T (|r − r′|)Ω̂ · ε (r′)
|r(− r′|2 1 1 2V ) ]} 2
+ T3(|
3
r − r′|)− T1(|r − r′|) ε (r′3 ) (2.34)
2
{∫ [ ]
ε (r) = 2π−3/22 ∫ A(r0) T3[(|r − r0|)− 2(Ω̂ ·U∞) T4(|r − r0|) Ω̂dΩ̂ω
dr′
+ ′( ′ T2(|r − r |)ε1(r
′) + T)3(|r − r
′
] |)Ω̂} · ε2(r
′)
V |r − r |2
+ T4(|r − r′|)−
3
T2(|r − r′|) ε (r′3 ) Ω̂ (2.35)
2
{∫ [ ( )
2
ε (r) = π−3/23 A(r0) T4(|(r − r0| −
3
) T2(|r − r0|)
3 ω 2 )]
− · 3∫ 2(Ω̂ U′ [∞(
) T5(|r − r0|)− T3(|r − r0|) dΩ̂
2 )
dr
+ T ′
3 ′ ′
| 3
(|r − r |)− T1(
V r(− r′|2 2 )|r − r |) ε1(r )
3
+ ′ ′ ′(T4(|r − r |)− T2(|r − r |) Ω̂ · ε2(r )2 ) ]}
′ ′ 9+ T5(|r − r |)− 3 T3(|r − r |)− T1(|r − r′|) ε ′3(r )
4
(2.36)
Note that we have integrated the reflection term over all molecular speeds, leaving
an integral over the solid angle ω.
Together, Eqns. (2.33–2.36) fully describe the problem. The equations are lin-
53
ear, which guarantees a unique solution for ε1(r), ε2(r), and ε3(r). These equations
are also valid for any geometry, not just for a sphere, provided the integration bounds
are appropriately specified. In theory, we should be able to solve these equations
directly; however, such an attempt would be very computationally expensive.
Fortunately, we can simplify the governing equations for ε1, ε2, and ε3 further.
Let us define the following vectors:
   ψ1(r) ε1(r)
  
ψ2(r) 
   
 √1 ε2r(r)2 
ψ(r) =  =  (2.37)ψ3(r)

 
 1√√ ε2θ(r)2 


ψ (r) 34 ε (r)2 3
  (0) ε1 (r)  √

(0) 
 2 ε2z (r)S(r) = π−3/2   = SA(r) + SU(r) (2.38)√  (0)  √2 ε

2x (r)
2 (0)ε
3 3
ε2z(r) and −ε2x(r) are the perturbations in the r- and θ-velocity, respectively.6 The
source term S(r) corresponds to the first term in Eqns. (2.34) and (2.35); SU(r)
is the part of the source term containing U∞ and SA(r) is the other part. These
definitions are consistent with the work of Cercignani and Pagani [70] and Law and
6The negative sign before ε2x is due to the fact that the positive x-axis used to define ε2x
points in the direction of decreasing θ. Thus, ε2x = −ε2θ. I recognize that this discrepancy can be
confusing, but I am simply retaining the nomenclature used by Lea [74].
54
Loyalka [76]. We can now write our problem as follows:
ψ(r) = Lψ(r′) + S(r) (2.39a)
∫ ′
Lψ(r′) ≡ π−3/2 Λψ(r′ dr) (2.39b)
 V |r − r
′|2
 ( )

 [T1 ε1 + T2 Ω̂ · ε
3
2 + T3− T1 ε3
 2 ]   √ ( ) 2 T ε + T Ω̂ · ε + T −3 2 1 3 2 4 T2 2 ε3 Ωz′  Λψ(r ) ≡   √ [ ( ) ] 

√ [ 2 T2 ε1 + T3 Ω̂ · ε2 + T
3
4− T2 ε3 Ωx 
( ) ( ) 2 ( ) ]
2 T −3 T ε + T −3 T Ω̂ · ε + T −3 T +93 1 1 4 2 2 5 3 T ε3 2 2 4 1 3
(2.39c)
Here, Ω̂ = (r−r′)/|r−r′|, Ω̂x and Ω̂z are the x- and z-components of the molecular
trajectory Ω̂, the argument of the Tn functions is |r− r′|, and the perturbations εn
are a function of r′.
Let us consider each of the terms in Eq. (2.39). Again, we can break up the
source term into the portion that involves U∞ and the portion that includes A(r0).
For a sphere, the source terms SU and SA are
 WU1(r) cosα

 
WU2(r) cosα
SU(r) = U   (2.40)WU3(r) sinα
WU4(r) cosα
55
 
WA1(r) cosα


 WA2(r) cosα S (r) = gU  A (2.41)WA3(r) sinα
WA4(r) cosα
where the W functions are defined in Appendix A. Refer to Appendix A for the
derivation of Eqns. 2.40 and 2.41.
The constant g in Eq. (2.41) is given by
∫
−1/2 2
∞ r
g = π + (ρ(r) ·a(r))dr (2.42)
U r r0 0
where a(r) is defined in Appendix A and ρ(r) is defined shortly. Refer to Ap-
pendix A for the derivation of g.
As I have mentioned earlier, the problem for ψ(r) is linear. Furthermore, the
source term is the only non-homogeneous part of the equation. In order for ψ(r)
to satisfy Eq. (2.39), it must have the same angular dependence as the source terms
[73, 74]. In other words, we can write ψ(r) as the product of an unknown function
of the distance r and a known angular dependence:
   cosα 0 0 0  ρ1(r) cosα 

ψ =  
  0 cosα 0 0 ρ2(r) cosα
·ρ(r) =   (2.43)

  
0 0 sinα 0  ρ 3(r) sinα
0 0 0 cosα ρ4(r) cosα
56
This greatly simplifies the solution, since we need only find the radial dependence
of ψ.
We can now rewrite the kernel given by Eq. (2.39b) as
   cosα 0 0 0  H11 H12 H13 H14∫ ∞ r′ 

0 cosα 0 0  
′ −   

H  21
H22 H23 H24
Lψ(r ) = π 1/2 ·
r   

 r0  0 0 sinα 0  H 31 H32 H33 H34
0 0 0 cosα H41 H42 H43 H44 ρ ′ 1
(r )


ρ2(r
′)
·    dr
′ (2.44)
ρ (r′3 )
ρ4(r
′)
where the matrix H(r, r′) is defined in Appendix A. If we substitute Eq. (2.43) into
Eq. (2.39), the angular dependence cancels out, leaving
∫ ∞ r′
ρ(r) = π−1/2 H(r, r′) ·ρ(r′)dr′ + U∞ [WU(r) + gWA(r)] (2.45)
r r0
Finally, we can divided this equation by U∞ to get
∫ ∞
q(r) = π−1/2 K(r, r′) · q(r′)dr′ +WU(r) + gWA(r) (2.46)
r0
where q = ρ/U∞ and
r′
K(r, r′) = H(r, r′) (2.47)
r
57
This is the equation that we will solve to find the density, velocity, and temperature
perturbations, which are related to q by
ε1(r) = U∞q1(r) cosα (2.48a)
√
ε2z(r) = 2U∞q2(r) cosα (2.48b)
√
ε2x(r) = 2U√ ∞
q3(r) sinα (2.48c)
2
ε3(r) = U∞q4(r) cosα (2.48d)
3
Again, α is the angle between the free stream velocity and r, ε2z(r) is the perturba-
tion in the radial velocity, and −ε2x(r) is the perturbation in the θ-velocity. Thus,
the local density and velocity profiles are (in non-dimensional form)
n(r, α) = n∞ [1 + q1(r)U∞ cosα] (2.49a)
[ ]
1
u2r(r, α) = 1 + √ q2(r) U∞ cosα (2.49b)
[ 2 ]
1
u2θ(r, α) = − 1 + √ q3(r) U∞ sinα (2.49c)
[ √2 ]
2
T (r, α) = T∞ 1 + q4(r)U∞ cosα (2.49d)
3
Since I am also interested in the drag force on the sphere, I must relate the
58
drag to q(r). Let us start by writing the expression for the stress tensor,
∫ ∫
ρ∞ 2
pij(r) = m cicjf(r, c)dc = δij + ρ π
−3/2
∞ cicj e
−c h(r, c)dc (2.50)
2
where ρ∞/2 is the static pressure far from the sphere and δij is the Kronecker delta.
(Note that pij must be multiplied by (2kbT∞/m) to get the appropriate dimensions of
the stress tensor.) The static pressure is related to the unperturbed distribution f∞,
while the shear stress distribution is related to the perturbation h. Going forward,
we can ignore the static pressure contribution to the stress tensor because it will
not contribute to the drag force.
To compute the drag force, we must compute the shear stress tensor at the
surface of the sphere:
∫
2
τij(r ) = ρ π
−3/2
0 ∞ c
−c
icj e h(r, c)dc
c · n̂<0 ∫
+ ρ π−3/2 −c
2
∞ cicj e h(r, c)dc (2.51)
c · n̂>0
Here, we have explicitly divided the bounds of the integral in Eq. (2.50) into the set
of molecules moving towards the surface and the set of molecules moving away from
the surface of the sphere. Substituting Eq. (2.25) into the integral for c · n̂ < 0 and
Eq. (2.26) into the integral for c · n̂ > 0, we get
∫ [ ( ) ]
τ −3/2
3 dr
ij(r0) =ρ∞π T3 ε1(r) + T4 Ω̂ · ε2(r) + T5− T3 ε3(r) ΩiΩj
V 2 |r − r0|
59
∫ ∫
+ ρ π−3/2
2 2
∞ A(r0) cic
−c
j e dc− ρ∞π−3/22U∞ · c c c e−ci j dc
c · n̂>0 c · n̂>0
(2.52)
where Ω̂ = (r0 − r)/|r0 − r|, the argument of the Tn functions is |r − r0|, and
V is the space above the plane tangent to the sphere at r0. This expression gives
the shear stress tensor in terms of the coordinate system (x, y, z) in Fig. 2.2, where
the z-axis is along the vector normal to the surface at r0 and the velocity U∞ is in
the xz-plane. However, we want to know the drag force in the coordinate system
(X, Y, Z) in Fig. 2.2, where U∞ is along the Z-direction. Thus, we must write the
shear stress tensor in terms of (X, Y, Z):
ΠXX =τ
2
xx cos α− 2τxz sinα cosα + τzz sin2 α
ΠY Y =τyy
ΠZZ =τxx sin
2 α + 2τ sinα cosα + τ cos2xz zz α (2.53)
Π 2 2XZ =τxz cos α + (τxx − τzz) sinα cosα− τxz sin α = ΠZX
ΠXY =ΠY X = ΠY Z = ΠZY = 0
Here, α is the angle between U∞ and n̂. (See Fig. 2.4).
The drag is then the integral over the surface of the sphere of the normal
component of the shear force,
( ) ∫
2kBT∞
F̃D = r̃
2
0 Π · n̂dSm S
60
In the (X, Y, Z) coordinate system, the drag is in the negative Z-direction, i.e. in
the direction opposite the flow, so we need only consider the Z-component of the
drag:
( ) ∫
2kBT∞
F̃D,Z =( ) r̃
2
0 (∫ΠZX ,ΠZY ,ΠZZ) · (− sinα, 0, cosα)dSm S1/2
2kBT∞
= r̃20 (−ΠZX sinα + ΠZZ cosα)dSm S
Substituting our expressions for Πij into the above equation for the drag, we get
( ) ∫
2kBT∞
F̃ 2D,Z = r̃0 (τxz sinα + τzz cosα)dSm S
Using Eq. (2.52) for the components τzx and τzz of the shear stress tensor and
performing a lot of algebra, we get the following expression for the drag:
( )1/2 { ∫
−ρ∞ 2πkBT∞ 2π
1/2 ∞
F̃D,Z = Ũ r̃
2
0 r
2[q1(r)WU1(r)− q2(r)WU2(r)
3 m r20 r0 }
− 2q 1/23(r)WU3(r) + q4(r)WU4(r)]dr + [8− gπ ] (2.54)
The details of this derivation can be found in Appendix A.
Taking the ratio of the drag we obtain from the BGK model to the free molec-
ular drag, ( )1/2
−ρ∞ 2πkBT∞F̃ = Ũ r̃2D,fm 0(8 + π)3 m
we get the following drag ratio:
61
{
2π1/2
∫ ∞
FD = r
2[q1(r)WU1(r)− q2(r)W2 U2(r)r0 r0 }
− 2q3(r)WU3(r) + q4(r)WU4]dr + [8− gπ1/2] (8 + π)−1 (2.55)
2.3.2 Numerical Methods
Eq. (2.46) cannot be solved analytically for q(r), so we must solve the problem
numerically at discrete radial locations. One can choose any set of points at which
to solve Eq. (2.46), but it makes the most sense to use points corresponding to the
Gaussian quadrature nodes.
A brief introduction to the Gaussian quadrature formula is warranted at this
point. The integral of any function f(t) in the interval [−1, 1] can be approximated
by ∫ 1 ∑N
f(t)dt ≈ Ajf(tj) (2.56)
−1 j=1
where t1, . . . , tN are nodes, Aj is the weight of the jth node, and f(tj) is the value
of f(t) at the jth node. The nodes and weights for a given N are available from any
number of sources (e.g. Table 25.4 of Abramowitz and Stegun [98]). We can change
the interval [−1, 1] to any interval [a, b], such that our approximation of the integral
becomes
∫ b ∫ ( )b− a 1 b− a a+ b
f(t)dt = f ( t+ dta 2 −1 2 2 )
≈b−
n
a∑ b− a a+ b
Ajf tj + (2.57)
2 2 2
j=1
62
The Gaussian quadrature formula works best when the function can be approxi-
mated by a polynomial in the interval [a, b]; particular care is needed when there are
singularities, as is the case for H ′ii when r = r . Nevertheless, the Gaussian quadra-
ture formula generally provides accurate results using a relatively small number
of points, especially when compared to other numerical integration formulas like
the trapezoidal rule or Simpson’s rule. A good, concise explanation of Gaussian
integration may be found in Kreyszig [100].
With this brief introduction concluded, we will return to the discussion of the
problem of flow around a sphere. We start by rearranging Eq. (2.46):
∫ R′
q(r)− π−1/2 K(r, r′∫) · q(r
′)dr′
r0
∞
= π−1/2 K(r, r′) · q̃(r′)dr′ +WU(r) + gWA(r) ≡ S′(r) (2.58)
R′
Here, we have broken the kernel integral into two separate integrals: one over our
region of interest from r ′0 to R , and another over the region farther away from the
sphere. The right-hand side of Eq. (2.58) can be thought of as an effective source
term S′(r) that includes both the contribution to q(r) from molecules reflected from
the sphere (i.e. the true source term) and the contribution of molecules that enter
the region of interest from farther away.
The function q̃(r) in Eq. (2.58) is a trial function for the perturbation far
from the sphere. Using the asymptotic solution of Takata et al. [78] for the density,
velocity, and temperature far from the sphere, our trial function for q on the right-
63
hand side of Eq. (2.58) is
 ( ) γc ( )

r 21 0
 − c [
3 
r0 ( r ) ] √ r r 3 0 02 c1 + c 2r r
q̃(r) = 
 [ ( ) ]

(2.59)
3
√1 r0 r0  c1 − c2  2 r( ) r r 2 0
c3
r
Here, c1, c2, and c3, are constants to be determined for each Kn number, while
γ = 1.270 is a constant appropriate for hard-sphere molecules. For r0  1 (i.e. for
continuum flow), c = −3 , c = 11 2 , and c3 = 0, as we can verify by solving the Stokes2 2
equation for flow around a sphere.
We must make one final modification to our governing equation:
[ ∫ ]R′ ∫ R′
q(r) 1− π−1/2 K(r, r′)dr′ − π−1/2 K(r, r′) · [q(r′)− q(r)] dr′ = S′(r)
r0 r0
∫ (2.60)
Here, we added and subtracted the term K(r, r′)q(r)dr′ from the left side. We have
done this to deal with the singularities in the kernel at r = r′. Because kernel the is
infinite at this point, it will cause problems when we use our Gaussian integration
formula to calculate the integral. By adding and subtracting a term, we end up
multiplying the singular points by zero, which minimizes the problems caused by
the singularities in K.
64
We can now write Eq. (2.46) as a linear algebra problem,
Bq = S′ (2.61)
where B is a 4N×4N matrix, q and S′ are 4N -element vectors, and N is the number
of nodes in the Gaussian integral formula. The vectors are constructed such that
the first N elements correspond to q1(r) and S
′
1(r), the next N elements correspond
to q2(r) and S
′
2(r), and so on for q3(r) and q4(r).
The nodes rj for a given sphere radius are
R′ − r0 R′ + r0
rj = tj + (2.62)
2 2
where tj are the nodes in the Gaussian quadrature formula (Eq. (2.56)) for the
specified N . Lea [74] uses N = 24 and R′ = 10 for all calculations, but we are free
to choose the number of nodes and the size of our region of interest on a case-by-case
basis. With that said, I am generally using R′ = 10 because this choice yields drag
results that compare well with Millikan’s experiments and with the computational
results of Cercignani et al. [73] and Lea and Loyalka [75].
Lastly, the elements of B are
 N [ ∑ R′
]
 − − r0δpq kpq(rm) + AlKpq(rm, rl), m = n2 m=6 lBij =

l=1 (2.63)
 [ ′ ]− R − r0 AnKpq(rm, rn), m =6 n
2
65
where δij is the Kronecker delta, rm and rn are the mth and nth nodes, Kpq is an
element of the 4× 4 matrix K,
∫ R′
k ′ ′pq(rm) = Kpq(rm, r )dr , (2.64)
r0
and the various indices are related as follows:
i = (p− 1)N +m; j = (q − 1)N + n;
1 ≤ p, q ≤ 4; 1 ≤ m,n ≤ N ; 1 ≤ i, j ≤ 4N
We can think of B as a 4× 4 matrix,
 
B′ B′ B′ B′ 11 12 13 14B
′ B′ ′ ′ 21 22 B23 B24
B =  

B
′ ′ ′ ′ 
31 B32 B33 B34
B′ B′ ′ ′41 42 B43 B44
where each B′pq is an N×N matrix. Notice that we must treat the diagonal elements
of each B′pq differently than the off-diagonal elements due to the singularities present
in the kernels for the diagonal elements (since the diagonal elements correspond to
rm = rn), as introduced in Eq. (2.60).
I have written a MATLAB function to solve Eq. 2.61. (See Section G.1 for
the source code.) The function first populates the matrix B and the vectors WU(r)
66
and WA(r), since the elements of these arrays are independent of the solution q(r).
Next, the function calculates S′(r) using the trial function Eq. 2.59 with an initial
guess for c1, c2, and c3 to compute g and the kernel integral from R
′ to infinity in
Eq. 2.58. The function then inverts Eq. 2.61 solve for r. Since g depends on q(r),
we must recompute g and solve for q(r) until the value for g converges.
The continues the above procedure until the solution for q(r) matches the trial
function at R′, i.e.
    
( )
 γc ( )

1 − r
2
0
q (R
′) c  31 r ′    0 R      √
[
r ( ]0 r )3 0
q ′2(R )  2 c1 ′ + c 2  R R′ = [  ( )
]
3

 ′ q3(R ) √1 r0   c1 ′ −
r0
c2 
   2 R
′
( ) R 
′ r 2

q 04(R ) c3
R′
It uses the MATLAB function fsolve to find values for c1, c2, and c3 that minimize
the error in the above equation.
2.4 Conclusions
In this chapter, I have presented the derivation of the Krook equation and
solved the equation for mass transfer to a sphere (Section 2.2) and uniform flow
around a sphere (Section 2.3) for an arbitrary Knudsen number. This work largely
mirrors previous work in the literature [73, 74, 76]. In the next few chapters, I will
describe how to use these results to determine the drag and torque on aggregates
67
consisting of N spheres in point contact and apply this method to various aerosol
physics problems.
68
Chapter 3: Friction Coefficient for Translating Particles
3.1 Introduction
Aerosol fractal aggregates formed from the coagulation of smaller, spherical
primary particles are found in many natural and industrial settings. Understanding
the forces on these aggregates is important in a number of science and engineering
disciplines, including combustion, fire safety, atmospheric and environmental sci-
ences, materials engineering [2], and nuclear reactor safety [21]. The translational
drag force for a particle moving slowly relative to the surrounding fluid – given
by F = −ζU0, where U0 is the particle relative velocity and ζ is the orientation-
averaged scalar friction factor – is particularly important because it influences the
transport properties of the particle, including its diffusion coefficient and electrical
mobility.
In many practical applications, the primary sphere radius a is significantly less
than the mean free path of the surrounding gas (λ ≈ 65 nm at standard temperature
and pressure and an order of magnitude higher near a flame), so that the primary
sphere is in or near the free molecule flow regime. At the same time, the radius of
gyration Rg for the agglomerate may be comparable to or larger than the mean free
path, so that the aggregate is in the transition flow regime. As one example, for
69
carbonaceous soot a ≈ 5− 30 nm and Rg ≈ 30− 1000 nm.
There are a number of theories and techniques for computing the translational
friction factor of macromolecules and particle aggregates in the continuum regime,
including Kirkwood-Riseman (KR) theory [28] and its extensions by Rotne and
Prager [57], Yamakawa [58], and Chen et al. [30], as well as algorithms that use the
Hubbard and Douglas analogy between the electrostatic capacitance and the friction
factor [31, 63, 101]. Likewise, there are established methods for computing ζ in the
free molecule regime that simulate the ballistic nature of interactions between gas
molecules and aggregates [34–36, 102].
In contrast, there are few approaches for the transition regime. Melas et al.
[79] estimated the friction coefficient in the near-continuum regime by solving the
Laplace equation with a slip boundary condition at the surface of the particle. In a
follow-up paper, the authors determined that their Collision Rate Method is valid
for Knudsen numbers less than 2 [80].
Dahneke [67] developed the adjusted sphere method for the transition regime,
which applies a slip correction factor to the continuum friction factor. The key to
this development is the identification of an aggregate Knudsen number that reduces
a problem involving two length scales (primary radius and aggregate radius of gyra-
tion) to a single dimensionless length. Dahneke’s approach is similar to the approach
used to calculate the drag on a sphere in the transition regime, but the adjusted
sphere method uses an adjusted Knudsen number based on geometric descriptions
of the particle in the continuum (hydrodynamic radius, RH) and free molecular
(projected area, PA) regimes.
70
Through scaling analysis, Zhang et al. [41] developed an approach analogous
to the adjusted sphere method and demonstrated that the approach yields friction
factors comparable to Direct Simulation Monte Carlo (DSMC) results for the ag-
gregates they studied (spheres, dimers, and dense and open 20-particle aggregates).
However, it requires knowledge of the hydrodynamic radius and the projected area
of the particle, which may take tens of minutes to a few hours to obtain compu-
tationally for a single particle. Obtaining RH and PA experimentally is possible,
but it requires painstaking TEM measurements [68]. More rigorous computational
techniques for calculating the transition regime friction factor – such as DSMC
or molecular dynamics – are time consuming: for instance, the reported DSMC
calculation times in Ref. [41] were on the order of one CPU week for a given Knud-
sen number and a given aggregate. Thus, a self-consistent field theory method for
quickly estimating the scalar friction factor of an aggregate across the Knudsen
range is highly desirable.
In this chapter, I present a new approach for computing the hydrodynamic
friction tensor H and the scalar friction coefficient ζ for fractal aggregates across the
entire Knudsen range. This approach involves solving for the velocity field around a
sphere in the transition regime and using the velocity field to compute the friction
factor for the aggregate. In essence, this approach is an extension of KR theory
[28] from the continuum regime to the transition regime. I will first present the
solution of the Krook equation for the velocity field around a sphere, which follows
the procedure developed by Lea and Loyalka [75] and Law and Loyalka [76]. I then
describe the extension of KR theory to the transition regime. Finally, I compare my
71
results to the DSMC results of Zhang et al. [41] and to the scaling theory in that
paper.
3.2 Velocity field
To determine the velocity around a sphere in the transition regime, I use the
kinetic theory approach provided by the Boltzmann equation. For this study, I will
use the Bhatnagar-Gross-Krook (BGK) model [71] instead of the full Boltzmann
collision operator.
Consider a sphere with dimensional radius a∗ in a gas moving at constant
velocity U ∗∞. I will define the viscosity in terms of the gas mean free path λ as
µ = 0.499ρc̄λ, where c̄ is the gas mean thermal speed, which is consistent with
Ref. [41]. For this study, the non-dimensional sphere radius is related to the Knudsen
√
number Kn = λ/a∗ by a = 0.501 πKn−1 [70, 73].
If the flow speed is very small compared to the thermal speed of the gas
molecules (U∞  1), then one can linearize the molecular velocity distribution
f(r, c),
2
f = π−3/2ρ −c∞ e [1 + 2c ·U∞ + h] (3.1)
where ρ∞ is the density far from the sphere, c is the molecular speed, and h is the
perturbation to the distribution function due to the sphere. With this linearization
and using the BGK model, one gets the non-dimensional Krook equation,
c · ∇h(r, c) = ε1(r) + c · ε2(r) + 2(c2 − 3)ε3(r)− h (3.2)3 2
72
where ε1, ε2, and ε3 are perturbations to the density, velocity, and temperature
fields around the sphere,
ρ(r) =ρ∞[1 + ε1(r)] (3.3)
U(r) =U + 1∞ ε2 2 (3.4)
T (r) =T∞[1 + ε3(r)] (3.5)
I followed the same general solution procedure for the perturbations as Lea and
Loyalka [75] and Law and Loyalka [76], with one exception related to the solution far
from the sphere, as discussed below. Notably, I assumed diffuse reflection between
the gas molecules and the sphere. This approach gives the r- and θ-components of
√ √
the velocity perturbation ε2 as U0 2q2(r) cos θ and −U0 2q3(r) sin θ, where r is the
distance from the origin and θ is the angle between r and U∞. The full velocity
field in spherical coordinates is
[ ] [ ]
1 1
U(r) = U∞ cos θ 1 + √ q2(r) êr − U∞ sin θ 1 + √ q3(r) êθ (3.6)
2 2
Far from the sphere (i.e. for r − a > 10), I fit q2(r) and q3(r) to the asymptotic
solution to the Krook equation given by Takata et al. [78],
√ a √ (a)3
lim q2(r) = 2c1 + 2c2 (3.7)
r→∞ r r
c a c (a)31 2
lim q2(r) = √ − √ (3.8)
r→∞ 2 r 2 r
73
Lea and Loyalka [75] and Law and Loyalka [76] used a slightly different form of the
solution for large distances from the sphere, but otherwise my approach is consistent
with the approach in Refs. [75, 76].
I present my solution of the drag as the ratio between drag F for the specified
Knudsen number and the free molecule drag FFM . As shown in Figure 3.1, my drag
results compare favorably (i.e within 2-3%) with a fit to Millikan’s oil drop data [64]
reported by Cercignani et al. [73],
F A+B
=
F 2π−
(3.9)
1/2
FM a+ A+B exp[−2π−1/2Ca]
where A = 1.234, B = 0.414, and C = 0.876. My results are also consistent with
previous calculations [73, 75, 76, 78], as shown in Table 3.1. (See Appendix B for
more detailed results.)
Figure 3.1: Ratio of the calculated drag from the Krook equation to the free molecule
drag. Results are compared to a fit to Millikan’s data [64]
74
Table 3.1: Comparison of my results for F/FFM to Millikan’s data and to results
from previous computational studies
a Kn Millikan Cercignani Law and This
[64] et al. [73] Loyalka study
[76]
0.05 17.8 0.9784 0.9778 0.9771 0.9769
0.075 11.8 0.9677 0.9651 0.9658 0.9654
0.10 8.88 0.9571 0.9529 0.9546 0.9540
0.25 3.55 0.8959 0.8864 0.8912 0.8884
0.50 1.78 0.8036 0.7900 0.8007 0.7916
0.75 1.18 0.7236 0.7088 0.7271 0.7104
1.00 0.888 0.6549 0.6404 0.6513 0.6423
1.25 0.710 0.5961 0.5824 0.5967 0.5850
1.50 0.592 0.5456 0.5332 0.5507 0.5363
1.75 0.507 0.5021 0.4910 0.5115 0.4947
2.00 0.444 0.4645 0.4546 0.4779 0.4588
2.50 0.355 0.4029 0.3951 0.4233 0.4001
3.00 0.296 0.3551 0.3488 0.3521 0.3545
4.00 0.222 0.2863 0.2818 0.2870 0.2884
5.00 0.178 0.2396 0.2360 0.2431 0.2429
6.00 0.148 0.2058 0.2029 0.2120 0.2099
7.00 0.127 0.1804 0.1779 0.1822 0.1848
8.00 0.111 0.1606 0.1583 0.1642 0.1650
9.00 0.0987 0.1447 0.1426 0.1501 0.1492
10.00 0.0888 0.1317 0.1297 0.1388 0.1361
75
3.3 Kirkwood-Riseman theory
Kirkwood and Riseman [28] demonstrated that the force on the ith element
of an N -element polymer chain is given by
∑n
Fi = −ζ0(U0 − ui)− ζ0 Tij ·Fj (3.10)
i 6=j
whereU0 is the unperturbed fluid velocity, ui is the velocity of the ith chain element,
ζ0 is the friction factor given by Stokes’ law, and Tij is the hydrodynamic interaction
tensor. The to∑tal force on the chain is the vector sum of the forces on the chain
elements, F = Ni Fi.
The original derivation used the Oseen tensor for Tij. Rotne and Prager [57]
and Yamakawa [58] derived a modified hydrodynamic tensor Tij that accounts for
the curvature of the chain elements and hydrodynamic interactions between two
elements, {[ ] [ ]}
1 r r 2a2ij ij 3rijrij
Tij = I + + I− (3.11)
8πµr 2 2ij rij 3rij r
2
ij
where rij is the vector from the ith element to the jth element and rij is the distance
between the elements. Chen, Deutch, and Meakin later applied this approach to
find the translational drag force on a fractal aerosol particle [30, 54, 55].
Rotne and Prager [57] and Yamakawa [58] noted the similarities between their
modified interaction tensor and the solution of Stokes flow around a stationary
sphere. One can write the perturbation to the velocity caused by the sphere in the
76
following form:
v(rij) = Vij ·U0 (3.12)
where [( ) ( )]
6πµa rijrij a
2 3rijrij
Vij(rij) = I + + I− (3.13)
8πµr r2 2 2ij ij 3rij rij
Written thus, the velocity perturbation is the dot product of the unperturbed ve-
locity U0 and a tensor Vij that describes the action of the sphere on the flow. Vij
is the product of the Stokes friction factor (the numerator of the leading coefficient
in Eq. (3.13)) and a hydrodynamic tensor that is the same as the modified hydro-
dynamic interaction tensor in Eq. (3.11), with the exception of the factor of 2 in the
r−3ij term in Tij. This suggests that I can replace the product ζ0Tij in Eq. (3.10)
with the tensor Vij with minimal error, since the term proportional to r
−3
ij decays
quickly as one moves further from the jth particle. The force on the ith particle is
now ∑N
Fi = −ζ0U0 − Vij ·Fj (3.14)
i 6=j
Here, I am assuming that each of the primary particles is translating at the same
velocity relative to the background gas, which is appropriate for an aerosol particle.
That relative velocity is now specified as U0.
To verify that the error in using the velocity perturbation tensor in place of
the product of the modified Oseen tensor and the monomer friction coefficient is
small, I calculated the drag on the open and dense 20-particle aggregates described
below using both approaches. The error in the drag calculated using Vij is less
77
than 2% for the dense aggregate and less than 1% for the open aggregate. This
error decreases as the number of primary spheres increases, as expected from the
r−3ij dependence.
To extend KR theory from the continuum regime to the transition regime,
I set ζ0 equal to the friction coefficient for a sphere that I calculated using the
Krook equation and write the hydrodynamic tensor Vij in terms of the velocity
perturbation ε2,
( )
−q2√(rij) rijrij − q3√(rij) − rijrijVij = I (3.15)
2 r2ij 2 r
2
ij
I obtain the orientation-averaged translational friction factor for the particle by
following the approach outlined in Happel and Brenner [49]: I calculate F /U0 for
three mutually-orthogonal particle orientations, compute the eigenvalues λm of the
resulting friction tensor, and set the friction coefficient equal to the harmonic average
of the eigenvalues, ( )−1
1 1 1
ζ = + + (3.16)
λ1 λ2 λ3
Note that the friction tensor is symmetrical (allowing for some numerical uncer-
tainty) in the transition regime, as it is in continuum flow.
Because my drag results from the Krook equation are non-dimensionalized
by the free molecule drag force, I obtain the dimensional scalar friction factor by
multiplying by the free molecule friction factor for the primary sphere,
∗ π(8 + π) µζ = ζ a2 (3.17)
2.994 λ
78
3.4 Results
Numerous experimental studies have been performed to determine the friction
coefficient – or a related quantity, the electrical mobility – for fractal aggregates in
the transition regime, as summarized in a review paper by Sorensen [4]. However,
most published data lacks the detailed description of particle morphology (i.e. the
hydrodynamic radius RH and the projected area PA) needed for a meaningful com-
parison with my theoretical calculations. Zhang et al. [41] compared DSMC results
to their own data [103], while Thajudeen et al. [68] compared mobility data to the
adjusted sphere method scaling law; both studies showed good agreement between
theory and experimental data. Thus, I will compare my results to the scaling law
and to published DSMC results [41] for well characterized particles over a wide
range of Knudsen numbers. Specifically, I have generated aggregates with similar
characteristics as the open and dense aggregates in Ref. [41]. These aggregates are
shown schematically in Figure 3.2. I generated the particles with a cluster growth
algorithm [36] where we specify the fractal prefactor and exponent and the number
of primary spheres. I verified that the particles have similar hydrodynamic radii
and projected areas as the particles described in Ref. [41] using the Zeno algorithm
[62] for RH and my own algorithm for the projected area. (The Zeno algorithm
uses a random walk approach to calculate the electrostatic capacity of an aggre-
gate; Hubbard and Douglas [31] have demonstrated that the hydrodynamic radius
is within 1% of the electrostatic capacity for shapes with analytical solutions for
both quantities. See Appendix C for a description of my Monte Carlo algorithm.) I
79
also compared my results for a dimer to the DSMC results.
Figure 3.2: Open (left) and dense (right) 20 particle aggregates used in this study.
The calculated RH and PA for these aggregates are very close to the values for the
open and dense aggregates in Ref. [41]. The colors represent the calculated ratio
of the drag on a primary sphere to the drag on an isolated sphere at the specified
Knudsen number. A ratio of unity suggests that a sphere behaves as if it is isolated.
Figure 3.2 illustrates the effects of the Knudsen number on the flow field and
drag on each primary sphere. The color of each sphere in the figure is the ratio be-
tween the calculated drag Fi on each sphere and the drag ζ0U0 on an isolated sphere
at the specified Knudsen number; alternatively, the color represents the fluid veloc-
ity at the center of each sphere. The open aggregate at a primary Knudsen number
of 10 has relatively little effect on the flow field. Monomers near the periphery of
the aggregate behave almost like isolated spheres, while monomers near the interior
of the particle experience a lower fluid velocity largely due to direct shielding by the
80
Figure 3.3: Comparison of my results for the slip correction factor to DSMC results
from Zhang et al. [41] for a dimer, an open 20 particle aggregate, and a dense 20
particle aggregate. The slip correction factor is the ratio of the continuum friction
factor 6πµRH to the calculated friction factor.
other spheres. This behavior is characteristic of free molecule flow.
For the same particle at a primary Knudsen number of 1, the velocity at each
monomer is much lower than in the Kn = 10 case. Clearly, all of the monomers are
affected to a larger degree by the presence of the neighboring spheres. The same
is true for the dense aggregates with a fractal dimension of 2.5: each monomer has
more neighbors, and thus each monomer behaves less like an isolated sphere than
in the case of an open aggregate with a fractal dimension of 1.78.
My results for the drag on the dimer and open and dense aggregates are
shown in Figure 3.3. Here, I have plotted the aggregate slip correction factor,
Θ−1 = 6πµR ∗H/ζ , versus the primary Knudsen number. Both my results and the
DSMC results assume diffuse reflection and full thermal accomodation between the
gas molecules and the particle. In general, my KR theory results compare well
81
with the DSMC results. My calculated slip correction factors are higher than the
DSMC slip factors at decreasing Knudsen numbers, though Zhang et al. [41] note
that their DSMC results tend to under-predict the slip correction factor due to
the finite size of the computational domain. The DSMC results are particularly
influenced by domain size at lower Knudsen numbers, which explains the larger
deviation between my results and the DSMC results in the near-continuum regime.
Note that I discarded one of the near-continuum dense aggregate DSMC points from
Ref. [41] because it fell significantly below the continuum limit Θ−1 = 1.
I used my extended Kirkwood-Riseman approach to test the observations put
forth in Refs. [41, 68, 69] that plots of the slip correction factor versus the aggregate
Knudsen number, defined by Zhang et al. [41] as
Kn = πλRH/PA, (3.18)
collapse to a single curve. Figure 3.4 shows my results for aggregates with a range of
fractal dimensions and number of primary spheres. I also include the DSMC results
from Zhang et al. [41] for comparison. My results and the DSMC results all follow the
same general curve, with relatively little deviation among the various calculations.
This provides further support to the theory of a universal slip correction factor
versus aggregate Knudsen number scaling law.
82
Figure 3.4: Calculated slip correction factors for a range of aggregate morphologies,
plotted versus the aggregate Knudsen number. DSMC results from Zhang et al. [41]
are included for comparison.
3.5 Discussion
I have introduced a new approach for computing the translational friction
coefficient for a fractal aerosol particle across the entire Knudsen range, given the
particle’s coordinates and primary sphere radius. Coordinates can be generated
using a cluster growth algorithm, as I have done for this study, or they can be
obtained from TEM images, using methods described in the literature (e.g. Ref. [68]).
The solution method is also very fast: it takes approximately 10 seconds on
a single processor to obtain the friction coefficient for approximately 50 Knudsen
numbers for a 20-particle aggregate. Furthermore, my Kirkwood-Riseman results
converge to the correct continuum and free molecule limits obtained using the
Hubbard-Douglas approximation for the continuum and a ballistic approach for
83
the free molecule aggregate friction factor.
Over the parameter range examined, my results support the validity of the
adjusted sphere/scaling method developed by Dahneke [67] and Zhang et al. [41] and
promoted by more recent studies [68, 69, 79, 80]. Because the Kirkwood-Riseman
approach can provide results quickly across the Knudsen range, this approach may
be preferable to DSMC for evaluating scaling laws (such as those developed by
Rogak et al. [37], Lall and Friedlander [38], and Eggersdorfer et al. [39]) that relate
the friction coefficient to the number of primary spheres in the aggregate.
While I have focused on the friction coefficient in this chapter, my method
also determines the friction tensor, which is important when considering particle
alignment in an external force field [5]. This is discussed further in Chapter 7.
Finally, I emphasize that my results assume diffuse reflection between the gas
molecules and the particle. This is consistent with past computational studies for
fractal aerosol particles (e.g. Refs. [36, 41]) and with experimental results, which
suggest that most collisions are diffuse [32]. With that said, the Kirkwood-Riseman
approach could be applied for alternative reflection models, provided one solves the
Krook equation for the velocity using the appropriate boundary condition at the
surface of the sphere.
84
Chapter 4: Analytical Expression for the Friction Coefficient of DLCA
Aggregates based on Extended Kirkwood-Riseman The-
ory
4.1 Introduction
Aerosol particles formed at high temperature are often fractal aggregates de-
scribed under the assumption of equally-sized spherical primary particles as
( )d
R fg
N = k0 (4.1)
a
where N is the number of primary spheres, Rg is the radius of gyration of the
agglomerate, a is the primary sphere radius, and df and k0 are the fractal dimension
and prefactor.
The transport properties of these particles (e.g. the diffusion coefficient, set-
tling velocity, and electrical mobility) can be related to the particle scalar friction
coefficient ζt, which is defined by the relationship between the drag force and the
relative velocity between the particle and the fluid,
Fd = ζt(uf − up) = ζtU (4.2)
85
where uf and up are the velocities of the fluid and the particle, respectively. Knowl-
edge of the friction coefficient is crucial to predicting particle diffusional, phoretic,
and electrostatic behavior in real-world applications.
For the simple case of a sphere with radius a, the friction coefficient is given
by Stokes’ law,
6πµa
ζt = (4.3)
Cc(Kn)
where µ is the gas viscosity, Kn = λ/a is the Knudsen number, λ is the gas mean
free path, and Cc is the Cunningham slip correction factor,
1
[ ( )]
C
Cc(Kn) = 1 + Kn A+B exp − (4.4)
Kn
Spheres that are very large compared to the mean free path (Kn → 0) are in the
continuum regime. In this case, the slip correction is unity, and the continuum
friction factor is simply
ζct,0 = 6πµa (4.5)
Spheres that are very small compared to the mean free path are in the free molecule
regime, where the friction coefficient is given by Epstein’s equation,
FM π(8 + απ) µζt,0 = a
2 (4.6)
2.994 λ
The momentum accommodation coefficient α is equal to unity for purely diffuse re-
1In this work, I define the viscosity by the relation µ = 0.499ρc̄λ, where ρ is the gas density
and c̄ is the mean thermal speed. This expression describes a hard sphere gas. Furthermore, I use
Davies’ coefficients (A = 1.257, B = 0.4, and C = 1.1) [65] in the slip correction factor.
86
flection and zero for purely specular reflection at the surface of the particle. Epstein
[32] determined that most collisions are diffuse.
Determination of the friction coefficient is much more complicated for fractal
aggregates. In the continuum regime, the friction coefficient is given by
ζt = 6πµRH (4.7)
where RH is the particle hydrodynamic radius, which may be obtained by applying
either the Kirkwood-Riseman [30, 54–56] or Hubbard-Douglas [31, 101] method.
For the free molecule regime, one can obtain the friction coefficient using a ballistic
approach [34–36], such that the friction coefficient is related to the orientation-
averaged projected area of the particle.
Both computational and experimental results seem to support power-law type
relationships between the number of primary spheres in the aggregate and the fric-
tion coefficient:
ζt = AN
η (4.8)
Sorensen [4] reviewed available experimental data for particles formed by diffu-
sion limited cluster aggregation (DLCA) and proposed exponents of 0.46 forN < 100
and 0.56 for N > 100 in the continuum regime and 0.92 for all N in the free molecule
regime.
In many practical applications, the primary sphere radius is smaller than the
gas mean free path, such that the primary spheres may be in the free molecule
87
flow regime. For situations in which the primary sphere Knudsen number is in
the free molecular regime, many researchers (e.g. [34–36]) have used free molecu-
lar techniques to compute the scalar friction coefficient for fractal aerosol particles.
However, the agglomerate size characterized by the radius of gyration may be com-
parable to or larger than the mean free path, which leads to some ambiguity about
the appropriate flow regime. Therefore, an alternate approach is needed to deter-
mine the friction coefficient for particles whose geometric measures (primary sphere
radius and radius of gyration) lie in the transition flow regime.
To date, most of the approaches for transition regime drag are based on extrap-
olation of free molecule or continuum methods to the transition regime or power-law
fits to experimental data. One exception is the adjusted sphere method (ASM) de-
veloped by Dahneke [67] and [41], which applies a slip correction to the continuum
drag based on an aggregate Knudsen number,
6πµRH
ζt,ASM = (4.9)
Cc(Knagg)
πλRH
Knagg = (4.10)
PA
where the hydrodynamic radius RH and the projected area PA are continuum
and free molecular measures of particle size, respectively. Zhang et al. [41] found
good agreement between the friction coefficient computed using the adjusted sphere
method and Direct Simulation Monte Carlo (DSMC) results for a dimer and for
open (df = 1.78, k0 = 1.3) and dense (df = 2.5, k0 = 1.5) 20-particle aggregates
88
for a range of aggregate Knudsen numbers. For this approach one must obtain the
hydrodynamic radius and projected area, either through TEM analysis or through
moderately expensive computational models mentioned previously.
Recently, I developed a self-consistent field method to compute the friction
coefficient for a fractal aggregate across the entire Knudsen range (Chapter 3).
This method is based on Kirkwood-Riseman theory for the drag on a particle or
macromolecule in continuum flow. Initial applications of the self-consistent method
show good agreement with DSMC results [41] and with the adjusted sphere method.
Here, I apply my self-consistent field method to compute the scalar friction
coefficient for a wide range of primary sphere radii and aggregate sizes. I compare
the results to experimental data in the literature [43, 44] and to the predictions
of other models that have been developed for the transition regime, including the
adjusted sphere method and the correlations developed by Rogak et al. [37], Lall
and Friedlander [38], and Eggersdorfer et al. [39].
4.2 Theoretical Methods
4.2.1 Kirkwood-Riseman Theory
Consider an aggregate consisting of N identically-size spherical particles of ra-
dius a. Kirkwood and Riseman [28] demonstrated that the force on the ith spherical
element can be obtained by considering the effects of all the other elements on the
89
fluid flow pattern, as described by
∑N
F = ζc ci t,0Ui − ζt,0 Tij ·Fj (4.11)
i=6 j
Here, ζct,0 = 6πµa is the friction coefficient on the primary spheres as given by Stokes’
law, Ui is the velocity of the ith sphere,
2 and Tij is the hydrodynamic interaction
tensor. The original version of the theory uses the Oseen tensor for Tij.
Later researchers extended this approach to fractal aerosol particles [30, 54, 55]
and colloids [56]. These later studies used the modified form of the Oseen tensor
derived independently by Rotne and Prager [57] and Yamakawa [58]:
[( ) ( )]
1 r 2ijrij 2a 3rijrij
Tij = I + + I− (4.12)
8πµr r2ij ij 3r
2 2
ij rij
Here, rij is the vector from the ith particle to the jth particle.
These applications of Kirkwood-Riseman theory involve objects in continuum
flow. We now wish to extend this approach to the transition flow regime, using
appropriate expressions for the friction coefficient ζt,0 and the hydrodynamic inter-
action tensor Tij.
We start by dividing Eq. (4.11) by the friction coefficient to give the fluid
velocity at a point ri: ∑N
u(ri) = Ui − Tij ·Fj (4.13)
i 6=j
In other words, the fluid velocity at a point is the sum of the free stream velocity
2If the particle is rigid and if it is not rotating, then Ui = U , where U is the particle velocity.
90
and the velocity perturbations caused by each primary sphere in the particle.
For uniform Stokes flow around an isolated sphere, the velocity obtained by
solving the Navier-Stokes equation can be written in the form
u(r) = U −V ·U (4.14)
where [( ) ( )]
3a r 2ijrij a 3rijrij
V(r) = I + + I− (4.15)
4r r2 3r2 2ij ij rij
is the velocity perturbation tensor at the point r and r is the distance of that point
from the origin (i.e. the center of the sphere). We can also write the velocity as
u(r) = U −T′(r) ·F (4.16)
where
[( ) ( )]
′ ≡ V(r) 1 rijrij a
2 3rijrij
T (r) = I + + I− (4.17)
ζct,0 8πµr r
2
ij 3r
2 2
ij rij
and F = ζct,0U is the drag force on the sphere.
The tensor T′ is the same as the Rotne-Prager-Yamakawa hydrodynamic in-
teraction tensor [Eq. (4.12)], with the exception of the factor of 2 in the r−3 term.
Since we are primarily concerned with the velocity perturbation at distances greater
than 2a from the sphere, we can ignore the factor of 2 with minimal error and re-
place Tij in Eq. (4.11) with T
′. Now, the drag force on the ith sphere of a fractal
91
particle is ∑N
Fi = ζ
c
t,0Ui − Vij ·Fj (4.18)
i 6=j
where Vij is the velocity perturbation at the ith sphere caused by the jth sphere.
Of course, there is no reason to make the approximation T′(r) ≈ Tij(r)
for continuum flow. However, this approximation allows us to extend Kirkwood-
Riseman theory to the transition regime because solving for the velocity profile
around an isolated sphere in the transition regime is considerably easier than explic-
itly considering the hydrodynamic interaction between two spheres in the transition
regime. Numerous solutions of the former problem are available in the literature
[75, 76, 78, 104], whereas we have not been able to find any reference to the latter
problem.
Before we proceed further with our derivation of the force on a fractal aggregate
in the transition regime, we will first consider the solution of the kinetic equation
for the velocity around a sphere.
4.2.2 Flow around a Sphere
Consider steady flow around a sphere in the transition regime.3 The gas den-
sity, velocity, and temperature far from the sphere are ρ∞, U∞, and T∞, respectively.
In the absence of external forces, the Boltzmann equation can be written as
c · ∇ δff(r, c) = ∣∣∣∣ (4.19)δt coll
3More information about solving the kinetic equation for flow around a sphere in the transition
regime can be found in Chapter 2.
92
where f is the velocity distribution function and c is the gas molecular velocity. The
right-hand side of Eq. (4.19) is the collision operator, which describes the evolution
of the distribution function as a result of collisions between gas molecules. The
full collision operator is exceedingly complicated, so we will consider the simplified
collision operator proposed by Bhatnagar, Gross, and Krook [71]:
∣
δf ∣∣∣ = ν[f0(r, c)− f(r, c)] (4.20)δt coll, BGK
Here, ν is the collision frequency and f0 is the Maxwellian velocity distribution at
point r, ( )3/2 ( )
m 2
f0(r, c) = n exp −
m|c−U |
(4.21)
2πkBT 2kBT
where m is the mass of a gas molecule, kB is the Boltzmann constant, and n, U ,
and T are the local gas number density, bulk velocity, and temperature. Essentially,
the BGK model assumes that the non-equilibrium distribution f relaxes to the
equilibrium distribution f0 after one collision, with the collision frequency given by
ν = p/µ, where p is the gas pressure.
If the velocity of the gas around the sphere is small relative to the thermal
speed of the gas molecules and the perturbation caused by the sphere is relatively
small, then the distribution function can be linearized, giving
f(r, c) ≈ f∞[1 + 2c ·U∞ + h(r, c)] (4.22)
The Maxwellian distribution f∞ represents a gas with zero velocity at the far-away
93
gas density and temperature. The first two terms of the linearization represent the
distribution far from the sphere, while the function h represents the perturbation
to the distribution caused by the sphere. Likewise, the linearized local Maxwellian
distribution can be written
f0(r, c) ≈ f∞[1 + 2c ·U∞ + ε1 + c · ε2 + (c2 + 3)ε3 (4.23)2
where ε1, ε2, and ε3 are perturbations to the density, velocity, and temperature of
the gas defined below as moments of the distribution function h.
Now define the following non-dimensional variables:
[ ( ) ]3/2 −1
m ( )
f ? = f −3/2 ?(n ) = π exp −|c −U
?|2
2kBT
1/2
? mc = c ( ) (4.24)2kBT 1/2 √
? r m πr = = r
ν 2kBT 1.996λ
The final expression for the non-dimensional radius makes use of the previously
defined expressions for the collision frequency and the viscosity of a hard sphere
gas. With these definitions, the linearized, non-dimensional BGK equation is
c? · ∇h = ε1 + c? · ε + (c?22 − 3)ε3 − h (4.25)2
94
with the moments related to the gas number density, velocity, and temperature by
∫
n
= 1 + ε = 1 + π3/21 h exp∫(−c
?2)dc?
n∞
U ? = U ? + 1∞ ε2 = U
? + π−3/2∞ ∫ hc exp(−c
?2)d2c? (4.26)
2
T
= 1 + ε = 1 + 2π−3/2 h(c?2 33 − ) exp(−c?2)d2c?
T 3 2∞
The integrals in the moment equations represent triple integrals over the entire
molecular velocity space. The boundary conditions for flow around a sphere are
diffuse reflection at the sphere surface and vanishing h far from the sphere.
Lea and Loyalka [75] solved the above problem numerically for the number
density and velocity perturbations around the sphere assuming isothermal conditions
(ε3 = 0). Their solution procedure involved solving for the perturbations using a
Gaussian quadrature out to a radius of a? + 10, or about 8.9 mean free paths from
the surface, then matching the numerical solution at a?+10 to a trial function based
on the continuum (Stokes flow) solution. They adjusted the numerical coefficients
of the trial function until the inner and outer solutions converged. Law and Loyalka
[76] applied this approach for non-isothermal conditions.
We follow the general approach of Loyalka and colleagues, but using the
asymptotic solution to the BGK equation for large r [78] as the trial function for
r? > a? + 10. Like in continuum flow, the velocity in the transition regime can be
written as separable radial and angular components,
√
ε ?2r = 2U∞q2(r) cos θ
95
√
ε2θ = − 2U?∞q3(r) sin θ (4.27)
where q2 and q3 are functions describing the radial dependence of the r- and θ-
components of the velocity perturbation obtained by solving the BGK equation.
These functions depend on the primary sphere radius, so that the solution procedure
applies to a specific Knudsen number. We can also obtain the friction coefficient from
the solution of the BGK equation. In general, our velocity results compare well with
the velocities reported by Takata et al. [78] based on their solution of the linearized
Boltzmann equation, and our drag results compare well with Millikan’s data [64].
Note, however, that our calculated friction coefficients for the near continuum regime
(Kn < 0.1) are less accurate, likely due to numerical error that is more prominent
for lower Knudsen numbers.4 We will discuss this point further in Section 4.3.2.
4.2.3 Application of BGK Results to Kirkwood-Riseman Theory
I now apply Kirkwood-Riseman theory to particles in the transition regime by
explicitly writing the friction coefficient and velocity tensor in Eq. (4.18) as functions
of the primary sphere Knudsen number,
∑N
Fi = ζt,0(Kn)Ui − Vij(Kn) ·Fj
i=6 j
4This is because the friction coefficient obtained from our solution of the BGK equation is
non-dimensionalized by the free molecule friction coefficient (Epstein’s equation). As a result, the
friction coefficient decays to zero for decreasing Knudsen number, meaning numerical errors are
more prominent for the near-continuum regime.
96
where the velocity perturbation tensor is
( )
−q2(r√ij,Kn) rijrij − q3(r√ij,Kn) rijrijVij(Kn) = I− (4.28)
2 r2 2ij 2 rij
For primary spheres separated by distances r?ij < a
?+10, q2 and q3 are tables of data;
for spheres separated by greater distances, q2 and q3 are the asymptotic solutions to
the BGK equation for large r with coefficients chosen to match the inner solution
for that Knudsen number.
Eq. (4.18) gives the force on each primary sphere for a given flow velocity.
(Dividing Eq. (4.18) by the friction coefficient ζt,0 gives the velocity at each primary
sphere.) The total force on the particle is the vector sum of the force on each primary
sphere. I obtain the friction tensor Ξt by solving Eq. (4.18) for the velocity in three
mutually orthogonal directions. The force on the particle for arbitrary fluid velocity
is then
Fd = Ξt ·U (4.29)
In the slow rotation limit, the scalar friction factor is the harmonic mean of the
three eigenvalues of the friction tensor [49]. In the fast rotation limit, the scalar
friction factor is the arithmetic mean of the eigenvalues [105].
4.3 Results and Discussion
I have calculated the scalar friction coefficient for a large range of primary
sphere sizes and number of primary spheres. All calculations involve particles with
97
df = 1.78 and k0 = 1.3, which are representative of aggregates formed by DLCA.
The particles have been generated with an algorithm that imitates cluster-cluster
aggregation. Due to limitations with our fractal generator, aggregate size is capped
at 2000 primary spheres. In this chapter, I am reporting the friction coefficient for
the slow rotation limit, meaning I am taking the harmonic average of the friction
tensor eigenvalues.
4.3.1 Comparison to Experimental Data and Power-Law Models
Figure 4.1 compares the results of my friction coefficient calculations for a pri-
mary sphere Knudsen number of 7 to tandem differential mobility analyzer (TDMA)
and combined DMA and aerosol particle mass analyzer (DMA-APM) results [43, 44].
The primary sphere size for the TDMA 80− 300 nm and DMA-APM curves was
experimentally-determined to be 19.5 nm with a standard deviation of 6.1 nm, while
the primary sphere size is assumed to be 19.5 nm for the TDMA 30− 100 nm curve
[44]. My self-consistent field results compare very well to the experimental data.
Furthermore, the self-consistent results support the observation of Shin et al. [43]
that deviations from a power-law relationship at high N may be due to hydrody-
namic interactions among the primary spheres in the aggregate.
Next, we compare our friction coefficient results to the results from three
models for DLCA particle drag in the transition regime.
The Lall and Friedlander [38] model,
c?Nµa
ζLF = (4.30)
Kn
98
Figure 4.1: Friction factor results for fractal aggregates with primary sphere diam-
eter 19.5 nm in ambient air (Kn = 7). TDMA and DMA-APM results [43, 44] are
shown for comparison.
is based on the calculations of Chan and Dahneke [34] for the drag on straight chain
aggregates in the free molecular regime. Here, c? = 9.17 is a dimensionless drag
force that assumes 93% diffuse reflection and 7% specular reflection. Chan and
Dahneke [34] argued that Eq. (4.30) should be valid for aggregates with N > 12
that have occasional kinks and branches. Implicit in Lall & Friedlander’s model
is that the aggregate behaves as if it is in the free molecule regime as long as the
primary spheres are in the free molecule regime. The model of Eggersdorfer et al.
[39] model relates the mobility radius – or the radius of a sphere that has the same
drag as the particle – to the number of particles through the relationship
( )1/2D
N α
rm,E = a (4.31)
kα
where kα = 1.1 and Dα = 1.08 are based on DLCA simulations. The friction
99
coefficient is obtained by substituting the mobility radius into Stokes’ law,
6πµrm,E
ζm,E = (4.32)
Cc(λ/rm,E)
Finally, Rogak et al. [37] noted that the mobility radius is approximately equal to
the orientation-averaged projected area radius for particles with mobility radii less
than 200 nm. Thus, I compare my results to the friction coefficient calculated using
the particle projected area:
√
6πµ√PA/πζm,R = (4.33)
Cc(λ/ PA/π)
I also compare my results to friction coefficients calculated using the adjusted sphere
method, Eq. (4.9). I computed the particle hydrodynamic radius using the Zeno code
[62], which uses the Hubbard-Douglas approximation, and I computed the projected
area using my own algorithm (Appendix C).
Figure 4.2 shows the comparison between my results and the aforementioned
models for primary sphere Knudsen numbers of 100, 10, 1, and 0.1, corresponding
to sphere radii of 6800 nm, 680 nm, 68 nm, and 6.8 nm, respectively. I also includes
free molecule results obtained with my own free molecule code and continuum results
obtained with Zeno on select figures. All of the models give results for Kn = 100 for
all N that are very similar to the free molecular limit, which is not surprising given
the very small primary sphere size. However, for large N all of the models – with the
exception of the Lall & Friedlander model – begin to diverge from the free molecular
100
limit for Kn = 10 primaries, suggesting that hydrodynamic interactions among the
primaries are important even at this primary Knudsen number. Interestingly, my
results and the ASM results approach the Zeno continuum results as N increases for
Kn = 1. Finally, my Kn = 0.1 results compare favorably to the continuum results
and to the ASM.
Figure 4.2: Comparison of self-consistent field results to other models for the scalar
friction factor for (a) Kn = 100, (b) Kn = 10, (c) Kn = 1, and (d) Kn = 0.1.
Results are for particles in ambient air. Where appropriate, free molecular results
from a ballistic algorithm and continuum results from the Zeno code are displayed
for reference.
Figure 4.3 shows the ratio between the predictions of the aforementioned mod-
els and my friction coefficient results for N = 2000. Values of unity represent perfect
agreement between my results and other models. Once again, there is very good
101
agreement with the adjusted sphere method across the entire Knudsen range. The
Eggersdorfer and Rogak friction coefficients are notably lower than our results at low
to moderate primary sphere Knudsen numbers, though it is important to reiterate
that this comparison is for large aggregates (N = 2000). The agreement between the
models is better for smaller aggregates at lower primary sphere Knudsen numbers,
as indicated in Figure 4.2.
Additionally, Figure 4.3 illustrates how the aggregate approaches the con-
tinuum limit for decreasing Knudsen number: the ratio of the continuum result
calculated using the Zeno code to my Kirkwood-Riseman results is near unity for a
primary sphere Knudsen number as high as 2. This figure explicitly shows the differ-
ence between using the monomer friction coefficient from the BGK model solution
and using the monomer friction coefficient from the Cunningham slip correction fac-
tor. Differences are largest for small Knudsen numbers, though results are in good
agreement with the continuum results at low Knudsen number whether one uses the
BGK friction coefficient or the Cunningham slip coefficient for ζt,0 in Eq. (4.18).
Figure 4.4 clearly illustrates how the friction coefficient diverges from the free
molecular limit and exhibits more continuum-like behavior as the particle size (both
in terms of N and a) increases. Here, calculated friction coefficients are normalized
to the monomer friction coefficient for several primary sphere Knudsen numbers in
the transition regime. The power law exponent [i.e. η from Eq. (4.8)] decreases
from a value of approximately 0.9 – corresponding to the free molecule regime – as
both the number of primary spheres and the primary sphere size increases, until it
reaches a limit of approximately 0.54 for the continuum regime. The free molecule
102
Figure 4.3: Ratio of friction coefficients from other models to my results for N =
2000. Free molecule and continuum results are calculated using my own Monte
Carlo algorithm and the Zeno algorithm, respectively. For the upper plot, myfriction
coefficient results are obtained using the calculated drag from the BGK model. For
the lower plot, my friction coefficient results use the Cunningham slip formula for the
monomer friction coefficient [ζt,0 in Eq. (4.18)]. Free molecule results for Kn < 15,
LF results for Kn < 35, and continuum results for Kn > 15 are more than twice our
self-consistent field results and thus do not appear in the plots above.
103
and continuum values are in agreement with previous observations [4]. The change
in the power law exponent reinforces the importance of accounting for hydrodynamic
interactions among primary spheres, even for fairly open aggregates with primary
spheres in the near-free molecular regime.
Figure 4.4: Normalized friction coefficient results for a range of aggregate sizes.
Previously researchers have looked at the evolution of the ratio between the
mobility radius and the radius of gyration as the number of primary spheres in-
creases. Figure 4.5 compares my self-consistent field results for this ratio (β =
Rm/Rg) to the same calculation in the continuum (where the mobility radius and
the hydrodynamic radius are equivalent) and free molecule regimes. Our results
agree with previous observations [54–56] that β approaches an asymptotic value in
the continuum regime. My results also agree qualitatively with the general obser-
vations of Sorensen [4], specifically Figure 2 of that work. However, my asymptotic
results for Kn = 0.01 and Kn = 0.1 are approximately 0.85, which is significantly
104
different (i.e. outside of numerical uncertainty) from the value of 0.75 recommended
by Sorensen for the continuum regime. (Note that the Zeno results for the hydro-
dynamic radius suggest an asymptotic value of β = 0.8 for large N .) Figure 4.5 also
notably shows that the Kn = 1, Kn = 3, and Kn = 10 curves also reach asymptotic
limits, again suggesting that aggregates approach the continuum regime behavior as
the number of primary spheres increases, even when the primary spheres are in the
near-free molecule or transition regime.
Figure 4.5: Relationship between the mobility radius and the radius of gyration for
several Knudsen numbers.
4.3.2 Uncertainty in the Calculated Friction Coefficients
I have demonstrated in this chapter and in the previous chapter that the
friction coefficient for DLCA aggregates computed using the extended Kirkwood-
Riseman method is in good agreement with experimental data, the continuum and
free molecule limits, the adjusted sphere method [41, 67], and Direct Simulation
105
Monte Carlo results [41]. But the question becomes, how accurate is the extended
Kirkwood-Riseman method? To answer this question, I provide a very rough esti-
mate of the error in my results.
There are two primary sources of error in my calculations: the BGK results
for the velocity around and drag on a sphere in the transition flow regime, and
the Kirkwood-Riseman method itself. There is ample discussion in the literature
about the accuracy of the Kirkwood-Riseman method for continuum flow; see Refs.
[4, 31, 106, 107] for a small sample. I refer the reader to the literature for a thorough
discussion. I simply note that in my experience, the Kirkwood-Riseman results
(using either the Stokes flow velocity perturbation or the Rotne-Prager tensor) is
within 3% of the Zeno results for DLCA aggregates with 102000 primary spheres.
For N = 10, the Kirkwood-Riseman method underpredicts the friction factor by less
than 3%. At N = 2000, the Kirkwood-Riseman result is approximately 2% greater
than the Zeno result. Thus, I estimate the error in our calculated transition regime
friction coefficients due to the Kirkwood-Riseman method itself is on the order of a
few percent.
The second source of error is related to the solution of the BGK equation. From
this solution, I obtain the velocity around a sphere and the ratio of the drag on the
sphere to the free molecule drag. One can easily estimate the error in the drag on
a sphere by comparing my results from the BGK equation to the drag from Stokes’
law with the Cunningham slip correction factor. Applying Davies’ coefficients in
the slip correction formula, the calculated error in the BGK drag results is less than
3% for Kn ¿ 0.2. (One obtains similar errors when using the coefficients of Allen
106
and Raabe [66] in the slip correction factor.) The error is greater at lower Knudsen
numbers, as noted in Section 4.2.2; for Kn = 0.01, the BGK drag is approximately
7% greater than the drag from Stokes’ law. For most of the Knudsen range, the
error in my calculated drag force is comparable to the error in Stokes’ law for the
slip regime (either due to the model parameters used in the slip correction factor
or to experimental uncertainties); the BGK results are only in significant error near
the continuum regime.
We can compare our velocity results to the linearized Boltzmann equation
results of Takata et al. [78]. The linearized Boltzmann model is more rigorous than
the BGK model, but its associated computational cost is much higher than that
required to solve the BGK model. Takata et al. present the velocities as a function
√
of the parameter k∞ = πKn/2. My velocity results are generally within 1-2%
of the linearized Boltzmann results for Kn = 0.11, 1.1, and 11 (k∞ = 0.1, 1, and
10). From these comparisons of the BGK velocity and drag results to the linearized
Boltzmann results and to Stokes’ flow in the continuum limit, I estimate that the
error in my aggregate friction coefficient results due to the use of the BGK model
is less than 5% for Kn > 0.2 and up to 10% for 0.01 < Kn < 0.2.
Combining the two sources of error, I would estimate the overall error in my
EKR results to be less than 10% for most of the Knudsen range. This estimate
is supported by comparing my friction coefficient results to the ASM results for
N = 2000 (Figure 4.3): the difference is less than 10% for 0.01 < Kn < 100.
Also, my calculated friction coefficient results for a primary sphere Knudsen number
greater than 5 are within 10% of the direct simulation Monte Carlo results of Zhang
107
et al. [41] for a 20-particle aggregate with a fractal dimension of 1.78 and a prefactor
of 1.3 (Chapter 3).
4.3.3 Analytical Expression for Friction Coefficients of Aggregates
While the Kirkwood-Riseman method is capable of providing the friction co-
efficient of an aggregate quickly – within seconds for N ∼ 100 and within minutes
for N ∼ 1000 – it is still not fast enough for use in an aerosol dynamics code. Thus,
it would be beneficial to use my friction coefficient results to develop a simple model
that provides the friction coefficient given only the number of primary spheres, the
primary sphere size, and the gas properties.
Sorensen and Wang [108] proposed computing the friction coefficient in the
transition regime as the harmonic sum of the continuum and free molecule expres-
sions,
ζ−1t = (ζ
c
t )
−1 + (ζFMt )
−1 (4.34)
For a sphere, the continuum and free molecule friction coefficients are given by
Stokes’ law [Eq. (4.5)] and Epstein’s equation [Eq. (4.6)], respectively. I adopt this
approach for my model of the friction coefficient of DLCA aggregates with fractal
dimension and prefactor of 1.78 and 1.3.
I start by writing the continuum and free molecule aggregate friction coeffi-
cients as power laws,
ζm m ηt,agg = ζ [AN + (1− A)] (4.35)
where ζm is the continuum (m = c) or free molecule (m = FM) monomer friction
108
coefficient from Eq. (4.5) or Eq. (4.6), and A and η are model parameters obtained
from fits to the continuum (Zeno) or free molecule (Monte Carlo) results for N = 1
to 2000. We include the 1 − A term in the power law fits to give the correct
friction coefficient for a monomer. The free molecule coefficients AFM = 0.843
and ηFM = 0.939 are in excellent agreement with Mackowski’s correlation for the
free molecule friction coefficient (AFM = 0.847 and ηFM = 0.94 for k0 = 1.3 and
df = 1.78) [36]. The continuum coefficients Ac = 0.852 and ηc = 0.535 from
my Zeno results are also in good agreement with previous studies, as reported in
Sorensen’s review article [4].
Taking the harmonic sum of the continuum and free molecule power law fits, I
obtain the following expression for the aggregate friction coefficient as a function of
the number of primary spheres, the primary sphere radius, and the gas properties:
ζt { }
= [A ηccN + (1− A )]−1 +BKn [A NηFMc FM + (1−
−1
AFM)]
−1 (4.36)
6πµa
Here, B = 1.612 for a hard-sphere gas with a momentum accommodation coefficient
of unity (i.e. pure diffuse reflection), consistent with my assumptions throughout
this chapter. For a monomer in the transition regime, the above relation reduces to
6πµa
ζt,0 = (4.37)
1 +BKn
Sorensen and Wang [108] point out that the monomer friction coefficient given by
the harmonic sum is up to 10% less than the friction coefficient given by Stokes’
109
law with the slip correction factor. Thus, I apply a correction factor to my model
to give the same monomer drag as Stokes’ law. The final result is Eq. (4.38), which
provides an easily deployed analytic result to compute the friction coefficient over a
wide range of aggregate and primary particle sizes.
ζt 1 + 1.612Kn
[( )
= 0.852N0.535
−1
+ 0.148
6πµa Cc(Kn) ( ) ]
0.939 −1
−1
+1.612Kn 0.843N + 0.157 (4.38)
Figure 4.6 plots the friction coefficient calculated from Eq. (4.38) as a function
of primary sphere Knudsen number and the number of primary spheres. Results are
normalized using Stokes’ law evaluated for a = λ/Kn. The figure shows a clear tran-
sition between continuum behavior, where the friction coefficient is proportional to
1/a for a given number of primary spheres, and free molecule behavior characterized
by a 1/a2 dependence. (The normalized coefficients have no dependence on a in the
continuum and a 1/a dependence in the free molecule regime.) This figure shows
that the transition from the continuum regime to the free molecule regime occurs at
larger Knudsen numbers as the number of primary spheres increases, demonstrating
once again that particles exhibit more continuum-like behavior as both the Knudsen
number and the number of primary spheres increase.
Figure 4.7 shows the error in my fit relative to my self-consistent field results,
ζt,fit − ζt,EKR
error = (4.39)
ζt,EKR
110
Figure 4.6: Normalized friction coefficient as a function of the primary sphere Knud-
sen number and the number of primary spheres, N , calculated using Eq. (4.38). Fric-
tion coefficients are normalized by Stokes’ law evaluated at the specified Knudsen
number.
with ζt,fit given by Eq. (4.38). The figure presents the error for a range of aggregate
sizes and primary sphere Knudsen numbers. Overall, Eq. (4.38) provides a good fit
to my self-consistent field results for all values of N and Kn that we have evaluated.
Note that I compare our fit to my EKR results using the semi-empirical slip cor-
rection for the monomer drag coefficient, instead of the drag coefficient we obtain
by solving the BGK model. As I have stated, this distinction is only significant for
monomers near the continuum limit.
4.4 Conclusions
I have presented my self-consistent field results for the translational scalar
friction coefficient of DLCA aggregates of 10 to 2000 primary spheres with primary
sphere Knudsen numbers between 0.01 and 100. My results compare well to the
111
Figure 4.7: Error of my harmonic sum model for the friction coefficient, Eq. (4.38),
relative to my extended Kirkwood-Riseman friction coefficient results for a range
of Knudsen numbers. Error is calculated with Eq. (4.39); the EKR results in this
equation use the monomer friction coefficient from Stokes’ law instead of the friction
coefficient computed from the BGK model.
112
experimental data of Shin et al. [43, 44] and to the friction coefficient from the
adjusted sphere method [41, 67]. I estimate that my results are within approximately
10% of the true friction coefficient for DLCA aggregates up to 2000 primary spheres
for 0.01 < Kn < 100, though I would need to compare my results to experimental
data over a wide range of primary sphere Knudsen numbers and aggregate sizes
to verify this estimate. These results have been obtained by taking the harmonic
mean of the eigenvalues of the translational friction tensor. The difference between
the harmonic and arithmetic averages of the eigenvalues is generally less than 1%,
which is consistent with previous calculations for low-aspect-ratio particles in the
free molecule regime [105]. This difference is minor compared to the estimated
uncertainty in my results.
One significant finding of this study is that aggregate drag becomes more
continuum-like as the number of primary spheres increases, even for primary sphere
Knudsen numbers near the free molecule regime. Thus, one should not use free
molecule techniques to compute the drag on an aggregate unless the aggregate size
is very small with respect to the gas mean free path. This finding supports the theory
behind the adjusted sphere method, that one can calculate the drag on an aggregate
using an aggregate Knudsen number instead of the primary sphere Knudsen number.
My method is fast, but not fast enough to implement in an aerosol dynamics
code. The same is true of the adjusted sphere method, unless one already knows the
hydrodynamic radius and projected area of an aggregate. For this reason, I have
compared my results to the harmonic sum of power laws for the friction coefficient in
the continuum and free molecule regimes. The result presented in Eq. (4.38) provides
113
an analytical expression for the drag over a range of aggregate and primary particle
size. The simple model is within 8% of my self-consistent field results for the entire
range of aggregate sizes and primary sphere Knudsen numbers that I have studied.
This analysis is for fractal clusters generated using a cluster-cluster aggregation
method for a fractal dimension of 1.78 and a prefactor of 1.3.
114
Chapter 5: Calculating the Rotational Friction Coefficient of Frac-
tal Aerosol Particles in the Transition Regime using Ex-
tended Kirkwood-Riseman Theory
5.1 Introduction
Nanoscale aerosol particles consisting of many spheres in point contact are
formed in many natural and synthetic processes. The size, shape, and orientation
of these particles greatly affect their transport properties [4, 5], optical properties
[90, 91, 109], degree of alignment in an external field [6, 90, 91, 109], filtration
efficiency [110], and their effects in biological systems, including lung deposition
[19, 20].
Much of the theoretical and experimental literature on the transport properties
of nano-scale aerosol particles focuses on the translational friction coefficient (or,
equivalently, the electrical mobility). There is comparatively little focus on the
rotational friction or diffusion coefficients, which affect particle alignment in an
external field and relaxation time from an aligned state to a fully random state
[6, 90, 91, 109]. Inclusion of rotational dynamics is also important when considering
particle coagulation rates in Brownian motion simulations [111].
115
There are analytical expressions available in the literature for the torque on
(or the rotational diffusion coefficient of) simple shapes – such as spheres, rods,
and ellipsoids – in both the continuum [29, 49, 112] and free molecule regimes
[3, 113, 114]. However, there are no such expressions for complicated shapes such
as fractal aggregates. Garcia de la Torre and colleagues have extensively studied
the rotational problem for rigid particles consisting of multiple spheres in point
contact in the continuum regime [51–53, 107, 115]. More will be said about their
work shortly. There are far fewer studies available for the free molecule regime. Li
et al. [6] approximate the torque on a fractal aggregate rotating in a quiescent fluid
by considering only the linear velocity of each sphere and neglecting the effects of
shielding by the other spheres in the cluster, thereby providing an upper bound for
the torque. I am unaware of any more detailed methods for calculating the torque
on a fractal aggregate in either the free molecule or the transition flow regime. This
is significant because in many aerosol applications the primary spheres are much
smaller than the mean free path of the gas.
In this chapter, I discuss the application of my extended Kirkwood-Riseman
(EKR) theory [92] to the translational and rotational motion of fractal aggregates
in the transition flow regime. In Section 5.2 I provide the equations for the drag
and torque on a rigid particle, as introduced by Brenner [29]; I describe how one
can apply Kirkwood-Riseman theory to the problem; and I employ Monte Carlo to
compute the drag and torque on a translating or rotating particle, which I use to
validate the EKR method. I present my results for the rotational friction coefficient
as a function of Knudsen number and compare my results for Kn 1 and Kn 1
116
to the continuum and free molecule limits in Section 5.3.
5.2 Drag and torque on a rigid particle
Consider a rigid particle with center of mass moving at velocity UO and rotat-
ing with angular velocity ω, where point O is the origin of the system. For particles
in Stokes (i.e. low Reynolds number) flow, the force F and torque TO on the particle
are given by
F = −Ξt ·UO −Ξ†O,c ·ω (5.1)
TO = −ΞO,c ·UO −ΞO,r ·ω (5.2)
where Ξt, ΞO,r, and ΞO,c are the friction tensors for translation, rotation, and
translation-rotation coupling, respectively, and Ξ†O,c is the transpose of the coupling
tensor. The coupling and rotation tensors are defined with respect to the origin, O,
while the translation tensor is independent of the origin.
Brenner [29] proved that these friction tensors are related to the translation,
rotation, and coupling diffusion tensors by the generalized Stokes-Einstein relation
DO = kTM−1O , (5.3)
where DO and MO are the 6× 6 grand diffusion and friction matrices given by
 D †O,t DO,cDO =  (5.4)
DO,c Dr
117

 
 Ξ Ξ†t O,cM = O  (5.5)
ΞO,c ΞO,r
Rewriting Eq. (5.3) as MO · DO = kT I, where I is the identity tensor, one can
show that the translation, rotation, and coupling diffusion tensors are related to the
friction tensors by [29, 52]
DO,t = kT (Ξ −Ξ† ·Ξ−1t O,c O,r ·Ξ
−1
O,c) (5.6)
Dr = kT (Ξ
−1
O,r −ΞO,c ·Ξt ·Ξ
†
O,c)
−1 (5.7)
D −1 † −1 −1O,c = −kT ΞO,r ·ΞO,c · (Ξt −ΞO,c ·ΞO,r ·ΞO,c) (5.8)
According to Brenner [29], the translation and coupling tensors are most mean-
ingful when computed at the center of diffusion. At this point D, the coupling tensor
ΞD,c is symmetrical. The vector from the origin to the center of diffusion rOD can
be expressed as [29, 52]
 −1  D22 +D33 −D12 −D13 r r r r  
D23 32O,c −DO,c
rOD =  −D12 D11 +D33 −D23  · D31 −D13  (5.9)r r r r O,c O,c
−D13 23 11 22 12 21r −Dr Dr +Dr DO,c −DO,c
The translation and coupling tensors at the center of diffusion are given by
Dt = DO,t − rOD ×Dr × rOD + D†O,c × rOD − rOD ×DO,c (5.10)
118
Dc = DO,c + Dr × rOD (5.11)
Finally, I can write the scalar translational diffusion coefficient as [52]
Dt = kT/ζ
1
t = Tr (Dt) (5.12)3
where ζt is the translational friction coefficient and Tr (Dt) is the trace of the trans-
lation diffusion tensor. Similarly, I can define scalar rotational diffusion and friction
coefficients as
Dr = kt/ζr =
1 Tr (D ) (5.13)
3 r
5.2.1 Kirkwood-Riseman Theory
Based on the preceding discussion, one can fully describe the translational and
rotational behavior of a rigid particle, provided one can obtain the translation, ro-
tation, and coupling friction tensors. I will now describe one approach for obtaining
those tensors for rigid particles consisting of N spherical elements in the continuum
regime. For this discussion, I will consider the case where all N elements have the
same radii ai = a, though this need not be the case when applying the general
framework described here. I will later discuss how to extend this approach to the
transition flow regime.
Kirkwood and Riseman [28] demonstrated that the drag on a particle in contin-
uum flow can be calculated by considering the hydrodynamic interactions between
each pair of spheres in the aggregate. Initially, hydrodynamic interactions between
spheres were calculated using the Oseen tensor. Later authors introduced more so-
119
phisticated hydrodynamic interaction tensors to account for the finite size of the
spherical elements [57, 58] and for rotational and translation-rotation coupling ef-
fects [59–61]. In all of these cases, the relationship between the linear velocity ui
and angular velocity ωi of the ith spherical element and the force Fj and torque Tj
at the center of each of the N elements is [53]
∑N ∑N
− u = Qt c †i ij ·Fj + (Qij) ·Tj (5.14)
j=1 j=1
∑N ∑N
− ω = Qc ·F ri ij j + Qij ·Tj (5.15)
j=1 j=1
where Qtij, Q
r
ij, and Q
c
ij are the translation, rotation, and coupling hydrodynamic
tensors between the ith and jth spherical elements. These tensors will be defined
shortly.
This linear system of equations can be written in matrix form as

UP

 

Qt (Qc)†
 
− = F  (5.16)
W Qc Qr TP
where UP ,W , F , and TP are the 3N -element vector containing the linear velocities,
angular velocities, forces, and torques on the N spherical elements; and Qt, Qr,
and Qc are the 3N × 3N matrices of the translation, rotation, and coupling tensors
for all ij-pairs. Note that subscript P indicates that the property is evaluated at
the center of each element. For example, the linear velocity of the ith sphere that
appears in UP is ui = uO + ω × ri, where ri = (xi, yi, zi) is the vector from the
120
origin to the center of the ith element. Inverting Eq. (5.16), I get
F 
   
 = −St (Sc)†UP (5.17)
T cP S Sr W
where
   −1St (Sc)† Qt (Qc)†=  (5.18)
Sc Sr Qc Qr
Carrasco and Garcıa de la Torre [53] show that the 3 × 3 submatrices of the
3N × 3N S matrices are related to the translation, rotation, and coupling friction
tensors in Eqs. (5.1) and (5.2) by
∑N ∑N
Ξt = S
t
ij (5.19)
i=1 j=1
∑N ∑N [ ]
Ξ = Sr − ScO,r ij ij ·Aj + A · (Sc † ti ij) −Ai ·Sij ·Aj (5.20)
i=1 j=1
∑N ∑N [ ]
ΞO,c = S
c
ij + Ai ·Stij (5.21)
i=1 j=1
where  
 0 −zi yi 
Ai = 

z 0 −x  (5.22)i i
−yi xi 0
Carrasco and Garcıa de la Torre [53] summarize the hydrodynamic theories
of Reuland et al. [59], Mazur and Van Saarloos [60], and Goldstein [61] and show
121
that the hydrodynamic interaction tensors Qtij, Q
r c −3
ij, and Qij all agree to order rij .
These tensors are given by
[ ( ) ( )]
t δij 3(1− δij) a rijr 3ij 2a 3rijrijQij = I + I + + I− (5.23)ζ 2 3t,0 4ζt,0 rij rij 3rij r2ij
[ ]
δ (1− δ ) a3r ij ij 3rijrijQij = I + − I (5.24)ζ 3r,0 2ζr,0 rij r2ij
3
Qc −(1− δij) aij =  · rij (5.25)ζ 3r,0 rij
where  
 0 z ij
−yij
 · rij = −zij 0 xij  , (5.26)
yij −xij 0
δij is the Kronecker delta, and ζt,0 = 6πµa and ζr,0 = 8πµa
3 are the continuum
friction and torque coefficients. Note that the second term in Eq. (5.23) is the
Rotne-Prager-Yamakawa tensor [57, 58] [Eq. (3.11)].
Carrasco and Garcıa de la Torre [53] determined that including terms of order
lower than r−3ij in the interaction tensors did not significantly improve results for
the simple shapes that they analyzed. Since the effect of these lower order terms
drop off rapidly for larger particles, one can safely ignore these terms for the larger
particles I will consider in this paper.
122
5.2.2 Extension to the Transition Regime
I now wish to extend this Kirkwood-Riseman framework to the transition flow
regime. I start by multiplying Eqs. (5.14) and (5.15) by the monomer friction coeffi-
cient ζt,0 and the monomer torque coefficient ζr,0, respectively. As Rotne and Prager
[57] and Yamakawa [58] have noted, the product of the Rotne-Prager-Yamakawa ten-
sor and the Stokes’ law friction coefficient is similar to the flow field Vij around a
translating sphere in Stokes flow,
v(rij) = Vij ·U0 (5.27)
where Vij is given by Eq. (3.13). The difference between Vij and ζt,0Tij is a factor
of 2 in the r−3ij term in ζt,0Tij.
Recently, Corson et al. [92] exploited the similarity between Tij and the flow
around a sphere to extend Kirkwood-Riseman theory to the transition flow regime by
solving for the velocity around a sphere as a function of Knudsen number (Kn = λ/a,
where λ is the mean free path of molecules in the gas) and substituting the resulting
Vij(Kn)/ζt,0(Kn) for the second term in Eq. (5.23). This gives the drag on the ith
element of a purely-translating N -element particle as
∑N
Fi = −ζt,0(Kn)UO − Vij(Kn) ·Fj (5.28)
i=6 j
123
In this case, the translation hydrodynamic interaction tensor is given by
1
Qtij(Kn) = [δijI + (1− δij)Vij(Kn)] (5.29)ζt,0(Kn)
Similarly, I show in Appendix D that the (1−δij) terms in the rotation and cou-
pling hydrodynamic interaction tensors are directly related to the flow field around
a rotating sphere. Thus, solving for the velocity around and torque on a rotat-
ing sphere in the transition flow regime would provide expressions for Qrij(Kn) and
Qcij(Kn). This approach should be accurate to order r
−3
ij , subject to the accuracy
of the numerical solution to the kinetic equation in the transition regime and the
small error introduced by omitting a factor of 2 in the r−3ij term in the translation
hydrodynamic interaction tensor. (These errors are discussed in Chapters 3 and 4).
To get the friction tensors for a given particle, one would populate and invert the
6N × 6N Q matrix and apply Eqs. (5.19)–(5.21).
Alternatively, one can apply a simplified approach to determine the friction
and diffusion tensors for a particle in the transition regime. Ignoring rotation and
coupling hydrodynamic interactions, the friction tensors are given by [51, 52]
∑N ∑N
Ξt = S
t
ij(Kn) (5.30)
i=1 j=1
∑N ∑N
Ξ = r × StO,c i ij(Kn) (5.31)
i=1 j=1
∑N ∑N
ΞO,r = − r × Sti ij(Kn)× rj (5.32)
i=1 j=1
124
Here, St is the inverse of the 3N×3N translation matrix Qt, rather than a 3N×3N
block of the 6N × 6N Q matrix in Eq. (5.16). This approach is equivalent to
considering only the linear velocity ω × ri of each spherical element and ignoring
their angular velocities. One obvious flaw of this method is that it predicts zero
torque on a rotating sphere and on a chain of spheres rotating around its long axis.
Garćıa de la Torre and Rodes [107] suggest adding Nζr,0 to the diagonal elements
of ΞO,r to partially compensate for this error.
For my transition flow regime calculations, I will apply the simplified approach
given by Eqs. (5.30)–(5.32). This avoids the need to solve the kinetic equation for
a rotating sphere in the transition flow regime and requires inverting a 3N × 3N
matrix instead of a 6N × 6N matrix. However, I will apply the volume correction
of Garćıa de la Torre and Rodes [107] to the rotational friction tensor, using the
approximate expression for the ratio of the torque to the free molecule torque given
by Loyalka [77] [Eq. (44) in that work]. As I will demonstrate, the simplified
approach is sufficiently accurate for larger particles, for which the O(r−3ij ) terms in
the interaction tensors become less important.
5.2.3 Monte Carlo Calculations for Free Molecule Drag and Torque
In Chapters 3 and 4, I compared my results for the translational friction coef-
ficient to published experimental data and analytical results for the transition flow
regime. Unfortunately there is very little information on the rotational diffusion ten-
sor in the transition flow regime. In order to test the extended Kirkwood-Riseman
method I must compare to results in the continuum and free molecule limits. Con-
125
tinuum results will be taken from published results in the literature (where available)
or obtained using the hydrodynamic interaction tensors given by Eqs. (5.23)-(5.25).
I now describe my approach for calculating the friction tensors in the free molecule
limit.
Previous authors [34–36] have used a ballistic approach to calculate the drag
on a translating particle in free molecule flow. I use the same approach, but now
I consider both translational and rotational motion, and I calculate both the drag
and the torque on the particle.
The procedure is as follows. Consider the general case in which the bulk gas
velocity is a combination of translational velocity UO in the positive x-direction and
angular velocity ω about the x-axis. (For small translational and angular velocities,
this is practically equivalent to a particle moving with translational velocity UO in
the negative x-direction and rotating with angular velocity −ω, but it is easier to
consider the case in which the particle is stationary [77].) Surround the particle by
a launch sphere with radius R, randomly select starting locations on the surface of
the launch sphere, and define local coordinates (x?, y?, z?), where x? is the inward
normal for the position on the launch sphere. To determine the momentum of gas
molecules leaving the launch sphere, sample from the distribution of velocities of
gas molecules entering the launch sphere,
? ? 2
f(c , c , c ) = Kc e−[c −(UO+ω×R) ] /2RTx? y? z? x? (5.33)
where c? = (cx? , cy? , cz?) is molecular velocity in the local coordinate system, R and
126
T are the gas constant and the gas temperature, the bulk gas velocity UO +ω ×R
is written in terms of local coordinates, and K is a normalization constant defined
such that ∫ ∞ ∫ ∞ ∫ ∞
dcx? d2cy? d2cz? f = 1
0 −∞ −∞
If the molecule trajectory intersects the particle, calculate the momentum transfer
for diffuse reflection from the surface. For diffuse reflection, the molecule direction
is sampled from a cosine-squared distribution for the polar angle and an isotropic
distribution for the azimuthal angle, while the molecule speed is sampled from the
Maxwell-Boltzmann distribution
√( )3
1 2 −c2f(c) = 4π c e /2RT (5.34)
2πRT
Continue to follow the molecule trajectory until it exits the launch and account for
multiple collisions between the gas molecule and the particle. After launching M
molecules, calculate the total drag and torque on the particle:
∑MA
F = φipi (5.35)
M
i=1
M
A ∑
T = φiri × pi (5.36)
M
i=1
Here, A is the launch sphere surface area, pi is the momentum transferred to the
particle by the ith molecule, and ri is the point at which the molecule collides with
the particle. The quantity φi is the flux of gas molecules entering the launch sphere
127
at Ri [40],
√ { }
RT −s2 √φ = n e cos2 θii + πs cos θi[1 + erf(s cos θi)] (5.37)
2π
where n is the gas number density, s = U/RT is the ratio of the bulk velocity to
the molecular velocity in the gas, and θi is the angle between the bulk velocity and
the inward normal to the launch sphere at Ri.
Using the above procedure, I determine the translation friction tensor by set-
ting the angular velocity of the flow field equal to zero and calculating the drag F x,
F y, and F z for flow in the x-, y-, and z-directions. For a translation velocity much
√
less than the thermal speed 2RT , the friction tensor is
 F x F y F z x x x1Ξt = x y z
UO Fy Fy Fy


F x F yz z F
z
z
where F yx signifies the x-component of the force on the particle for flow in the y-
direction. The pure translation calculation (i.e. ω = 0) also gives the coupling
tensor from the torque on the particle per Eq. (5.2). Finally, I calculate the rotation
friction tensor by setting the translation velocity to zero and calculating the torque
for rotation about the x-, y-, and z-axes.
I have tested my Monte Carlo drag and torque code by comparing my results
to published calculation results of Mackowski [36] for the translational friction coef-
ficient (taken as the harmonic average of the eigenvalues of the friction tensor) and
128
by comparing to simple test cases (e.g. a sphere rotating about its center, a sphere
rotating about an axis at a fixed distance from its center) for the torque problem.
All of my Monte Carlo results are in excellent agreement with results from alternate
calculation methodologies. (See Appendix C.) Furthermore, my calculated transla-
tion and rotation friction tensors are symmetrical to within the error in the Monte
Carlo calculations. Thus, I can use my Monte Carlo code to evaluate the results of
my EKR results in the free molecule limit.
5.3 Results
To verify that the extended Kirkwood-Riseman method produces reasonable
results across a wide range of Knudsen numbers, I will compare my calculated
rotational friction coefficient ζr to its values in the continuum and free molecule
limits. I first discuss the continuum and free molecule results.
5.3.1 Continuum regime
Before presenting my results for the rotational friction coefficient in the transi-
tion flow regime, it is appropriate to consider the effect of neglecting the rotational
and coupling hydrodynamic interaction tensors on ζr in the continuum. This is-
sue is discussed in depth in the works of Garcia de la Torre and colleagues (e.g.
Refs. [53, 115]). I will be using the EKR method to calculate the translation and
rotation friction coefficients of a dimer, a linear hexamer, and an octrahedral hex-
amer, so I will briefly discuss the results of Carrasco and Garcıa de la Torre [53]
129
Figure 5.1: Representations of the fractal aggregates used in this study.
for these aggregates. I will also provide continuum results for four different fractal
aggregates: N = 20, df = 1.78; N = 20, df = 2.5; N = 100, df = 1.78; and
N = 100, df = 2.5. These aggregates are shown in Figure 5.1. Note that these
are the same aggregates that I used in Chapter 3. Also note that particles formed
by diffusion-limited cluster aggregation processes – such as soot – have a fractal
dimension of approximately 1.78.
Carrasco and Garcıa de la Torre [53] provide results of various hydrodynamic
interaction models for a dimer, linear hexamer, and octrahedral hexamer. The EKR
method in the continuum limit is nearly the same as the KRMV method described
in that paper, with the only difference being the factor of 2 in the O(r−3ij ) term in
Qtij. Presumably, the most accurate computational results are obtained using the
130
shell method, where each spherical element in the aggregate is replaced by a large
number of frictional units and hydrodynamic interactions are described using the
Oseen tensor. I will compare my extended Kirkwood-Riseman results in the con-
tinuum limit to the shell method results for the linear and octrahedral hexamers.
Exact results are available for a dimer in continuum flow [49], so I will compare
my extended Kirkwood-Riseman results to the exact values. Based on Table II of
Carrasco and Garcıa de la Torre [53], I would expect the EKR results to underpre-
dict the translational friction coefficient (or overpredict the translational diffusion
coefficient) and overpredict the rotational friction coefficient. I would expect better
agreement for less compact aggregates, like linear chains or fractals with df = 1.78,
and better agreement for the translational friction coefficient than for the rotational
friction coefficient.
Table 5.1 shows translational and rotational friction coefficients for the four
fractal aggregates mentioned previously. The Table includes ζt and ζr computed us-
ing terms up to order O(r−3ij ) in the interaction tensors (the 3RD method described
by Carrasco and Garcıa de la Torre [53]) and using the extended Kirkwood-Riseman
method (EKR, where I use the Stokes flow solution around a sphere for the trans-
lation interaction tensor and set the coupling and rotation hydrodynamic interac-
tion tensors to zero). Translational friction results are normalized by Stokes’ law,
ζt,0; rotational results are normalized to the monomer rotational friction coefficient,
ζr,0 = 8πµa
3.
The difference between ζc,3RD and ζc,EKRt t is small (< 2%) for the cases shown
here. The difference in the rotational friction coefficient is much larger, with the
131
Table 5.1: Continuum friction coefficient for fractal aggregates, normalized by the
monomer friction results. Friction coefficients are calculated using terms up to
order r−3ij in the hydrodynamic interaction tensors (3RD) or my extended Kirkwood-
Riseman theory (EKR) in the continuum limit.
Case ζc,3RD ζc,EKR ζc,3RD ζc,EKRt t r r
N = 20, df = 1.78 4.35 4.31 94.5 110.8
N = 100, df = 1.78 10.3 10.2 1292.0 1376.7
N = 20, df = 2.5 3.41 3.37 42.5 57.5
N = 100, df = 2.5 6.71 6.64 313.6 398.2
greatest difference (35%) occurring for N = 20, df = 2.5. The difference decreases
as the average distance between spheres increases due to the reduced importance
of the O(r−2ij ) and O(r−3ij ) terms in the coupling and rotation interaction tensors,
respectively. These trends are consistent with the results of Carrasco and Garcıa
de la Torre [53] and Garćıa de la Torre et al. [115].
It is important to note that the 3RD method tends to underpredict the ro-
tational friction coefficient (or overpredict the rotational diffusion coefficient) com-
pared to more computationally-intensive methods like the shell model [53, 115]. On
the other hand, the EKR method appears to overpredict the friction coefficient. In
other words, the difference between my EKR results in the continuum and the true
value of the rotational friction coefficient may be less than that suggested by the
results in Table 5.1. I shall return to this subject in Section 5.4.
5.3.2 Free Molecule Regime
I have computed the free molecule translational and rotational friction coef-
ficients for the seven aggregates described in the previous section. The results are
shown in Table 5.2; the values in the table are normalized to the free molecule
132
Table 5.2: Free molecule results for fractal aggregates, normalized by the monomer
friction results
Case ζFMt ζ
FM
r
N = 2 1.832 3.829
N = 6, df = 1 5.056 16.46
N = 6, octahedron 4.157 21.57
N = 20, df = 1.78 14.48 443.9
N = 100, df = 1.78 64.37 12320
N = 20, df = 2.5 11.64 196.2
N = 100, df = 2.5 43.38 2713
monomer translational and rotational friction coefficients,
π(8 + π) µ
ζFMt,0 = a
2 (5.38)
2.994 λ
FM 2π µζ 4r,0 = a (5.39)1.497 λ
where I have substituted the viscosity for a hard-sphere gas, µ = 0.499ρc̄λ, into the
expressions for ζFM FMt,0 and ζr,0 [3, 32] and assumed diffuse reflection at the surface of
the sphere.
My translation friction coefficient results are in excellent agreement with pub-
lished computational results for linear chains [34] and for fractals with df = 1.78 [36].
I are unaware of any published results for the denser particles or for the rotational
friction coefficients of any of the particles in Table 5.2.
For my free molecule calculations, I sample 109 molecular trajectories to ensure
good statistical results. Each calculation takes less than three hours on a single
processor, and the CPU time increases linearly with the number of trajectories. In
general, my results are accurate to three or four significant figures, based on multiple
133
calculations performed for each case. This level of accuracy is more than sufficient
for most practical applications.
5.3.3 Transition Regime
I have performed my extended Kirkwood-Riseman calculations for the seven
particles discussed above for Knudsen numbers ranging from 0.01 to 100. In Chap-
ter 3, I reported the translational friction coefficient as a function of Knudsen number
for the fractal particles, calculated using the harmonic mean of the eigenvalues of
the translational friction tensor Ξt. The difference between ζt computed using this
approach and ζt computed using Eq. (5.12) is less than 1%.
Figure 5.2 presents my results for the scalar rotational friction coefficient ζr
[defined in Eq. (5.13)] of a dimer, linear hexamer, and octahedral hexamer. Results
are presented as a slip correction factor,
ζc
C (Kn) ≡ rr (5.40)
ζEKRr (Kn)
where the continuum rotational friction coefficient ζcr is calculated using the best
available method. (Cr is analogous to the Cunningham slip correction factor, which
represents the ratio between Stokes’ law and the friction coefficient for a sphere in
the transition regime. It is also analogous to the parameter Θ−1, defined by Zhang
et al. [41] as the ratio of the continuum friction coefficient to the transition regime
friction coefficient for an aggregate.) For the dimer, ζcr is given by the exact solution
to the Stokes equation [49]; for the hexamers, ζcr is taken as the shell method solution
134
from Carrasco and Garcıa de la Torre [53]. The free molecule limit for each particle
is shown as a dashed line. Note that the curves representing the dimer and the
linear hexamer nearly coincide due to the chosen normalization used in the plot.
At small Knudsen numbers, the rotational friction coefficient approaches a
constant value that differs from the continuum value for the aggregate ζcr because
my calculations neglect rotational and coupling hydrodynamic interactions, as dis-
cussed in Section 5.3.1 and illustrated by the results in Table 5.1. At large Knudsen
numbers, ζEKRr is in excellent agreement with my Monte Carlo calculations for the
free molecule rotational friction coefficient (within 5% at Kn=100). The slight dis-
crepancy between the solid and dashed lines in the free molecule limit are due to
numerical uncertainty in the Monte Carlo calculations and interpolation error in ap-
plying my results for the velocity around a sphere to Vij(Kn) in Eq. (5.28). (Note
that I use a Gaussian quadrature to solve for the velocity within approximately 10
mean free paths of the sphere surface. Thus, interpolation errors are most significant
near the free molecule regime, where the sphere radius is comparable to the node
spacing.) These results suggest that rotational and coupling interactions between
primary spheres are negligible at large Knudsen numbers, as one would expect due
to the nature of free molecule flow.
Figure 5.3 presents my results for ζEKRr for the fractal particles. Again, the
results are plotted as a slip correction factor, but in this case the continuum rota-
tional friction coefficient is calculated using the 3RD method. Note that the two
N = 20 curves appear to lie on top of each other, as do the two N = 100 curves;
again, this is due to the chosen normalization.
135
Figure 5.2: Calculated rotational slip correction factor [defined by Eq. (5.40)] for
a dimer, linear hexamer, and octahedral hexamer. For the dimer, the continuum
value in the slip correction is the exact solution from Happel and Brenner [49]; for
the hexamers, the continuum values are the shell method values (SHM) from Table
II in Carrasco and Garcıa de la Torre [53]. The free molecule limit for each case
(dashed lines) is calculated using my Monte Carlo algorithm.
As with the dimer and hexamers, my results for the fractals are in excellent
agreement in the free molecule limit (dashed line), while the errors in the contin-
uum regime are up to 40% because my method neglects rotational and coupling
interactions between monomers. This error decreases significantly for larger, less
dense particles: for example, the difference between the EKR results and the 3RD
results for 100-sphere soot-like fractal is less than 10%. The decrease can be at-
tributed to the reduced importance of the O(r−2ij ) and O(r−3ij ) terms in the coupling
and rotational interaction tensors, respectively, relative to the O(r−1ij ) term in the
translational interaction tensor. For larger, less dense particles, the monomers are
on average spaced further apart than the monomers in a smaller, denser aggregate,
such that the translational hydrodynamic interactions dominate.
136
Figure 5.3: Calculated rotational slip correction factor [defined by Eq. (5.40)] for
four fractal aggregates. The continuum rotational friction coefficient that appears in
the slip correction is calculated using the 3RD method, and the Knudsen-number-
dependent friction coefficient is calculated using my EKR method. The free molec-
ular limit for each aggregate (dashed lines) is calculated using my Monte Carlo
algorithm.
Dahneke [67] and Zhang et al. [41] posited there exists for the translational
friction coefficient a universal relationship between the friction coefficient in the
transition flow regime and an aggregate Knudsen number,
ζct = Cc(Knagg) (5.41)
ζt(Knagg)
where Cc is the Cunningham slip correction factor. Zhang et al. [41] showed using
dimensional analysis that the appropriate aggregate Knudsen number for transla-
tional friction is
πλRH
Knagg = (5.42)
PA
where RH and PA are the hydrodynamic radius and projected area of the aggregate,
137
which characterize particle size in the continuum and free molecule regimes, respec-
tively. In other words, the aggregate Knudsen number is proportional to the ratio
between the continuum and free molecule friction coefficients for the aggregate.
This Adjusted Sphere Method implies that plots of the aggregate translational
slip correction factor [Eq. (5.41)] versus the aggregate Knudsen number [Eq. (5.42)]
fall on the same universal curve, regardless of particle shape. Experimental and com-
putational studies [41, 68, 69, 92, 93] suggest that this is indeed the case. Based on
this evidence, I propose that the rotational slip correction factor [Eq. (5.40)] should
exhibit similar behavior when plotted against an appropriate aggregate Knudsen
number. Since the translational aggregate Knudsen number is proportional to the
ratio of continuum to free molecule friction coefficients, I posit that the rotational
aggregate Knudsen number is
ζcr 23.952 ζ
c?
Kn rr,agg = = Kn (5.43)
ζFMr 8 + π ζ
FM?
r
where Kn is the primary sphere Knudsen number and ζc? cr ≡ ζr/ζc and ζFM?r,0 r ≡
ζFM/ζFMr r,0 are the dimensionless continuum and free molecule rotational friction co-
efficients for the aggregate.
Figure 5.4 shows my rotational friction coefficient results plotted as the rota-
tional slip correction factor versus the aggregate Knudsen number [Eq. (5.43)]. The
dimensionless continuum friction coefficients are calculated with the same reference
method used in Figs. 5.2 and 5.3 (i.e. the exact solution for the dimer [49], the shell
method for the hexamers [53], and the 3RD method for the 20- and 100-particle ag-
138
Figure 5.4: Rotational slip correction factor plotted versus an aggregate Knudsen
number. The aggregate Knudsen number is the ratio of the friction coefficient cal-
culated as if the aggregate is in continuum flow to the friction coefficient calculated
as if the aggregate is in free molecule flow. Continuum friction coefficients are calcu-
lated using the same reference method used to calculate the slip correction factor Cr,
while the free molecule coefficients are calculated using my Monte Carlo algorithm
gregates). The dimensionless free molecule friction coefficients are calculated using
my Monte Carlo algorithm. Roughly speaking, all of the aggregates exhibit the same
behavior when plotted in this manner. The differences among the curves near the
continuum regime are likely due to neglecting rotational and coupling hydrodynamic
interactions when calculating the rotational friction coefficient, as discussed previ-
ously. Errors in the calculated continuum friction coefficient may also contribute to
the spread among the curves. My results suggest that the aggregate rotational fric-
tion coefficient follows some universal function of the rotational aggregate Knudsen
number.
139
5.4 Discussion
I have applied my extended Kirkwood-Riseman theory to calculate the rota-
tional friction coefficient for aerosol particles in the transition flow regime. This
approach ignores rotational and translation-rotation coupling interactions between
spheres. These effects become less important as the number of primary spheres
increase and as the primary sphere size decreases. The former effect is due to the
dominance of the O(r−1ij ) term in the translational interaction tensor over the lower
order terms in the rotational and coupling interaction tensors. The latter effect
occurs because smaller particles perturb the flow field less than large particles.
Consistent with this discussion, my EKR results are in excellent agreement
with my Monte Carlo results for large Knudsen numbers (i.e. within 5% for Kn =
100). The agreement is not as good in the continuum regime: I have observed errors
as high as 40% for dense aggregates relative to the rotational friction coefficient
computed considering terms up to order O(r−3ij ) in the interaction tensors. The EKR
results are in better agreement with the 3RD results for less dense fractal aggregates.
It is also worth mentioning that the EKR and 3RD methods respectively under-
and over-predict the rotational friction coefficient compared to the computationally-
intensive shell method, so the rotational friction coefficient computed using the EKR
method is mostly likely in better agreement with the true friction coefficient than
my results in Table 5.1 and Figure 5.3 suggest.
My results also suggest that there is a universal relationship between the ro-
tational friction coefficient and an aggregate Knudsen number. This is analogous to
140
relationship between the translational friction coefficient and the aggregate Knudsen
number introduced by Dahneke [67] and Zhang et al. [41], which is supported by
experimental data and computational results [41, 68, 69, 92, 93]. For the rotational
friction coefficient, the appropriate aggregate Knudsen number is the ratio of the
aggregate continuum and free molecule friction coefficients.
I could improve the accuracy of my method – particularly near the contin-
uum regime – if I considered pairwise rotational and coupling interactions between
primary spheres in the aggregate. As I have demonstrated (Appendix D), the rota-
tional and coupling interaction tensors are related to the flow field around a rotating
sphere; one could solve the kinetic equation for flow around a rotating sphere as a
function of Knudsen number to obtain the appropriate interaction tensors in the
transition flow regime. With that said, my simplified method is sufficiently accu-
rate for most practical purposes – particularly for larger aggregates with a fractal
dimension of 1.78.
Finally, I will note that while I have focused exclusively on the scalar friction
coefficient, my method also provides the translation, rotation, and coupling friction
tensors. Thus, the extended Kirkwood-Riseman method can be used when consid-
ering alignment of aerosol particles in an external field [6, 90, 91, 109, 114] or when
simulating Brownian diffusion of small particles.
141
Chapter 6: Analytical Expression for the Rotational Friction Co-
efficient of DLCA Aggregates over the Entire Knudsen
Regime
6.1 Introduction
The rotational behavior of an aerosol particle can be characterized by the
rotational friction and diffusion coefficients ζr and Dr, which are related by the
Stokes-Einstein relationship ζr = kBT/Dr. The rotational behavior can be impor-
tant to evaluating the average drift velocity [105, 116] and particle alignment in
external electric field and subsequent relaxation [5, 6, 90, 91, 109].
For fractal aggregates that consist of many nano-sized spheres in contact,
determining the rotational friction/diffusion coefficient is a difficult problem. This
is due to two principal factors: the complicated, fractal-like shape of the aggregates,
and the fact that the particle size is often comparable to the mean free path of the
molecules in the gas. The latter complication means that the particles are in the
transition flow regime, where one must use kinetic theory to solve for the forces
and torques exerted by the gas on the aggregates. For these fractal-like particles,
the relationship between the number of spheres in the aggregate and its radius of
142
gyration is ( )d
R fg
N = k0 (6.1)
a
where df and k0 are the fractal dimension and prefactor. For particles formed
by diffusion-limited cluster aggregation (DLCA), which is the focus of this work,
k0 ≈ 1.3 and df ≈ 1.78.
In this chapter, I build on my the work on the rotational friction coefficient of
aggregates described in the previous chapter. Here, I apply my extended Kirkwood-
Riseman (EKR) theory to determine the rotational friction coefficient of DLCA
aggregates over a parameter range of interest to aerosol scientists. I use my results
to generate a simple analytical expression for the rotational friction coefficient, as a
function of primary sphere size and the number of spheres in the aggregate.
6.2 Theoretical Methods
The force and torque on a rigid particle moving slowly relative to the sur-
rounding fluid can be expressed as
F = −Ξt ·UO −Ξ†O,c ·ω (6.2)
TO = −ΞO,c ·UO −ΞO,r ·ω (6.3)
where point O is the center of mass of the particle; UO and ω are the translational
and rotational velocities of the particle; and Ξt, ΞO,r, and ΞO,c are the translational,
rotational, and translation-rotation coupling friction tensors. The translational,
143
rotational, and coupling friction tensors relate the particle translational velocity to
the force on the particle, the particle angular velocity to the torque on the particle,
and the particle translational or angular velocity to the torque or force on the
particle, respectively. Note that the subscript O indicates that the property is
described relative to the particle’s center of mass, while the dagger symbol represents
the transpose of a tensor. These equations apply for creeping flow in the continuum,
free molecule, and transition regimes, characterized by very small, very large, and
intermediate Knudsen numbers, respectively. For spheres, the Knudsen number is
defined as Kn = λ/a, where λ is the gas mean free path and a is the sphere radius.
Brenner [29] demonstrated that a particles friction and diffusion tensors are
connected by a generalized Stokes-Einstein relationship,
DO = kBTM−1O (6.4)
where the grand mobility and diffusion tensors DO andMO are defined as
  Ξ Ξ†t O,cMO =  (6.5)
ΞO,c ΞO,r
 
D =  D D†t O,cO  (6.6)
DO,c DO,r
144
The scalar friction and diffusion coefficients are obtained from the trace of the
rotational diffusion tensor:
Dr = kBT/ζ =
1
r Tr(D3 r) (6.7)
Analytical expressions are available for the rotational friction/diffusion coef-
ficient of simple shapes in the continuum [29, 49, 112] and free molecule regimes
[3, 113, 114], but more approximate methods are needed for fractal-like aggregates
of touching spheres. The rotation problem has been studied extensively by Garcia
de la Torre and colleagues for rigid particles in continuum flow [51, 53, 115]. Li et al.
[6] used a simplified approach in the free molecule regime that ignored shielding by
other spheres in the aggregate. In principle, one could use a Monte Carlo approach
in the free molecule regime analogous to that used to compute the translational
friction coefficient [see e.g. [34, 36, 117, 118]] by replacing the linear velocity field
by a rotating velocity field, though no one appears to have published any results
using this approach.
Unfortunately, in many practical situations aerosol particles are fractal-like
aggregates in the transition flow regime, so a different approach is needed to analyze
their rotational behavior. In the previous chapter, I demonstrated that my extended
Kirkwood-Riseman (EKR) method can be applied to the rotational problem.
6.2.1 Extended Kirkwood-Riseman Method for the Rotational Friction Coefficient
Kirkwood and Riseman [28] developed a method to determine the force exerted
145
by a fluid on a particle or m acromolecule consisting of N spherical elements in
continuum flow, whereby the force on each element is equal to the force on an
isolated sphere minus the perturbations to the flow field caused by the other spheres:
∑N
F = −ζci t,0Ui − ζct,0 Tij ·Fj (6.8)
i=6 j
[( ) ( )]
1 rijrij 2a
2 3rijrij
Tij = I + + I− (6.9)
8πµr r2ij ij 3r
2 2
ij rij
Here, ζr,0 = 6πµa is the monomer translational friction coefficient, Ui is the velocity
of the ith sphere in the aggregate, and Tij is the hydrodynamic interaction tensor
that quantifies the effects of the jth sphere on the ith sphere. Applications of KR
theory often use the Rotne-Prager-Yamakawa (RPY) tensor [57, 58], Eq. (6.9), for
Tij, which is accurate to O(r−3ij ), where rij is the distance between the ith and jth
sphere.
The product of the monomer friction coefficient and the RPY tensor is similar
to the solution for Stokes flow around a sphere Vij, where u(rij) = Vij ·Uj is
the velocity around a sphere moving with velocity Uj. Noting this, I introduced
my extended Kirkwood-Riseman (EKR) method by replacing ζt,0Tij in Eq. (6.8)
with the velocity tensor for flow around a sphere in the transition regime, Vij(Kn)
(Chapters 3 and 4): ∑N
Fi = −ζt,0(Kn)Ui − Vij ·Fj (6.10)
i=6 j
As noted in the above equation, both the velocity tensor and the monomer friction
coefficient are functions of the primary sphere Knudsen number. These functions are
146
obtained by solving the Bhatnagar-Gross-Krook equation [71] using the approach
of Loyalka and colleagues [75, 76].
To determine the rotational friction coefficient, I first apply Eq. (6.10) by
substituting Ui = ω × ri, where ω is the angular velocity of the aggregate and ri
is the vector from the center of mass to the center of the ith sphere. This gives the
force on each sphere in the rotating aggregate. Next, I calculate the torque on each
sphere about the center of mass and sum up these torques to determine the total
torque on the aggregate: ∑N
TO = ri × Fi (6.11)
i=1
Performing this procedure for angular velocities about three mutually orthogonal
axes, I obtain the rotational friction tensor. Finally, I obtain the rotational diffusion
tensor and the rotational friction/diffusion coefficient from Eqs. (6.4) and (6.7). Note
that the translational and coupling friction tensors that appear in the grand mobility
tensorMO are obtained by solving Eq. (6.10) for a uniform translational velocity in
three orthogonal direction for the force and torque [via Eq. (6.11)] on the particle.
One significant flaw in applying the above method to calculate the torque on
a rotating particle is that it yields a value of zero for a sphere or for a straight chain
rotating about its long axis. To address this flaw, Garćıa de la Torre and Rodes [107]
proposed adding the torque of N rotating spheres to the torque computed using KR
theory for a rotating aggregate in continuum flow (the so-called volume correction).
I take the same approach with the EKR method, except now the monomer rotational
147
friction coefficient is a function of Knudsen number:
∑N
TO = −Nζr,0(Kn)ω + ri × Fi (6.12)
i=1
There is very little experimental data available for the torque on a rotating sphere
as a function of Knudsen number. Tekasakul et al. [119] and Bentz et al. [120] state
that their experimental results are in good agreement with the linearized Boltzmann
results of Loyalka [77] but do not provide sufficient data for a meaningful comparison.
Thus, I use the published computational results of Loyalka and fit those results to
an analytical expression for ζr,0(Kn). Specifically, I posit that the functional form
of the monomer rotational friction coefficient is analogous to the Cunningham slip
correction factor that appears in the monomer translational friction coefficient:
8πµa3 [ 8πµa3ζr,0 = = ( )] (6.13)
Cr(Kn) 1 + Kn A + A A3r1r 2r exp − Kn
The numerator of the above expression represents ζr,0 in the continuum limit. For
very large Knudsen numbers, Eq. (6.13) must reduce to the free molecule expression
for ζr,0 [3, 32],
FM 2πα 4 2πα µζr,0 = ρc̄a = a
4 (6.14)
3 3(0.499) λ
where α is the fraction of gas molecules that are reflected diffusely from the sphere
surface. In the last equality, I use the viscosity of a hard sphere gas, µ = 0.499ρc̄λ,
where ρ is the gas density and c̄ the mean speed of gas molecules. Using the results
from Table IV (Present column) and Eq. (33) from Loyalka [77] and noting that for
148
diffuse reflection A1r +A2r = 5.988, I get the following values for the coefficients in
the rotational slip correction factor: A1r = 3.930, A2r = 2.058, and A3r = 0.3277.
A second concern in employing the EKR method is that it ignores rotational
and translation-rotation coupling interactions between spheres. Simply put, the lin-
ear and rotational motions of a sphere can induce a rotation or torque in another
sphere, in accordance with Faxns second law [49]. The rotational and coupling hy-
drodynamic interaction tensors are (to order r−3ij ) equal to vorticity field around
a rotating sphere and the vorticity field around a translating sphere, respectively.
Thus, one would need to compute these fields as a function of Knudsen number to
account for rotational and coupling effects. In addition, one would now need to in-
vert a 6N -by-6N matrix (instead of a 3N -by-3N matrix, as in the current method)
to obtain the translational, rotational, and coupling friction tensors. (See Chap-
ter 5 for further discussion.) Rotational and coupling hydrodynamic interactions
are weaker (i.e. lower order in rij) than the translational hydrodynamic interactions
described by Tij, so I ignore these effects in the EKR method. The resulting error is
appreciable (around 30-40%) for very small, dense (i.e. high fractal dimension) ag-
gregates but decreases as the aggregate size (and thus the average distance between
spheres) increases (Chapter 5). I will discuss this further later in this chapter.
To apply my method for calculating the rotational friction coefficient, one
simply needs the coordinates of the primary spheres in the particle (either from a
cluster-cluster aggregation algorithm, as I use for this study, from a detailed Brow-
nian simulation, or from a TEM image) and the velocity field around a sphere. (See
Appendix B for velocity results at select Knudsen numbers.) Given this informa-
149
tion, one forms a 3N-by-3N matrix where each 3-by-3 block is Qij = [δijI + (1 −
δij)Vij(Kn)]/ζr,0(Kn) and δij is the Kronecker delta. The tensor Vij is a function
of primary size and the vector connecting the ith and jth spheres. In other words,
one solves Eq. (6.10) as a linear algebra problem to obtain the force on each sphere
and uses these results and Eq. (6.12) to determine the torque on the particle. Re-
peating this procedure for three mutually orthogonal angular velocities, one obtains
the rotational friction tensor. See the previous chapter for further discussion.
6.2.2 Adjusted Sphere Method for the Rotational Friction Coefficient
The extended Kirkwood-Riseman method can be applied to determine the
friction tensors of an aggregate in the transition flow regime, given the coordinates
of the spheres in the aggregate and the velocity around a sphere as a function of
the primary sphere Knudsen number. From these friction tensors, one can obtain
the rotational friction or diffusion coefficient. Here, I propose a parametric method
of determining the rotational friction coefficient that does not rely on the EKR
method. This parametric method is based on the adjusted sphere method (ASM) of
Dahneke and Zhang et al. for determining the translational friction coefficient. My
preliminary results showed that the adjusted sphere approximation can be applied
to the rotational friction coefficient (Chapter 5).
Dahneke [67] and Zhang et al. [41] posited that there is a universal relation-
ship between the translational friction coefficient of an aggregate and an aggregate
Knudsen number:
6πµRH
ζt,ASM = (6.15)
Cc(Knt,agg)
150
πλRH
Knt,agg = (6.16)
PA
Here, the hydrodynamic radius RH and orientation-averaged projected area PA
are continuum and free molecular measures of the particle size, respectively. The
adjusted sphere method allows one to compute the translational friction coefficient
of an aggregate given its hydrodynamic radius and projected area.
Melas et al. [80] show that this aggregate Knudsen number is proportional to
the ratio of continuum and free molecule expressions for the translational friction
coefficient. I propose an analogous approach for the rotational friction coefficient:
ζc
ζ = rr,ASM (6.17)
Cr(Knr,agg)
ζc
Kn = rr,agg (6.18)
(A1r + A FM2r)ζr
Here, ζc and ζFMr r are the aggregate rotational friction coefficients computed using
continuum and free molecular methods, respectively. The rotational slip correction
factor formula is the same as Cr defined in Eq. (6.13) and used to compute the
monomer rotational friction coefficient; likewise, coefficients A1r +A2r = 5.988, just
as in the monomer formula.
The adjusted sphere method is useful if one wants to obtain the rotational
friction coefficient without worrying about the details of the EKR method. However,
ASM incorporates both the continuum and free molecule friction coefficients for an
aggregate, and obtaining these expressions typically requires methods at least as
complex as the EKR method.
151
6.2.3 Scaling Laws for the Rotational Friction Coefficient in the Continuum and
Free Molecule Regimes
Given the complications involved in determining the rotational friction coef-
ficient of fractal aggregates, it is useful to develop a simple relationship between
the aggregate size and structure (i.e. the monomer size, the number of monomers,
and the fractal dimensions) and its rotational friction coefficient. Here, I present
some theoretical considerations for this relationship in both the continuum and free
molecule regimes and use those considerations to develop upper bounds for the ro-
tational friction coefficient. Later, I will use my results for the rotational friction
coefficient of DLCA aggregates to improve these simple relationships in the contin-
uum and free molecule regimes and to introduce a new expression for the transition
regime.
I begin with the continuum regime, where I will use an analogy with the
translational friction coefficient to develop the relationship between the aggregate
size and structure and the rotational friction coefficient. As mentioned in the previ-
ous section, one can define the hydrodynamic radius as the radius of a sphere with
the same translational friction coefficient as the aggregate. Computational studies
have shown that this hydrodynamic radius is roughly proportional to the radius of
gyration [4, 54, 93], such that the translational friction coefficient can be estimated
as ζct ∼ 6πµRg. By the same rationale, one can replace the sphere radius in the
expression for the rotational friction coefficient with the radius of the gyration of
152
the aggregate:
ζcr ∼ 8πµR3g (6.19)
Using Eq. (6.1), I obtain a scaling relationship between the aggregate size and
structure and the rotational friction coefficient:
c ∼ −3/dζ k f ζc N3/dfr 0 r,0 (6.20)
For DLCA aggregates with df ≈ 1.78, ζcr would scale with N1.685 based on this
simplified analysis. For the free molecule regime, I use the approach introduced by Li
et al. [6] to estimate the rotational relaxation time of an aggregate. In this approach,
one assumes that the drag on each sphere in a rotating aggregate is equal to the
drag on an isolated sphere moving with linear velocity Ui = ω× ri. Computing the
torque Ti = r × Fi on each sphere and summing over all spheres in the aggregate,
the torque becomes T ∼ ζFMt,0 NR2g1ω, where Rg1 is the radius of gyration about
the axis of rotation and ζFMt,0 is the free molecule translational friction coefficient.
Replacing Rg1 with Rg and using Eq. (6.1), I obtain a scaling relationship between
the aggregate size and structure and the rotational friction coefficient in the free
molecule regime:
ζFMr ∼
−2/d
k f ζFMN1+(2/df )0 r,0 (6.21)
Note that I have also substituted the free molecule rotational friction coefficient
for the product a2ζFMt,0 , since the two expressions are related by a constant of order
unity. For DLCA aggregates, ζFMr would scale with N
2.124 based on this simplified
153
analysis.
The above expressions are analogous to the power-law relationship between
the translational friction coefficient and the number of spheres in the aggregate
observed in numerous experimental and computational studies, as summarized by
Sorensen [4]. However, the exponents in Eqs. (6.20) and (6.20) are much higher
than the exponents in the translational friction power laws [approximately 0.54 and
0.94 in the continuum and free molecule regimes, respectively [4, 93]], resulting in a
much larger variation in the rotational friction coefficient for increasing N .
6.3 Results and Discussion
I have used my EKR method to determine the rotational friction coefficient
for DLCA aggregates (k0 ≈ 1.3 and df ≈ 1.78) with between 5 and 2000 primary
spheres and for primary sphere Knudsen numbers between 0.01 and 100. I have also
independently calculated ζr,ASM for these aggregates. The free molecule rotational
friction coefficients that appear in the aggregate rotational Knudsen number are cal-
culated using a Monte Carlo algorithm (Appendix C, while the continuum friction
coefficients are calculated using a KR-based method that accounts for translational,
rotational, and coupling hydrodynamic interactions between spheres in the aggre-
gate. (Specifically, I use the 3RD method described by Carrasco and Garcıa de la
Torre [53], which includes terms up to order O(r−3ij ) in the hydrodynamic interaction
tensors. See Appendix D for more information about rotational and coupling hydro-
dynamic interactions.) For each aggregate size, I determine the friction coefficients
154
of 20 different aggregates generated by a cluster-cluster algorithm. Thus, each data
point in the following graphs represents the average of 20 realizations of aggregates
with the same fractal dimension, prefactor, and number of primary spheres.
6.3.1 Comparison to Experimental Data
There is unfortunately a very limited databased for comparison. Colbeck et al.
[91] determined the rotational relaxation time of soot from the combustion of gaso-
line and other fuels using electro-optic scattering. In this technique, one compares
the intensity of scattered light for particles aligned in an electric field to that of
randomly-oriented particles. The measured relaxation time is related to the rota-
tional diffusion coefficient by τr = 1/(6Dr). For gasoline combustion generated soot
the measured relaxation time was about 4 ms.
Colbeck et al. provide SEM images of soot from the various fuels they used in
their study. Based on these images, the maximum aggregate length for the gasoline
soot [Figure 2(a) of [91]] is approximately 4000 nm. Since the largest aggregates
dominate the light scattering, I base my calculations on the larger clusters. The
authors do not specify the primary sphere size or the number of spheres in the
aggregate, so I must estimate these properties based on information available in the
literature. Köylü and Faeth [42] measured the primary sphere diameter of soot from
several gaseous and liquid hydrocarbons, including n-heptane (35 nm), isopropanol
(30 nm), benzene (50 nm), and toluene ((51 nm)). Thus, the mean primary sphere
diameter for soot from gasoline combustion is likely in the range of 30− 50 nm,
depending on the fraction of aromatics in the fuel blend.
155
Table 6.1: Comparison of EKR results to experimental data from Colbeck et al.
[91].
Primary Radius of Number of Rotational
Sphere Gyration Primary Relaxation
Diameter (nm) Spheres Time (ms)
(nm)
N/A N/A N/A 4a
35 1000 508 0.96
40 1000 400 0.96
45 1000 325 0.91
50 1000 269 0.88
35 1500 1045 3.6
40 1500 824 3.5
45 1500 668 3.6
50 1500 554 3.4
aExperimental result for gasoline (petrol), from Table 3 of [91]
Using primary sphere sizes of 35, 40, 45, and 50 nm; fractal dimension and
prefactor of 1.78 and 1.3; and a radius of gyration equal to 25% or 37.5% of the
maximum aggregate length,1 I can estimate the number of primary spheres in the
aggregate. Results for the rotational relaxation times for aggregates generated with
the above properties are listed in Table 6.1. It is seen that the results are weakly
dependent on the primary sphere size for a fixed radius of gyration. My results
based on these simple estimates of the aggregate properties for the larger radius of
gyration are in good agreement with the experimental results.
6.3.2 Comparison to Results in the Continuum and Free Molecule Limits
The continuum and free molecule results exhibit the following power-law rela-
tionship between the rotational friction coefficient and the number of spheres in the
1These are bounding estimates. Note that the algorithm we use to generate our aggregates
yields particles whose radii of gyration are ∼ 30− 33% of the maximum length.
156
aggregate:
ζc = 0.713ζc N1.627r r,0 (6.22)
ζFMr = 1.184ζ
FM
r,0 N
2.019 (6.23)
The exponents in these equations are within 5% of the exponents derived from the
simple scaling analysis in the previous section. In both the continuum and free
molecule regimes, the exponent obtained from my KR and Monte Carlo results
are slightly lower than the exponents from my scaling analysis. In the continuum,
this may be due to the fact that the hydrodynamic radius of a rotating aggregate
increases more slowly than the radius of gyration with increasing aggregate size.
Note that this behavior is observed for the translational friction coefficient. In the
free molecule regime, the exponent based on the MC calculations is lower than the
exponent from my scaling analysis because the scaling analysis does not account for
the effects of shielding by other spheres in the aggregate on the drag on each sphere.
The effects of shielding increase with aggregate size, which explains the reduced
exponent in Eq. (6.23) versus Eq. (6.21).
The large variation in the friction coefficient with N shown in Eqs. (6.22)
and (6.23) is evident in Figure 6.1. This figure also shows my EKR and rotational
adjusted sphere method results at Knudsen numbers of 0.1, 1, 2, and 10. All results
are normalized to the monomer rotational friction coefficient at the primary sphere
Knudsen number specified in each plot. At Kn = 0.1, the aggregates behave as if
they are in the continuum, as expected. Likewise, at Kn = 10 the aggregates exhibit
free molecular behavior, though the EKR and ASM results begin to diverge from
157
the free molecular limit at large N . At Kn = 1 and Kn = 2, the rotational friction
coefficient approaches the continuum limit at large N . These trends are analogous
to the trends in the translational friction coefficient as a function of primary sphere
Knudsen number and the number of spheres (Chapter 4); the notable difference
is that for the translational friction coefficient, the transition to continuum-like
behavior occurs at a higher Knudsen number and for aggregates with fewer primary
spheres. The reason for this is as follows. First, in the absence of any hydrodynamic
interactions or any shielding by other monomers, the torque on each monomer is
proportional to r2i cos θ, where θ is the angle between ri and the axis of rotation.
Second, the spheres furthest from the center of mass are less influenced by the other
spheres, and thus behave more as if they are isolated. As a result, the spheres
in the extremities of the aggregate experience a higher drag force than spheres in
the interior of the particle. Combining these two effects, the spheres furthest from
the center of mass have the largest effect on the rotational friction coefficient, and
these spheres behave most like they are isolated spheres whose drag force is given
by ζt,0(Kn)ωri. The shielding effect is also evident for the translational friction
coefficient, but since all the spheres move at the same translational velocity and
are weighted equally (not by the distance from the center of mass), the behavior of
the interior spheres dominates the aggregate translational friction coefficient. This
explains why the rotational friction coefficient transitions to continuum-like behavior
at lower Kn and higher N relative to the translational friction coefficient.
Given the nature of log-log plots in general and the six-order-of-magnitude
variation in the rotational friction coefficient between N = 1 and N = 2000, it is
158
Figure 6.1: Rotational friction coefficient results for Kn = 0.1, 1, 2, and 10. Re-
sults are normalized by the monomer rotational friction coefficient for each Knudsen
number.
159
Figure 6.2: Ratio of the rotational friction coefficients for N = 2000 calculated using
the rotational adjusted sphere method and in the continuum and free molecule limits
to the rotational friction coefficient calculated using the EKR method.
difficult to ascertain the differences between the calculation results from the methods
shown in Figure 6.1. For this reason, I have plotted the ratio of the adjusted
sphere method, continuum, and free molecule rotational friction coefficients to the
rotational friction coefficient calculated using the EKR method (Figure 6.2). Results
are shown for N = 2000. For small Knudsen numbers, my EKR results are in good
agreement with the continuum limit, while for very large Knudsen numbers, the
EKR results are in good agreement with the free molecule limit. The EKR and
ASM results are in good agreement for the entire Knudsen number range, which
provides further evidence that there is a universal relationship for the rotational
friction coefficient.
160
6.3.3 Relative Importance of Translational and Rotational Diffusion
One consideration in many aerosol studies is the relative importance of rota-
tional effects on aerosol transport properties. Studies often ignore rotational effects
[e.g. on coagulation rates and the resulting shapes of aggregates [121, 122]]. This
approach is valid if rotational diffusion is negligible relative to translational diffu-
sion, i.e. if particle orientation changes very little in the time it takes for the particle
to diffuse an appreciable distance. The relative importance of rotation on particle
transport is captured by the ratio of a characteristic translation time to a charac-
teristic rotation time. Defining the characteristic translation and rotation times as
the time required for the particle to diffuse one radius of gyration and rotate one
radian, this ratio becomes
τ R2t g/6Dt R
2
gζt
= = (6.24)
τr 1/6Dr ζr
Figure 6.3 shows this ratio as a function of primary sphere Knudsen number
and the number of primary spheres, calculated using both my EKR results and
analytic expressions for the translational and rotational friction coefficients (intro-
duced in Section 6.3.5 below). Note that the choppy nature of the EKR plot is due
to large variations in the rotational friction coefficient for a given set of aggregate
parameters, as discussed in the next section. The exact magnitude of the results in
this figure is insignificant since I am choosing arbitrary translational and rotational
diffusion distances; nevertheless, the results show that in the time it takes an ag-
gregate to diffuse one radius of gyration, it has rotated significantly. Simply put,
translational and rotational diffusion are equally important. This is true for all of
161
Figure 6.3: Ratio of the characteristic translational diffusion time to the character-
istic rotational diffusion time for DLCA aggregates as a function of primary sphere
size and the number of primary spheres. Each curve represents the results for the
specified primary sphere Knudsen number. The plot on the left shows results of
my EKR calculations, while the plot on the right shows results obtained using the
analytic fits for the translational and rotational friction coefficients [Eqs. (6.25) and
(6.26)].
the primary sphere and aggregate sizes we have included in this study. These results
do not imply that ignoring rotational behavior has a significant impact on the re-
sults of aerosol transport calculations (e.g. coagulation rates or filtration efficiency);
nevertheless, my results suggest that this issue deserves further attention.
It is also interesting to note that the ratio of translational to rotational relax-
ation times increases as a weak function of N in the continuum (low Kn) and free
molecule (large Kn) limits. One can also arrive at this conclusion by substituting
the power laws for the continuum and free molecule rotational friction coefficients
[Eqs. (6.22) and (6.23)] and the power laws for the translational friction coefficients
(N0.54 and N0.94 in the continuum and free molecule regimes, respectively) into
162
Eq. (eqn:rsoot:relaxation).2 In between, the situation is more complicated: the
transition to continuum-like behavior happens at lower N and higher Kn for trans-
lational transport compared to rotational transport, which explains the decrease in
the characteristic diffusion ratio at moderate Knudsen numbers.
6.3.4 Uncertainty in the Calculated Rotational Friction Coefficients
In Chapter 4, I discussed the uncertainty in my calculated translational friction
coefficients. There, I identified two main sources of uncertainty: the uncertainty in
the calculated velocity tensor and monomer friction coefficient that appear in the
EKR method, and KR theory itself. These factors also contribute to the uncertainty
in the rotational friction coefficient. However, I must add two additional sources of
uncertainty: the effects of neglecting rotational and coupling hydrodynamic interac-
tions and variations in the friction coefficient for aggregates with the same number
of primary spheres and the same fractal dimension and prefactor.
As I have mentioned, the linear and rotational velocities of each sphere in the
aggregate can induce a torque in the other spheres. In the continuum, the rotational
and coupling interactions are order O(r−3) and O(r−2ij ij ), respectively, as compared to
the translational hydrodynamic interaction tensor, which has leading terms of order
O(r−1ij ). As a result, I expect that the error in the EKR method would decrease
with increasing aggregate size due to the increase in the average distance between
monomers. My results confirm this expectation: the difference between the EKR
2Note that if one applies the simple scaling arguments from Section 6.2.3 [translational friction
coefficient proportional to the radius of gyration in the continuum and to the number of primary
spheres in the free molecule regime, and rotational friction coefficients given by Eqs. (6.20) and
(6.21)], one predicts that the ratio is independent of N in the continuum and free molecule regimes.
163
and continuum results at Kn = 0.01 is approximately 25% for N = 5 and 8% for N =
2000. That 8% difference at N = 2000 mirrors the difference between my calculated
monomer translational friction coefficient and the friction coefficient given by Stokes
law (Chapter 4), suggesting that the effects of coupling and rotational hydrodynamic
interactions are negligible for large aggregates. Expressions for the rotational and
coupling interaction tensors are unavailable for non-continuum flow, so it is difficult
to estimate the effects of neglecting these effects in the transition regime. However,
note that the EKR and free molecule results at Kn = 100 are within 5% for the
entire aggregate size range we studied. This suggests that rotational and coupling
interactions decrease in importance with increasing Knudsen number, as well as
for increasing aggregate size. As a result, the uncertainty in the rotational friction
coefficients computed using the EKR method is comparable to the uncertainty in my
computed translational friction coefficients (i.e. ∼ 10%) for all aggregates near the
free molecule regime and for large aggregates for any flow regime. This uncertainty
is larger (i.e. ∼ 25%) for small DLCA aggregates near the continuum regime.
The sources of uncertainty discussed above might better be termed “sources
of error,” since they represent differences between my results and the true rota-
tional friction coefficient. The final source of uncertainty is just that: variability in
the resulting rotational friction coefficient for the same input (namely, the primary
sphere Knudsen number, the number of primary spheres, and the fractal dimension
and prefactor). My reported results represent averages of results from 20 trials for
each set of inputs; here, the variability is due to differences in the coordinates of the
spheres that comprise the aggregate. For the translational friction coefficient, this
164
uncertainty is present, but it is very small: the standard deviation in the calculated
friction coefficient is less than 2% of the mean for all primary sphere and aggregate
sizes. However, for the rotational friction coefficient the standard deviation is as
high as 20% of the mean, and the standard deviation varies with Kn: it is lowest
near the continuum regime and largest near the free molecule regime. These varia-
tions can be understood in the context of my earlier discussion on the influence of
peripheral spheres on the rotational friction coefficient: spheres far from the center
of mass have a much larger impact on the rotational friction coefficient than spheres
near the center of mass. Thus, variations in the coordinates of the spheres in the
aggregate that have little impact on the translational friction coefficient are ampli-
fied when determining the rotational friction coefficient. Figure 6.4 illustrates this
point. In the figure, I show the torque on each sphere in two rotating 20-sphere
aggregates for rotation about the three principal axes (i.e. the axes for which the
rotational friction tensor is diagonal). The calculations are performed at Kn = 10.
The scale is the same for all six cases shown: the values indicate the torque on the
sphere, divided by the largest torque among the six cases. These two aggregates
have the same (i.e. within 0.5%) translational friction coefficient, but the rotational
friction coefficient – essentially the harmonic average of the friction coefficient for
each principal axis – of aggregate 2 is only 75% of the rotational friction coefficient
of aggregate 1. Thus, even though the torque is highest for aggregate 2 rotating
about the y- and z-axes, the torque about the x-axis is so low that the harmonic
average is less than the harmonic average of the torque coefficients for aggregate 1.
Again, these effects are dominated by spheres near the periphery of the aggregate,
165
Figure 6.4: Torque on each sphere of two 20-particle aggregates rotating about the x-
, y-, and z-axes (left, middle, and right, respectively) for Kn = 10. The rotation axis
is out of the page. The torque is normalized by the maximum torque among the six
cases. The rotational friction coefficient for aggregate 2 (bottom) is approximately
75% of the rotational friction coefficient for aggregate 1 (top).
as indicated by figure.
The effect of the peripheral spheres on the rotational friction coefficient is
more significant near the free molecule regime because these spheres behave almost
as if they are isolated, whereas in the continuum even the peripheral spheres are
somewhat affected by the particles overall effect on the flow field. This behavior
is illustrated in Figure 6.5, which shows the ratio of the drag on each sphere in
a 20-sphere aggregate to the drag on an isolated sphere (i.e. Fi/[ζt,0(Kn)Ui]) for
three different Knudsen numbers. Note that the uncertainty due to variations in
particle shape (as illustrated in Figure 6.4) does not affect the EKR results for
166
Figure 6.5: Ratio of the drag on each sphere in a 20-particle aggregate to the drag
on an isolated sphere (i.e. the monomer momentum shielding factor). Here, the
particle is rotating about the z-axis (out of the page). A value of unity indicates
that the sphere behaves as if it is isolated, while a value near zero indicates that the
perturbations caused by the other spheres have a significant impact on the drag.
individual particle realizations; rather it represents the variation in the rotational
friction coefficient for the same set of inputs.
6.3.5 Analytical Expression for Rotational Friction Coefficients of DLCA Aggre-
gates
Previously, I developed an analytical expression for the translational friction
coefficient of DLCA aggregates [Eq. (4.38) in Chapter 4] that I verified with my
EKR results:
ζt 1 + 1.612Kn
[( )
= 0.852N0.535
−1
+ 0.148
6πµa Cc(Kn) ( ) ]−1 −1
+1.612Kn 0.843N0.939 + 0.157 (6.25)
167
I will use the same approach to develop an analytical expression for the rotational
friction coefficient, which may be used to quickly estimate the characteristic rota-
tional relaxation time for a Brownian particle.
My expression is based on the harmonic sum of the continuum and free
molecule power-law expressions for the rotational friction coefficient [Eqs. (6.22)
and (6.23)]. In addition, I have added a term to each power law to give the correct
value for the monomer friction coefficient, and I have corrected for the fact that
the harmonic sum of the continuum and free molecule monomer friction coefficients
differs from the monomer friction coefficient given by Eq. (6.13). The resulting
expression is
ζr 1 + 5.988Kn
[( )−1
= 0.713N1.63 + 0.287
8πµa3 Cr(Kn) ( ) ]−1 −1
+5.988Kn 1.184N2.02− 0.184 (6.26)
Figure 6.6 shows the error in the fit relative to my EKR results:
ζr,fit − ζr,EKR
error = (6.27)
ζr,EKR
The error is within approximately 15% for much of the primary sphere Knudsen
number and aggregate size range studied here, though the error for smaller aggre-
gates near the continuum regime is higher. Again, this is because the continuum
power-law expression is based on results from a more rigorous method of calculating
the rotational friction coefficient, where I account for rotational and coupling hydro-
168
Figure 6.6: Error in the analytical expression for the rotational friction coefficient
[Eq. (6.26)] relative to my EKR results. This error is defined by Eq. (6.27).
dynamic interactions. Thus, the error in Eq. (6.26) relative to the true rotational
friction coefficient near the continuum regime is likely much lower than indicated
by Figure 6.6, which shows the error relative to the EKR results that overestimate
the rotational friction coefficient for small aggregates near the continuum regime.
However, my fit does not account for uncertainties in the rotational friction coeffi-
cient due to variations in the aggregate shape. Thus, the true error in the fit may
be as high as 30% for a given aggregate. This is still acceptable given the lack of
alternative methods for estimating the rotational friction coefficient in the transition
regime.
169
6.4 Conclusions
I have presented my self-consistent field results for the rotational friction co-
efficient of DLCA aggregates consisting of 5 to 2000 primary spheres with primary
sphere Knudsen numbers between 0.01 and 100. My results are in good agreement
with the continuum and free molecule limits for small and large Knudsen numbers,
respectively. The computed relaxation times from an aligned to randomly oriented
agglomerate are consistent with the measurements by Colbeck et al. [91] for soot
produced from gasoline. I estimate that my calculated rotational friction coefficients
are within 30% of the true value for the range of parameters I have studied, with
the greatest errors occurring for small aggregates near the continuum regime and
significantly better agreement for all aggregates near the free molecule regime and
for large aggregates in any flow regime.
I have used the EKR method to calculate the ratio of translational to rota-
tional characteristic diffusion times. A potentially important finding is that this
ratio is nearly independent of cluster size. My results show that aggregates rotate
significantly in the time it takes to diffuse one radius of gyration. This finding sug-
gests that further study is needed to assess the importance of rotation on aerosol
coagulation and deposition behavior.
I have introduced an analytic expression for the rotational friction coefficient
of DLCA aggregates (df = 1.78 and k0 = 1.3). This simple model can be applied to
quickly estimate the characteristic rotational relaxation time for studies involving
particle alignment in an external field. I have extended scaling analyses for the con-
170
tinuum and free molecule friction coefficients to the rotational problem, which can
also be used to estimate the rotational friction/diffusion coefficient of an aggregate
in these limits.
I have also provided an expression for the monomer rotational friction coef-
ficient as a function of Knudsen number, based on the computational results of
Loyalka [77]. This expression includes a slip correction factor with the same general
form as the Cunningham slip correction factor used for the translational friction
coefficient of a sphere.
171
Chapter 7: The Effect of Electric Field Induced Alignment on the
Electrical Mobility of Fractal Aggregates
7.1 Introduction
The transport behavior of nano-scale particles depends on particle size, shape,
and orientation. In the absence of an external field, Brownian motion randomizes
the particle orientation, such that the measured transport property (e.g. intensity
of scattered light, particle mobility) represents an average over all equally-likely par-
ticle orientations. In a strong field, particles become aligned in an orientation that
minimizes their energy in the field [1, 105]. This effect has been demonstrated ex-
perimentally by placing particles in an external electric field and measuring changes
in scattered light intensity [90, 91] or electrical mobility [5, 6, 87, 123] as the field
strength changes.
One common experimental technique for sizing nanoparticles involves using a
differential mobility analyzer (DMA) to determine the mobility of particles in an
electric field. The particle transport behavior is often expressed in terms of the
mobility diameter, which is the diameter of a sphere that has the same mobility as
the particle. For spherical particles, the measured mobility diameter is equal to the
172
geometric diameter and is independent of field strength. However, for non-spherical
particles, the mobility is a function of field strength. Plots of mobility versus field
strength are typically S-shaped, with the lower plateau at low fields representing fully
random particle orientation and the upper plateau at high fields representing fully
aligned orientation [5, 6, 87, 123]. The increase in mobility (decrease in drag and
mobility diameter) with increasing field strength is due to the electrical polarizability
of the particles. This means that particles tend to align such that the longest particle
dimension is parallel to the electric field. For example, a long, thin rod orients its
long axis parallel to the electric field direction at high field strengths.
Researchers have proposed experimental methods for obtaining shape infor-
mation or separating particles with different shapes by exploiting the dependence
of particle mobility on orientation in a DMA [6, 87, 88]. Such procedures involve
size-selecting particles in consecutive DMAs operated at different field strengths (or,
equivalently, at different sheath flow rates). The observed change in mobility may
give some clues about the shape of particles in the tandem DMAs.
The present study applies the theory of Li et al. [105] for the average particle
mobility as a function of field strength to calculate the mobility of aggregates with a
fractal dimension of 1.78, which is characteristic of soot and other particles formed
by diffusion-limited cluster aggregation. In the present study, I apply my extended
Kirkwood-Riseman (EKR) method (see Chapter 3) to obtain the translational fric-
tion tensor that appears in the theory of Li et al. [5, 105]. I compare my results to
experimental data [6], and I show how the particle mobility changes with electric
field strength for a wide range of primary sphere diameters and aggregate sizes. I
173
also use the EKR method to estimate the particle rotational relaxation time (see
Chapter 6) to evaluate the range of particle sizes for which it is appropriate to apply
the orientationally-averaged drift velocity method of Li et al. [105] to compute to
particle mobility. Finally, I discuss the implications of my results for obtaining shape
information by measuring the effect of electric field strength on particle mobility.
7.2 Theoretical Methods
Before discussing my theoretical methods in detail, I will provide an overview of
its various components. First, I compute the velocity field and the monomer friction
coefficient as a function of Knudsen number by solving the Bhatnagar-Gross-Krook
model equation [71] using the method of Loyalka and colleagues [75, 76]. The BGK
equation is a simplified, linearized version of the Boltzmann transport equation,
valid for near-equilibrium situations such as creeping flow of a sphere. This is done
once for each Knudsen number, with the velocity results saved for future use. (See
Appendix B.)
The second component involves computing the friction tensor for a cluster of
monomers by self consistently computing the flow field at each monomer resulting
from the flow field arising from all the other monomers (Chapter 3). The low
density of the aggregates is key to carrying out the calculations of large clusters in
a short time. This approach was initially used by Kirkwood and Riseman [28] for
computing the friction tensor for macromolecules in continuum flow. By using the
BGK results for the flow field in the transition regime, I can compute the friction
174
tensor of clusters composed of equally-sized monomer units. I have also applied the
theory to determine the rotational friction tensor (Chapter 5), which is necessary
for assessing the possible effect of rotation on the mobility.
Another element of the analysis is the calculation of the cluster polarizability
tensor, which is needed for computing the potential energy associated with align-
ment. I have obtained the polarizability tensor for the aggregates in this study from
ZENO (Mansfield et al., 2001), which uses a random walk algorithm [63] to compute
(among other things) the polarizability tensor for a perfectly conducting particle of
arbitrary shape. I assume that aggregate particles (e.g. soot) are perfectly conduct-
ing.
The final element involves the matrix manipulations and the ensemble aver-
aging [49, 105] to obtain the drift velocity (mobility) in the direction of the electric
field.
I now discuss the theory in more detail in the following sections.
7.2.1 Particle Orientation in an Electric Field
The probability distribution of a particle’s orientation in an electric field is
given by the Boltzmann distribution [1],
∫ ∫ ∫ e−U/kBTf(φ, θ, ψ) = 2π 2π 2π (7.1)
e−U/kBT sin θdφd2θd2ψ
0 0 0
where U is the energy of the particle in the electric field for the particle orienta-
tion given by the Euler angles (φ, θ, ψ). This equation shows that the probability
175
distribution is affected by the competition between randomizing Brownian forces
from collisions of gas molecules with the particle and electrical forces that tend to
align the particle in a particular direction. For non-polar materials, the interaction
energy includes contributions from free charges on the particle and from an induced
dipole due to polarization in the electric field [124]. The interaction energy from a
fixed charge is
Ue = −qre ·E (7.2)
where re is the vector from the center of mass to the point charge and E is the
electric field. For a conducting particle, the interaction energy from an induced
dipole is given by [124]
U = −1p E ·α ·E (7.3)2
where α is the electrical polarizability tensor. According to Fuchs [1], aerosol par-
ticles can be assumed to be conductors, even when comprised of non-conducting
materials, due to the ever presence of surface contaminants.
From Eqs. (7.2) and (7.3, the free charge and induced dipole interaction en-
ergies increase linearly and quadratically, respectively, with electric field strength.
Furthermore, the charge interaction energy increases linearly with particle charac-
teristic length, while the induced dipole interaction energy increases linearly with
particle volume. Because the polarization energy increases with a3 while the charge
energy increases with a, and because the particle orientation depends on the Boltz-
mann factor e−U/kBT , one can often ignore the effects of point charges when com-
puting the probability distribution for the particles orientation. For example, Li
176
et al. [5] determined that the ratio of polarization energy to fixed charge energy is
greater than 10 for carbon nanotubes with mobility diameters greater than 100 nm,
while Zelenyuk and Imre [87] observed no effects of particle charge on the mobility
of aligned doublets with primary sphere diameters of 240 nm. Also, for a conduct-
ing particle the charge can move rapidly, so that one can consider the charge to be
distributed evenly throughout the particle [124]. Based on these considerations, I
will consider only the polarization energy when computing the particle mobility.
To evaluate the probability distribution [Eq. (7.1)], it helps to define two coor-
dinate systems: a body-fixed coordinate system (x′, y′, z′) that rotates with the par-
ticle, and a space-fixed (or laboratory) coordinate system (x, y, z). (See Figure E.1
in Appendix E.) For convenience, I will choose the body-fixed axes to coincide with
the principal axes of the polarizability tensor. In this representation, the minimum
polarization energy occurs when the electric field is along the z′-direction. I will set
the space-fixed axes so that the z-axis is parallel to the electric field.
The relationship between a vector in laboratory coordinates and a vector in
body-fixed coordinates is given by the following relationship:
b = A · b′ (7.4)
The rotation matrix A represents three successive rotations from the body-fixed
system to the space-fixed system. For the ZXZ sequence of rotations, the rotation
177
matrix is given by
 
cosφ cosψ − cos θ sinφ sinψ − cosφ sinψ − cos θ sinφ cosψ sinφ sin θ 

A = sinφ cosψ + cos θ cosφ sinψ − sinφ sinψ + cos θ cosφ cosψ − cosφ sin θ
sinψ sin θ cosψ sin θ cos θ
(7.5)
where φ, θ, and ψ are the angles of the first, second, and third rotations, respectively.
One useful property of the rotation matrix is that its inverse is equal to its transpose
(Gel’fand, et al., 1963). Because of this property and my choice of laboratory
coordinates, the electric field in body-fixed coordinates is given by
 sinψ sin θ
E′ = cosψ sin θE (7.6)
cos θ
where E is the field strength. This shows that the probability distribution is a
function of only two of the three Euler angles. Using the above expression for the
electric field and noting that the polarizability tensor in body-fixed coordinates is
diagonal, I can explicitly write the interaction energy as
U = −1(α1 sin2 ψ sin2 θ + α2 cos2 ψ sin2 θ + α cos2 θ)E23 (7.7)2
where α3 > α2 > α1 are the eigenvalues of the polarizability tensor.
178
7.2.2 Average Drift Velocity of a Particle in an Electric Field
We use the probability distribution in Eq. (7.1) to calculate the average drift
velocity – and thus the mobility – of a particle in an electric field [105]. The drift
velocity is obtained by balancing the electric force on the particle with the aerody-
namic force for a given particle orientation:1
Vd = qΞ
−1
t ·E (7.8)
Here, Ξt is the translational friction tensor and q is the charge on the particle.
Combining Eqs. (7.1) and (7.8), we get the following expression for the particle
orientation-averaged drift velocity for a given electric field strength:
∫ 2π ∫∫2π ∫ 2π (q Ξ− )1 ·E e−U/kBT sin θd2φd2θd2ψ〈 0 0 0 tVd〉 = 2π ∫ 2π ∫ 2π (7.9)
e−U/kBT sin θd2φd2θd2ψ
0 0 0
In general, the orientation-averaged drift velocity is not parallel to the electric
field. However, the component of the drift velocity parallel to the electric field is
typically much larger than the components of the velocity perpendicular to the field.
For example, the perpendicular components of the drift velocity for soot-like fractal
aggregates are typically less than 5% of the parallel component. Thus, we can define
the particle mobility in terms of the parallel component of the orientation-averaged
drift velocity,
Z = 〈Vd,z〉/E (7.10)
1The linear relationship between the velocity and the drag force is valid in the creeping flow
regime, which applies for all of the conditions considered in this study.
179
Again, I have positioned the laboratory-fixed coordinate system so that the electric
field is in the z-direction. The z-component of the drift velocity can be written in
terms of the Euler angles and the components of the friction tensor in body-fixed
coordinates:
(
〈Vd,z〉 = qE M 2 2 2 2 233〈cos θ〉+M22〈cos ψ sin θ〉+M11〈sin ψ sin θ〉 )
+M12〈sin 2ψ sin2 θ〉+M13〈sinψ sin 2θ〉+M23〉 cosψ sin 2θ〉 (7.11)
Here, the angle brackets indicate orientation averages based on the distribution given
by Eq. (7.1) and the Mijs are components of the mobility tensor (i.e. the inverse of
the friction tensor) in body-fixed coordinates, i.e.
 
M 11
M12 M13
M ≡ M M M 

 −1 = (Ξ
′
t) (7.12)12 22 23
M13 M23 M33
In going from Eq. (7.9) to Eq. (7.11), I use the relation between the body-fixed
(Ξ′t) and space-fixed (Ξt) friction tensors, Ξ
−1
t = A · (Ξ′
−1
t) ·A†, where the dagger
symbol denotes the transpose of the rotation matrix. Note that Eq. (7.11) reduces
to the expressions given by Li et al. [5] for the special case of an axisymmetric body,
where M12 = M13 = M23 = 0, M11 = M22 = M⊥, and M33 = M‖.
For a randomly-oriented particle, the averaged mobility is
( )
q 1 1 1
Zrand = + + (7.13)
3 ζ1 ζ2 ζ3
180
where ζ1 > ζ2 > ζ3 are the eigenvalues of the translational friction tensor. At very
high field strengths, the particle will be oriented in the direction that minimizes the
electric field interaction energy. Thus, the high-field mobility is
−1
Z ′align = qk̂ · (Ξt) · k̂ = qM33 (7.14)
where k̂ is the unit vector in the z-direction.
7.2.3 Friction Tensor for an Aggregate
To calculate the orientation-averaged mobility of soot-like particles, one must
be able to determine the translational friction tensor for fractal aggregates con-
sisting of N primary spheres with radius a, where the Knudsen number of the
primaries (Kn = λ/a) is in the transition regime between continuum (Kn 1) and
free molecule (Kn  1) limits. To do so, I will use my extension (Chapter 3) of
Kirkwood-Riseman theory [28] from the continuum regime to the transition regime.
Kirkwood and Riseman proposed a method for calculating the translational
friction coefficient for a macromolecule or particle consisting of spherical subunits.
The drag on each sphere in the aggregate is obtained by considering the effects of
the other spheres in the particle on the flow field. The resulting force is the sum of
the drag on an isolated particle and the perturbations due to the other spheres:
∑N
F = −ζ0Ui − ζ0 Tij ·Fj (7.15)
i=6 j
181
Here, ζ0 = 6πµa is the friction coefficient for a sphere, given by Stokes law; Ui is the
velocity of the ith sphere; and Tij is the hydrodynamic interaction tensor. Carrasco
and Garcıa de la Torre [53] discuss some of the hydrodynamic interaction tensors
that have been proposed in the past and the relative accuracy of the various forms of
Tij. They conclude that the Rotne-Prager-Yamakawa (RPY) tensor [57, 58] – which
is accurate to order r−3ij , where rij is the distance between spheres – is sufficiently
accurate for practical purposes.
Noting that the product of the RPY tensor and the monomer friction coeffi-
cient is similar to the tensor Vij describing the flow field around a sphere moving
with velocity Ui (i.e. u(rij) = Vij ·Ui), I proposed replacing ζ0Tij with Vij, the
velocity field around a moving sphere in the transition flow regime:
∑N
F = −ζ0(Kn)Ui − Vij(Kn) ·Fj (7.16)
i=6 j
This extended Kirkwood-Riseman (EKR) approach is valid for creeping flow for any
Knudsen number, provided one can accurately solve for the velocity field around a
sphere as a function of Kn.
I obtain the velocity field and the monomer friction coefficientthat appear
in Eq.(7.16) by solving the Bhatnagar-Gross-Krook model equation [71] using the
method of Loyalka and colleagues [75, 76]. The BGK equation is a simplified,
linearized version of the Boltzmann transport equation, valid for near-equilibrium
situations such as creeping flow of a sphere. (See Chapter 2.) I solve Eq. (7.16) for
unit particle velocity in the x-, y-, and z-directions to determine the translational
182
friction tensor. My EKR results for the translational friction coefficient of fractal
aggregates compare well to published experimental data and calculational results
(Chapters 3 and 4).
7.3 Results
To determine the orientation-averaged mobility of fractal aggregates, I solve
Eq. (7.11) with the translational friction tensor calculated using the EKR method
and the polarizability tensor [which appears in the potential energy term, Eq. (7.7),
that affects particle orientation] obtained from ZENO [62]. ZENO uses a random
walk algorithm [63] to compute (among other things) the polarizability tensor for
a perfectly conducting particle of arbitrary shape. Again, I assume that soot par-
ticles are perfectly conducting. The polarizability and friction tensors are specified
in terms of the body-fixed axes, which correspond to the principal axes of the po-
larizability tensor, as discussed previously. I obtain the orientation averages in
Eq. (7.11) by integrating numerically using a 2D quadrature method (MATLAB
function integral2). I generate the aggregates using a cluster-cluster algorithm
[36]. For each N , I generate 20 clusters and present the average results of the 20
cases.
7.3.1 Comparison to Experimental Data
Li et al. [6] used a pulsed-field differential mobility analyzer (PFDMA) to de-
termine the electrical mobility of soot aggregates composed of 5-nm-radius primaries
183
(Kn = 13.5 for λ = 67.3 nm). They size-selected aggregates with mobility diame-
ters of approximately 129 nm, 154 nm, and 200 nm in a DMA operated at high field
(∼ 7000− 8000 V/cm), then used the PFDMA to measure the mobility of these
aggregates as a function of electric field strength.
For my calculations, I must first estimate the aggregate size and structure from
the reported mobilities. Like Li et al., I assume that the aggregates have a fractal
morphology, ( )d
R fg
N = k0 (7.17)
a
with fractal dimension df = 1.78 and prefactor k0 = 1.3. I determine the number of
primaries iteratively until I obtain a set of particles whose average random mobility
(as calculated using my EKR method) is in good agreement with the experimental
mobility at low field. I repeat this procedure for the three data sets, corresponding to
mobility diameters of 129 nm, 154 nm, and 200 nm. As an initial guess for N , I solve
for N in my expression for the friction coefficient of DLCA aggregates [Eq. (4.38)
of Chapter 4]:
ζt 1 + 1.612Kn
[( )
= 0.852N0.535
−1
+ 0.148
6πµa Cc(Kn) ( ) ]−1 −1
+1.612Kn 0.843N0.939 + 0.157 (7.18)
The friction coefficient is related to the mobility by ζt = q/Z.
Figure 7.1 compares the results of my calculations to data from Li et al. [6].
Overall, the results are in good agreement with the data enabling me to consider a
184
parametric study outside the bounds of available experimental data.
7.3.2 Effects of Aggregate Size and Field Strength on Mobility
Now that I have shown that the theory of Li et al. used in concert with my
EKR method can be used to calculate the orientation-averaged mobility of soot
as a function of electric field strength, I will use this approach to calculate the
mobility of DLCA aggregates over a wider range of primary sphere and aggregate
sizes. Again, my mobility results represent the average of 20 realizations from the
fractal generator. My calculations assume that Brownian rotation is slow compared
to the translational relaxation time of the particles. This means that the drag
force and the electric force are immediately balanced at each particle orientation.
Mulholland et al. [116] show the slow rotation limit applies for reduced rotational
velocity αnr = 2Dr,minτt < 0.05. Here, τt = m/ζh is the translational relaxation
time, ζh is the translational friction coefficient computed as the harmonic mean
of the eigenvalues of the translational friction tensor, m is the particle mass, and
Dr,min is the rotational diffusion coefficient of the particle about the axis that yields
the minimum Dr. I will examine the validity of this assumption in the Discussion
section below.
Figure 7.2 shows the effect of electric field strength on the normalized mobility
Z/Zrand of N = 100 and N = 1000 aggregates at Knudsen numbers corresponding
to primary sphere radii of 25 nm (Kn = 2.7), 13.5 nm (Kn = 5), 9.6 nm (Kn = 7),
6.7 nm (Kn = 10), and 5 nm (Kn = 13.5). Particles become aligned at lower electric
fields as the primary size and the number of primaries increase. For N = 1000,
185
Figure 7.1: Comparison of my calculated orientation-averaged mobilities to experi-
mental data from Li et al. [6] for mobility diameters of ∼ 129 nm, ∼ 154 nm, and
∼ 200 nm (based on the high-field mobilities). The number of primaries used for
the calculations represent the best fits to the data.
186
particles are fully aligned at fields as low as approximately 500 V/cm for the 25 nm
primaries versus approximately 5000 V/cm for the 5 nm primaries. The physical
basis of this result will be discussed later in this chapter. The normalized fully-
aligned mobility for the 1000-sphere aggregates increases slightly with decreasing
Knudsen number (increasing primary radius). The maximum increase in mobility
from random orientation to fully-aligned is approximately 8%.2
Figure 7.3 shows normalized mobility versus electric field strength for Kn = 2.7
and Kn = 13.5 at various aggregate sizes. All but the smallest aggregates with
Kn25nm radius primaries (Kn = 2.7) are fully aligned at 8000 V/cm, while only
the larger aggregates with 5 nm radius primaries (Kn = 13.5) are fully-aligned at
this field strength. The latter point is consistent with the data of Li et al. [6] and
my results shown in Figure 7.1. Note that several of the lines in the Kn = 2.7 plot
cross each other. This is due to statistical variations in the fully-aligned mobility,
caused by the finite number of particles I use to generate the results for each N . I
will return to this issue in the final paragraph of this section.
From Figures 7.2 and 7.3, it is clear that particle orientation may not be fully
random even at low field strengths. It is useful to determine the maximum field at
which the particle orientation is random; I show this maximum field as a function
of N and Kn in Figure 7.4. Here, I consider the particle orientation to no longer be
random when the mobility increases by 0.5% from the mobility in the limit of zero
field. Again, particles begin to partially align at lower field strengths as particle size
2For comparison, the change in the intensity of scattered light between aligned and random
states has been demonstrated to be as large as ∼ 50% for soot [90, 91].
187
Figure 7.2: Normalized mobility as a function of electric field strength for 100-sphere
(top) and 1000-sphere (bottom) aggregates.
188
Figure 7.3: Normalized mobility as a function of electric field strength for aggregates
with primary sphere radii of 25 nm (Kn = 2.7, top) and 5 nm (Kn = 13.5, bottom).
Note that the N = 100 and N = 1000 curves in this figure correspond to the
Kn = 2.7 and Kn = 13.5 curves in Figure 7.2.
189
(both N and a) increase.
Figure 7.4: Maximum electric field strength at which particles are randomly ori-
ented, defined as having a mobility within 0.5% of the mobility in the limit of zero
field strength. Note that results are capped at an upper limit of E = 10, 000 V/cm.
Finally, Figure 7.5 shows the mobility ratio, Zalign/Zrand, as a function of N for
several Knudsen numbers. The random and fully-aligned mobilities are calculated
using Eqs. (13) and (14), respectively. Generally speaking, the mobility ratio is
constant with increasing N near the continuum regime and decreases with N in
the free molecule regime. At intermediate Knudsen numbers, the aligned-versus-
random behavior becomes more continuum-like at large N ; this is analogous to
the behavior I have observed for the translational friction coefficient of soot-like
aggregates (Chapter 4). I will explain this behavior in the Discussion section below.
Note that each point in the figure represents an average over 20 particle realizations.
To give an idea of the uncertainty in the mean values shown in the figure, we show
bounds of one standard deviation of the mean for several N in the continuum and
190
free molecule limits. See Appendix E for further discussion about the variability in
the mobility ratio results.
Figure 7.5: Ratio of fully-aligned to random electric mobilities for wide range of
primary sphere Knudsen numbers and the number of primaries. The Kn = 0 and
Kn = ∞ curves represent the continuum and free molecular limits, as calculated
using the standard KR theory with the RPY tensor [see e.g. Chen et al. [30]] and
using a Monte Carlo code (Chapter 6), respectively. Uncertainties of one standard
deviation of the mean (based on 20 samples with the same fractal dimension but
different morphologies) are shown for the continuum and free molecule results for
several N .
The choppiness in the plots in Figure 7.5 can be explained by the statistics of
my results: the standard deviation of Zalign/Zrand is approximately 0.03 for all cases,
and thus the standard deviation of the mean3 is approximately 0.007. Assuming the
samples are normally distributed about the population mean, we would expect 68%
of the samples to be within one standard deviation of the population mean. Indeed,
most of the mean mobility ratios for Kn = 0, 0.1, and 1 are within 0.007 of an
estimated population mean of 1.085, so it is reasonable to conclude that the spread
3For a sample√of n trials having a sample standard deviation s, the standard deviation of the
mean is σx̄ = s/ n.
191
in Zalign/Zrand is partially due to my finite sample size.
7.4 Discussion
7.4.1 General Observations
My results show that particle alignment occurs at decreasing electric field
strengths as the primary sphere Knudsen number decreases (primary sphere radius
increases) and as the number of primaries increases. This occurs because polariz-
ability is proportional to volume, so that interaction energy between the electric
field and the induced dipole increases with volume. The particle becomes fully
aligned when the magnitude of the interaction energy is significantly greater than
the Brownian energy kBT .
I also show that the ratio of fully-aligned to random mobility is a function of
the number of primary spheres and the primary sphere size. Near the continuum
regime, the mobility ratio is approximately constant with N ; near the free molecule
regime, the mobility ratio decreases with N . I discuss this topic in some detail in Ap-
pendix E, but the brief explanation is as follows: the mobility of an aggregate in the
continuum and free molecule regimes is roughly inversely proportional to the radius
of gyration [4, 54, 93] and the orientation-averaged projected area [41], respectively.
Similarly, the continuum and free molecule aligned mobilities are correlated to the
inverses of the radius of gyration about the major axis of the polarizability tensor
(i.e. the z′-axis), R ′ , and the projected area in the plane normal to the z′gz -axis,
PAz′ . (See Appendix E.) Averaged over 20 cases, the ratio Rg/Rgz′ is approximately
192
constant (after accounting for the statistical fluctuations described above) with N ,
while PA/PAz′ decreases with N , mirroring the trends in the mobility ratios in the
continuum and free molecule limits.
7.4.2 Validity of the Slow Rotation Assumption
My calculations assume that Brownian rotation is slow compared to transla-
tional relaxation, i.e. the aggregates are in the slow rotation limit. To assess the
validity of this assumption, I use the EKR method to determine the translational
friction coefficient (ζh, the harmonic average of the eigenvalues of the friction ten-
sor) and the minimum rotational diffusion coefficient. Using these calculated friction
and diffusion coefficients, I compute the reduced rotation velocity αnr, as shown in
Figure 7.6.
Figure 7.6: Reduced rotation velocity for a range of primary sphere sizes and Knud-
sen numbers. Soot density is taken as 2 g/mL [125, 126]. Particles with a reduced
rotation velocity less than 0.05 (the dotted line) are in the slow rotation limit.
193
The figure shows that particles with Knudsen numbers less than 13.5 and more
than approximately 50 primary spheres are in the slow rotation limit (αnr < 0.05
[116]); none of the particles in this study are in the fast rotation limit (αnr > 10).
Note that for fast rotation, one should use an orientation-averaged drag approach to
calculate the mobility, as opposed to the orientation-averaged drift velocity approach
in the slow rotation limit. At low field strength, the fast rotation limit yields a scalar
friction coefficient equal to the arithmetic average of the eigenvalues, instead of the
harmonic average in the slow rotation limit [105]. (The two approaches yield the
same result for the fully aligned case.) Thus, some of the particles included in
this study are not in the slow rotation limit. However, the maximum difference
between the mobility calculated using the averaged drift velocity is at most 1% less
than the mobility calculated using the averaged drag force for the particles in this
study (i.e. those with fractal dimensions of 1.78). In other words, it makes little
difference which approach one uses to calculate the mobility of DLCA aggregates.
The distinction becomes more significant for particles with a large aspect ratio, such
as long, thin rods, where there is a large difference between the largest and smallest
eigenvalues of the friction tensor.
7.4.3 Polarizability Versus Friction
A conducting particle in an electric field will orient itself to minimize its in-
teraction energy with the field, in the absence of any Brownian thermal forces. The
minimum energy occurs when the charge separation in the particle is greatest. This
means that a rod or a chain of spheres will orient itself such that its long axis is
194
parallel to the electric field, while the fractal particles in this study are oriented such
that the axis through the most widely separated two spheres is parallel to the field.
One would expect that the minimum drag force also occurs when it moves along its
most elongated direction. This is exactly the case for axisymmetric particles like
rods and cylinders, so that the most likely orientation of the particle in an electric
field is the orientation with the minimum drag. For fractals, the situation is more
complicated: there is a finite angle between the eigenvector of the friction tensor
corresponding to the minimum drag and the principal axis of the polarizability ten-
sor (the most favorable orientation). In most cases, this angle is small (< 10◦),
though I have observed angles as large as 25◦ between these two eigenvectors. This
means that the most favorable orientation does not necessarily minimize the drag
for fractals.
A somewhat related issue is that translating particles with arbitrary shape
experience a torque, where the relationship between the particle velocity and the
torque exerted by the fluid on the particle is governed by the coupling tensor [49]. For
non-skew particles (such as rods and cylinders), there is no translational-rotational
coupling, but for skew particles like fractals the coupling tensor is non-zero. The
question is whether or not the hydrodynamic torque is sufficient to overcome the
interaction energy between the induced dipole and the field and reorient the particle.
To answer this question, I calculated the coupling torque on 1000-sphere ag-
gregates with a primary sphere Knudsen number of 13.5, using the EKR method
to determine the coupling tensor and the orientation-averaged drift velocity at a
field strength of 4000 V/cm (roughly corresponding to the minimum field strength
195
at which the particle is fully aligned). The resulting torque is more than two or-
ders of magnitude lower than the interaction energy. I repeated this calculation for
N = 100, Kn = 2.7, and E = 200 V/cm; again, the coupling torque is significantly
lower than the interaction energy. This shows that the coupling torque has no effect
on the particle orientation at high field strength.4
7.4.4 Using Field-dependent Mobility to Evaluate Particle Shape
My results clearly show that particle mobility increases with electric field from
a fully random state at low fields to a fully oriented state at higher fields. The
transition occurs at decreasing fields for increasing particle size (both in terms of
primary sphere size and the number of primaries), as expected since polarizability
is proportional to particle volume.
This behavior has prompted some researchers to propose methods to separate
particles with different shapes by exploiting the changes in mobility at different
electric fields, such as by size-selecting particles in a DMA followed by separation
with second DMA operated at a different field strength [87, 88]. Using this method,
one can distinguish between spheres (or aggregates with fractal dimension near
3) and more elongated particles like rods, chains, prolate ellipsoids, or soot-like
aggregates, since the mobility of a sphere does not change with field strength. In
practice, this technique may not be feasible for some particle sizes due to limitations
4At low field strength, translational-rotational coupling has a small but noticeable effect on the
orientation-averaged drag force. For this reason, one must account for rotational and translation-
rotation coupling effects when determining the translational diffusion of skew particles [29]. To
fully account for rotational and coupling effects at intermediate fields – where both hydrodynamic
torques and induced-dipole energies affect particle orientation – one could use a Brownian approach
similar to that of Fernandes and Garćıa de la Torre [127].
196
of current commercially available DMA operating conditions and configurations. For
example, Li et al. [6] estimate that they cannot operate their experimental system
at fields below 1000 V/cm. At this field strength, larger particles are fully-aligned
(see Figures 7.2 and 7.3), so measuring the mobility at fields of 1000 V/cm and
e.g. 8000 V/cm would yield the same result and lead to the erroneous conclusion
that the particle is spherical. At the other end of the size spectrum, small fractals
experience minimal changes in mobility over the range of electric fields studied here.
It may also be difficult to operate a DMA at a low enough field to ensure that the
particle orientation is fully random. One can consult Figure 7.4 to determine if it
is possible to select randomly oriented particles for the operating conditions of ones
DMA setup.
There are also issues distinguishing between two non-spherical particles with
different shapes. For example, the fully-aligned mobility of a doublet in continuum
flow is 8% greater than the mobility of a randomly oriented doublet [49, 53]. This
is comparable to the increase in mobility from random to fully-aligned orientations
for the soot particles included in this study. Thus, a doublet with primary size
near the continuum regime and a soot-like particle with the same low-field mobility
will behave similarly at higher voltages, making it difficult to distinguish between
these particles. As another example, I looked at the effect of the prefactor [k0 in
Eq. (7.17)] on the mobility of DLCA aggregates. While the prefactor does affect
the mobility (with lower prefactors resulting in decreased mobility for the same
number of primary spheres), it has little effect on the ratio of the fully-aligned to
fully random mobilities.
197
Beyond the experimental issues mentioned above, it is also difficult to accu-
rately calculate the mobility of a fractal aggregate in the transition regime. I have
estimated that the EKR method yields orientation-averaged translational friction
coefficients within 10% of the true value (Chapter 4), which translates to uncertain-
ties in any estimate of the primary sphere size or the number of primary spheres.
An obvious example is my attempt to fit my results to the data of Li et al. (2016),
as illustrated by Figure 7.1: my estimated aggregate sizes (in terms of N) are likely
within about 10% of the true value (though the actual error estimate depends on
the relationship between N and the friction coefficient). This situation is simplified
by the fact that we know the primary sphere size from TEM measurements, and we
have a good estimate of the fractal dimension and prefactor from numerous studies
of soot. (See Sorensen [4] for a review of these studies.)
In total, the above factors mean that while it may be theoretically possi-
ble to extract shape information from DMA measurements made at different field
strengths, it is difficult, and probably further consideration should be given to de-
velopment of DMA configurations optimized for this purpose.
7.5 Conclusions
I have applied the EKR method for calculating the translational friction tensor
of fractal aggregates to verify the theory of Li et al. [105] for the average mobility
of a particle in an electric field. My results compare well to published experimental
data for soot [6]. Furthermore, I use the EKR method to calculate the average
198
mobility of aggregates over a range of primary sphere sizes in the transition and near
free molecule regimes with up to 2000 primary spheres. The maximum increase in
mobility from random to fully-aligned orientations is approximately 8% for the soot-
like aggregates (df = 1.78, k0 = 1.3) included in this study. While my calculations
cover the Knudsen number range of 2.7 to 13.5 – which represents a representative
range of primary sphere sizes in soot particles (see, e.g. [6, 42]) – this approach is
valid for any primary sphere size and number of primaries, provided the particle is
in the slow rotation limit. See Chapters 4 and 6 for translational and rotational
friction coefficient results at larger and smaller Knudsen numbers.
While it is theoretically possible to use the relationship between mobility and
field strength to obtain size and shape information about particles or to separate par-
ticles with similar mobility but different shapes, my results suggest there are several
practical issues related to the experimental setup and to the accuracy of the meth-
ods used to relate the data to size and shape information. It is especially difficult
to obtain shape information for either very large or very small soot-like aggregates
because large aggregates are fully aligned at even very low field strengths and small
aggregates require very high field strengths to align. In these limits, the measured
mobility at low (∼ 1000 V/cm) and high (∼ 8000 V/cm) field strengths would be
nearly equal, which would suggest – incorrectly – that these fractal aggregates are
actually spherical.
199
Chapter 8: Hydrodynamic Interactions between Particles
8.1 Introduction
The vast majority of the literature on aerosol particle transport treats the
particles as if they are isolated when calculating the drag. This approach is valid
for very dilute aerosols, where the average separation between particles is large, and
thus hydrodynamic interactions between particles are negligible. For example, the
mean settling velocity of a system of spheres in continuum flow is U0(1 − 6.55φ),
where φ is the particle volume fraction and U0 is the velocity each sphere would
have if it was alone in an infinite fluid [128]. As another example, the viscosity of a
suspension of spheres in continuum flow is µ(1 + 2.5φ), where µ is the fluid viscosity
[2]. Clearly, the interaction effects on the settling velocity and suspension viscosity
are negligible for aerosol systems consisting of spheres at typical volume fractions.
However, there are situations where one might expect interactions between
particles to be more significant. Sorensen et al. [129] has demonstrated that aerogels
can form under certain conditions in sooting flames. In this process, fractal aggre-
gates formed by diffusion limited cluster aggregation (DLCA) reach a critical size
and begin to fill the entire physical volume because the aggregate fractal dimension
(df ≈ 1.78) is less than the spatial dimension. One would expect hydrodynamic in-
200
teractions between the soot aggregates to be more significant than between spheres
at the same volume fraction, due to the stringy nature of the DLCA aggregates.
More generally, for coagulating systems aerosol particles must approach each other,
so that interparticle hydrodynamic interactions may affect the coagulation rate.
Much of the literature on hydrodynamic interactions between particles focuses
on spheres in the continuum regime [1, 128, 130–136]. Little attention has been
paid to interactions between particles in the transition (or kinetic) flow regime,
when the particle size is comparable to the gas mean free path, as is often the case
in aerosol systems [2, 4]. Such an investigation is now possible using my theory for
hydrodynamic interactions in the kinetic regime [92].
The present study examines the forces exerted by aerosol particles on their
neighbors due to hydrodynamic interactions between the particles and the surround-
ing fluid. I consider both spheres and soot-like fractal aggregates moving parallel,
anti-parallel and perpendicular to their line of centers. (See Figure 8.1.) I employ
the extended Kirkwood-Riseman (EKR) method [92] to solve for the hydrodynamic
forces on the particles in the transition flow regime as a function of the separation
distance between particles, the Knudsen number (Kn ≡ λ/a), where λ is the gas
mean free path and a is the sphere radius) of the primary sphere(s) that comprise
the particles, and the number of primary spheres in the particle. As an example of
a situation where hydrodynamic interactions may be significant, I present calcula-
tions for the settling velocity of a cloud in an unbounded medium. Throughout this
study, I assume the particles are in the creeping flow regime (Re 1, Ma
1).
201
Figure 8.1: Two spheres in the parallel, anti-parallel, and perpendicular flow con-
figurations, and two 10-sphere aggregates with random orientations in parallel flow.
The descriptor refers to the direction of movement relative to the line connecting
the center of mass of each particle.
8.2 Theoretical methods
Let us consider two arbitrarily-shaped particles immersed in a viscous fluid.
Particles 1 and 2 are subjected to external forces F1 and F2, respectively; the centers
of mass of the particles are connected by vector r. (The forces may result from the
presence of the particles in a gravitational or electromagnetic field.) I shall assume
that we are in the creeping flow regime, such that inertial forces on both the fluid
and the particles are negligible compared to viscous forces. In this regime, there is
a linear relationship between the forces on the particles and their velocities,
     U1 M11 M12=  F1·  (8.1)
U2 M21 M22 F2
202
where Mij is the mobility tensor that relates the force on particle j to the velocity of
particle i. The mobility tensors are functions of the size and shape of both particles
and the distance between them. As r → ∞, hydrodynamic interactions between
particles become negligible. In this case, M12 = M21 ≈ 0, and the mobility tensors
M11 and M22 are simply the inverses of the translational friction tensors Ξt for
particles 1 and 2, defined by the relation
F = −Ξt ·U = −M−1ii ·U (8.2)
My goal is to determine the mobility tensors for particles in the transition
flow regime as a function of particle size, shape, and separation distance. Before
tackling this problem, it is worth considering the simpler case of two spheres in
continuum flow. From there, I will describe how one can approach the problem for
fractal aggregates in the continuum, before discussing how to apply the extended
Kirkwood-Riseman method for particles in the transition regime.
8.2.1 Two spheres in continuum flow
The instantaneous velocity of two spheres subjected to external forces F1 and
F2 has been studied by a number of authors [49, 97, 130, 131, 134, 137]. Exact
solutions are available for two spheres moving parallel and perpendicular to their
line of centers. For arbitrary r, the problem can be solved using the method of
reflections, whereby the mobility tensors are determined as expansions in powers
of the inverse of interparticle separation distance r. (See Happel and Brenner [49]
203
for more information about the method of reflections, including its application to
the two-sphere problem and to the problem of a particle in a tube.) Since the
mobility tensors are symmetric about the line of centers connecting the spheres, we
can formally write
rr ( rr)
Mij = Aij +Bij I− (8.3)
r2 r2
where the non-dimensional coefficients Aij and Bij can be written as expansions in
powers of r−n, ∑∞
A −nij = aij,nr (8.4a)
n=0
∑∞
B −nij = bij,nr (8.4b)
n=0
One obtains the coefficients aij,n and bij,n from the method of reflections.
One can include successive terms in the expansions for Aij and Bij to obtain
increasingly accurate results for the mobility tensors. The simplest approach is the
point force approximation, where M = ζ−1ii i I, Mij is given by the Oseen tensor (the
r−1 term in Eq. (8.5) below), and ζi = 6πµai is the monomer friction coefficient for
a sphere with radius ai. The next level of approximation yields M
−1
ii = ζi I and
[( ) ( 2 2)( )]1 rr ai + aj − 3rrMij = [(I + + I8πµr r2 3r2 r2
1 a2 2
) ( ) ]
− i
+ aj 2rr a
2
i + a
2 ( )
j rr
= 1 + 1 + I− (8.5)
8πµr 3r2 r2 3r2 r2
for i 6= j. For ai = aj, Eq. (8.5) reduces to the Rotne-Prager-Yamakawa tensor
[57, 58]. Batchelor [134] provides expressions for the mobility tensors accurate to
204
order r−5, while Felderhof [97] uses the method of reflections to derive expressions
valid to order r−7 for mixed slip-stick boundary conditions. Results from Felderhof’s
interaction tensor are equal to available exact solutions to the two-sphere problem
except for very small separation distances (i.e. nearly touching spheres), where
higher order terms become important [97].
Note that if sphere 2 is much smaller than sphere 1 (i.e. a2  a1) and if r is
large enough that terms higher than order r−3 are negligible, sphere 2 will simply
be advected by the Stokes velocity field generated by sphere 1. This becomes clear
solving Eq. (8.1) for U2 with F2 = 0, F1 = 6πµa1U1, and using Eq. (8.5) for M21
with a2  a1:
[
3a ( ) ( )]rr a31 3rr
U2 = I + +
1 I− ≡ V(r) ·U1 (8.6)
4r r2 4r3 r2
Here, V(r) ·U1 is the Stokes velocity field around a sphere moving with velocity
U1. I will refer to V(r) as the Stokes tensor.
Any of the above expressions for the mobility tensors can be used to solve the
two sphere problem for arbitrary sphere sizes a1 and a2 (provided the continuum
approximation is still valid), center-to-center distance and orientation r, and exter-
nal forces F1 and F2. Tables 8.1-8.3 show results for the speed of each sphere as a
function of separation distance for equal-sized spheres and equal force magnitude.
(For a1 = a2, the speeds of the spheres are equal.) The tables include results for the
exact solutions to the two-sphere problem [130, 132, 133], the mobility expression
of Felderhof for stick boundary conditions [97], and my EKR method (i.e. using
205
r/a Exact Felderhof EKR
2 1.550 1.623 1.688
2.01 1.549 1.618 1.685
2.1 1.536 1.582 1.660
2.5 1.486 1.493 1.568
3 1.432 1.433 1.481
3.5 1.386 1.386 1.417
4 1.347 1.347 1.367
5 1.287 1.287 1.296
7 1.210 1.210 1.213
10 1.149 1.149 1.150
15 1.100 1.100 1.100
Table 8.1: Speed of two spheres moving parallel to their line of centers, relative
to the speed of isolated spheres subjected to the same external force. The EKR
or Stokes tensor solution refers to M = ζ−1ii i and M
−1
ij = ζi V(r), where V(r) is
defined in Eq. (8.6). The Felderhof tensors [97] are accurate to order r−7. The exact
solution is given by Stimson and Jeffery [130].
the Stokes tensor) for the three cases shown in Figure 8.1. In all cases, the Stokes
tensor results are in very good agreement (less than 1% difference) with the exact
results for > 5. As expected, the error is more significant for smaller separation
distances because the Stokes tensor does not include terms of order r−4 and higher
and because there is a missing factor of 2 in the r−3 term.
8.2.2 Aggregates in continuum flow
Experiments and simulations have shown that particles formed by diffusion-
limited cluster aggregation have a fractal morphology with fractal dimension df ≈
1.78 and prefactor k0 ≈ 1.3 [4]. These parameters relate the radius of gyration to
the number of spheres in the aggregate,
( )d
R fg
N = k0 (8.7)
a
206
r/a Exact Felderhof EKR
2 0.000 0.080 0.313
2.01 0.019 0.089 0.315
2.1 0.135 0.161 0.340
2.5 0.361 0.360 0.432
3 0.491 0.490 0.519
3.5 0.570 0.570 0.583
4 0.626 0.626 0.633
5 0.702 0.702 0.704
7 0.787 0.787 0.787
10 0.851 0.851 0.851
15 0.900 0.900 0.900
Table 8.2: Speed of two spheres moving anti-parallel to their line of centers, relative
to the speed of isolated spheres subjected to the same external force. The exact
solution is given by Brenner [132].
r/a Exact Felderhof EKR
2 1.380 1.421 1.406
2.01 1.403 1.419 1.404
2.1 1.392 1.399 1.384
2.5 1.326 1.328 1.316
3 1.267 1.267 1.259
3.5 1.225 1.225 1.220
4 1.195 1.195 1.191
5 1.154 1.154 1.152
7 1.109 1.109 1.108
10 1.075 1.075 1.075
15 1.050 1.050 1.050
Table 8.3: Speed of two spheres moving perpendicular to their line of centers, rela-
tive to the speed of isolated spheres subjected to the same external force. The exact
solution is given by Goldman et al. [133].
207
Notable examples of DLCA aggregates include soot, as well as titania and other
ceramic powders synthesized in aerosol flame reactors [7].
In the previous section, I reviewed how one can solve the two sphere problem
to various levels of approximation. I now describe how to use a similar approach for
aggregates.
In approaches based on Kirkwood-Riseman theory [28], one uses any of the
above expressions for the interactions between pairs of spheres to determine the force
on a macromolecule or particle consisting of N spherical subunits. The resulting
system of equations is analogous to Eq. (8.1):
∑
−Ui = Mij ·Fj (8.8)
j=1
Here, Fj is the force exerted by the fluid on the jth primary sphere in an aggregate,
which is unknown, whereas in Eq. (8.1) Fj represents the known external force on
the jth particle in a two-particle system. (The negative sign appearing in Eq. (8.8)
is a consequence of the shift from the force exerted by the particle – as in Eq. (8.1)
– to the force exerted on the ith primary sphere by the fluid.) To determine the
force on a macromolecule or particle whose subunits have velocities Ui, one solves
the above linear system for the force on each sphere, then sums over all spheres
to get the total force. Alternatively, if the total external force on the particle is
given (e.g. gravity or the force on charged particle in an electric field), one can
solve Eq. (8.8) iteratively to get the particle velocity such that the calculated force
distribution is consistent with the total force.
208
KR-based treatments rely on the underlying linearity of the Stokes flow equa-
tions. In their original formulation, Kirkwood and Riseman [28] set Mij = δijζ
−1
i +
(1 − δij)Tij, where δij is the Kronecker delta and Tij is the Oseen tensor. Later
applications of KR theory [30, 56, 107] replace the Oseen tensor with the Rotne-
Prager-Yamakawa tensor. One can use higher order approximations for the mobility
tensors (e.g. the Felderhof expressions); one can also account for multi-body effects
using mobility tensors derived by Mazur and Van Saarloos [60]. However, Carrasco
and Garcıa de la Torre [53] have shown that using these more rigorous methods to
calculate a particles friction and diffusion tensors offer little improvement over using
KR-theory with the Rotne-Prager-Yamakawa tensor.
One can use the same KR-based methods to determine the forces between two
aggregates: one again solves the system of equations given by Eq. (8.8), but now
one sums over the spheres in each aggregate to determine the total hydrodynamic
force. Again, one can perform this calculation for aggregates with specified velocities
or specified external forces. By comparing this result to the force on an isolated
aggregate, one can determine the effect of the second aggregate on the mobility of
the first.
8.2.3 Extended Kirkwood-Riseman theory
The previous subsections have focused on situations where the continuum ap-
proach is valid. For particles consisting of spheres with diameters on the order
of tens of nanometers moving through air at standard temperature and pressure
(λ ≈ 67 nm), one cannot use the expressions for interactions between particles in
209
the continuum, but one can use the same general Eq. (8.8) relating the velocity of
particles in the transition regime to the drag forces exerted on the particles by the
fluid. One only needs to determine the mobility tensors as a function of the primary
sphere Knudsen numbers.
Noting that the Stokes velocity tensor provides a reasonable approximation for
the mobility tensor between two spheres in the continuum (see Tables 8.1-8.3), I [92]
proposed replacing the Stokes velocity tensor with a Knudsen-number-dependent
velocity tensor obtained by solving the Bhatnagar-Gross-Krook (BGK) equation
[71] for flow around a sphere. (The BGK equation is a simplified, linearized version
of the Boltzmann transport equation that is valid for near-equilibrium situations
[72]). Velocities calculated using the BGK equation compare well to solutions of the
linearized Boltzmann equation [77, 78].) The resulting set of equations relating the
sphere velocities to the drag is
∑N
Fi = −ζ (Kn −1i i)Ui − ζi(Kni) ζj (Knj)Vij(Knj) ·Fj (8.9)
i 6=j
Note that the force on sphere i depends on the friction coefficient for sphere i and
the velocities around the other j spheres.
My previous work [92, 93] has demonstrated that using the velocity and drag
results from the BGK equation and solving Eq. (8.9) yields accurate results for
the translational friction coefficient of fractal aggregates across the entire Knudsen
regime. This suggests that the EKR method can be used to determine the hydro-
dynamic forces between particles in the kinetic regime.
210
8.2.4 Point force approach
If one is only concerned about interactions between widely separated particles,
the analysis can be greatly simplified by treating the particles as point forces. In the
continuum, the flow field generated by the point force is often called the Stokeslet
and can be written as
1 ( rr)
u(r) = I + ·F (8.10)
8πµr r2
where the term multiplying the point force is the Oseen tensor. If the object is a
sphere, F is given by Stokes law, and Eq. (8.10) gives the flow field for large r.
Interestingly, this flow field is valid regardless of the shape of the object exerting
the force on the fluid. This means that the flow field far from an object looks like
the flow around a sphere exerting the same force on the fluid as the object, even
when the object is highly non-spherical. This behavior is due to viscous effects in
the fluid.
Based on the preceding discussion, we can determine the interactions between
the aggregates using a point force approach, such that the total hydrodynamic force
on each aggregate can be obtained by simultaneously solving the following equations:
∑N [ ( )]
Fi = −Ξi,0 ·
1 rr
Ui −Ξi,0 · I + ·F (8.11)
8πµr r2
j
ij
i 6=j
Here, N is the number of aggregates in the system, rij is the vector between the
center of mass of ith and jth aggregates and Ξi,0 is the translational friction tensor
for aggregate i when it is alone in an infinite fluid. Mackaplow and Shaqfeh [138]
211
demonstrated that the point force approach applied to the problem of sedimentation
of fibers in a semi-dilute homogeneous suspension, yields results similar to those of
a more sophisticated Monte Carlo simulation of the problem.
The point force approach should also be valid in the transition regime, albeit
with a Knudsen-number-dependent hydrodynamic interaction tensor in place of the
Oseen tensor. In fact, the transition regime interaction tensor should have the same
form as the Oseen tensor but with a different coefficient in place of 1/8. This is
because the fluid velocity far from a sphere has the general form
c1(Kn) a
( rr) c1(Kn) a ( rr)
u(r) = − I + ·U = − I + ·F (8.12)
2 r r2 2ζi,0(Kn) r r2
where U is the sphere velocity, F is the force exerted by the sphere on the fluid,
and c1(Kn) is a coefficient that can be obtained by solving the BGK or linearized
Boltzmann equation for flow around a sphere [78, 92]. In the continuum, c1 = −3/2,
and Eq. (8.12) reduces to Eq. (8.10).
Substituting Eq. (8.12) for the Oseen tensor in Eq. (8.11), we get the following
result for a system of N aerosol aggregates:
∑N [ c (Kn ( )]1 j,eff) rr
Fi = −Ξi,0(Kni) ·Ui −Ξi,0(Kni) · − I + ·F (8.13)
2ζ r2
j
6 j,0i=j
Here, aj,eff is the radius and Knj,eff = λ/aj,eff the Knudsen number of a sphere with
friction coefficient ζj,0 = |Ξj,0(Knj) ·Uj/Uj|, where Ξj,0(Knj) is the friction tensor
for aggregate j alone in an infinite fluid and Uj is the velocity of aggregate j. One
212
solves implicitly for the effective radius of the cluster,
6πµaj,eff
ζj,0(Knj,eff) = (8.14)
Cc(λ/aj,eff)
where Cc is the Cunningham slip correction factor. I have explicitly stated the de-
pendence of the aggregate friction tensors on the Knudsen number of the primary
spheres in the aggregate; of course, the friction tensor also depends on the num-
ber of spheres in the aggregate and the configuration of those spheres. Note that
Eq. (8.14) accounts for the orientation of each cluster through the use of the friction
tensors, and by determining the effective radius based on the drag force for a par-
ticular aggregate orientation relative to the flow. However, one can further simplify
Eq. (8.13) by replacing the friction tensors with the scalar friction coefficients based
on orientation averages for the particles. This simplification is justified for aerosol
particles due to the randomizing effects of Brownian forces.
To apply the point force approximation for a system of aggregates in the
transition regime, one must first determine the friction tensor for each aggregate
[e.g. by solving Eq. (8.9)]. Next, one must determine the effective sphere radius
for each aggregate. One obtains the coefficient c1(Knj,eff) for each aggregate from a
table of BGK or linearized Boltzmann results. With this information, one can solve
Eq. (8.13) for the hydrodynamic forces on the system of aggregates. Again, one can
replace the friction tensors with the scalar friction coefficient for each aggregate.
One can use the adjusted sphere method [41, 67] or my EKR method [92] to quickly
determine the friction coefficient for each aggregate, which significantly reduces the
213
complexity of the problem.
Using the point force approach is beneficial because one can significantly reduce
the number of simultaneous equations that one must solve compared to considering
the pairwise interactions among all of the spheres in all of the aggregates. This
is especially useful if one is considering a system consisting of identical aggregates
(such that one need only determine the friction tensor once) or if one is considering
interactions between coagulating aerosols (such that the friction tensors for the ag-
gregates remain constant until two particles coagulate, at which point one need only
calculate the friction tensor for the new particle). While the point force approach is
strictly valid only for widely separated particles, I will show in the following section
that this approach provides reasonable results even when the aggregates are fairly
close together.
8.3 Two particle results
I have used my extended Kirkwood-Riseman method [Eq. (8.9)] to determine
the hydrodynamic forces between two aggregates moving parallel, anti-parallel, and
perpendicular to their line of centers in the transition flow regime. I start by consid-
ering two spheres, then move to the case of two fractal aggregates. For simplicity,
I will focus on cases where the two particles are identical, though I do consider the
effects of orientation for the aggregate calculations.
214
8.3.1 Sphere results
Figure 8.2 shows the results for two spheres moving parallel, anti-parallel,
and perpendicular to their line of centers. Results are plotted as a function of
separation distance for three different Knudsen numbers. Here, I show the ratio of
the velocity of each sphere subjected to the same external force (albeit in opposite
directions for the anti-parallel case) to the velocity it would have if was alone in
an infinite medium subjected to the same force. Note that one can obtain the
effect on the hydrodynamic force for specified velocity by taking the inverse of the
results in Figure 2. I also present the velocities calculated using Felderhofs mobility
tensors [97] for mixed slip-stick boundary conditions for Kn = 0.1. These tensors
are accurate to order r−7 and are expected to yield nearly exact results except for
very small separation distances for this near-continuum situation.
The figure shows that my EKR results are in good agreement with the Felder-
hof mobility tensor results for near-continuum conditions (or near-stick boundary
conditions, in the case of the Felderhof results). For r/a > 3, the two methods yield
results within about 2% of each other; this is not surprising, since the higher order
terms in the expansions representing the mobility tensors decay quickly with in-
creasing separation. The error is still fairly low (< 8%) for touching spheres moving
parallel and perpendicular to their line of centers but is much more significant for
the anti-parallel configuration. This is because my EKR method ignores lubrication
forces between the particles, whereas Felderhofs mobility tensors partially account
for these forces through the higher order terms in the expansion.
215
Figure 8.2: Speed of two spheres moving (a) parallel, (b) anti-parallel, and (c) per-
pendicular to their line of centers, relative to the speed of isolated spheres subjected
to the same external force. Results using the mobility tensors of Felderhof [97] for
mixed slip-stick boundary conditions are shown for comparison to the Kn = 0.1
results. Note that for the Felderhof calculations I use a slip length of 0.9875 times
the gas mean free path based on the best-estimate results of Loyalka [139].
216
As expected, the strength of the hydrodynamic interactions between spheres
decreases with increasing Knudsen number. For the perpendicular configuration, the
two spheres behave almost as if they are isolated even when they are in contact at
high Knudsen number; the effect is more important in the parallel and anti-parallel
configurations, likely due to direct shielding of incoming gas molecules by the other
sphere and molecules that collide with both spheres before colliding with other gas
molecules.
As we saw in the previous section, the long-range hydrodynamic interactions
have a r−1 dependence in the continuum, as given by the Oseen tensor. The r−1
dependence is also evident in my transition regime results in Figure 8.2. The coef-
ficient of the r−1 term is a function of the Knudsen number. This result is directly
related to the asymptotic behavior of the velocity field at large distances from a
sphere, as I have discussed in Section 8.2.4 above.
8.3.2 Aggregate results
Figure 8.3 shows the results for two aggregates moving parallel to their line of
centers. Results are shown for two aggregates with 10 spheres each and for two ag-
gregates with 1000 spheres. The results are presented as the ratio of the drag on one
of the aggregates to the drag on that aggregate when it is alone in an infinite fluid.
Solid lines represent the full EKR solution [Eq. (8.9)], while the dotted lines repre-
sent the point force results [Eq. (8.13)]. From the figures, the point force solution
is sufficiently accurate for all but the closest separations, which greatly simplifies
any analysis of the effects of hydrodynamic interactions between aggregates. Also,
217
Figure 8.3: Hydrodynamic force on an aggregate as a function of the distance be-
tween its center of mass and the center of mass of an identical aggregate. The left
plot shows results for N = 10, while the right plot shows results for N = 1000. The
solid lines represent the full EKR results [Eq. (8.9)], while the dotted lines represent
the point force results [Eq. (8.13)]. The drag force is normalized by the drag on one
of the aggregates in an infinite fluid, while the separation distance is presented as
the number of effective sphere radii (i.e. the radius of a sphere experiencing drag
force F0) between the centers of mass of the particles.
the aggregates exhibit more continuum-like behavior as the number of spheres in
the aggregate increases, which is analogous to the behavior I have observed for the
translational and rotational friction coefficients of DLCA aggregates (Chapters 4
and 6).
Figure 8.3 presents the results for aggregates with a fixed orientation. Fig-
ure 8.4 explores the effect of orientation on the reduction in the drag force for
N = 500 and a primary sphere Knudsen number of 2.7. Results are shown for two
aggregates with a shape anisotropy A31 = 2.0 and for two aggregates with A31 = 6.5,
where A31 is the ratio of the largest to smallest eigenvalues of the inertia tensor of
the particle. Since each cluster in the first pair is fairly isotropic (A31 near unity),
there is little change in the hydrodynamic force on each cluster as the particles ro-
tate. In contrast, there is a noticeable difference in hydrodynamic force as the more
218
Figure 8.4: Effects of orientation on the hydrodynamic force on one of two 500-
sphere aggregates with primary sphere Kn = 2.7 (sphere radius of 25 nm at room
temperature). Results are presented as the ratio of the force on an aggregate for the
specified separation distance to the force on the aggregate alone in an infinite fluid
at a particular orientation. Each line represents a randomly chosen orientation for
each aggregate in the two-aggregate system. The left figure is for an aggregate with
anisotropy A31 = 2.0, while the right figure is for an aggregate with A31 = 6.5.
anisotropic particles rotate. This is especially true for small separation distances.
Note that in each case I normalize by the force on the particle for that particular
orientation.
8.4 Discussion
My results show that hydrodynamic interactions between particles in the tran-
sition flow regime are significant for small separation distances and still noticeable
at separations of more than 10 sphere diameters or 20 times the radius of gyra-
tion of an aggregate. The effect is stronger near the continuum than near the free
molecule regime and stronger for parallel and anti-parallel configurations than for
perpendicular configurations for all particle sizes.
With that said, hydrodynamic interactions likely have little effect on mea-
219
surable quantities of interest (e.g. deposition and coagulation rates) except in very
dense aerosol systems, due to the large average particle separations in most practical
systems. As an example, consider an aerosol consisting of spherical particles with
volume fraction φ. The average spacing between the particles is L̄/a = (4π/3φ)1/3,
so at a relatively high volume fraction of 10−4, the average spacing approximately
35 sphere radii. Near the continuum, the hydrodynamic force between spheres is
less than 5% of the external force generating the sphere motion; this ratio decreases
with increasing Knudsen number.
Of course, aerosol particles are not arranged in a grid, with equal spacing
between particles. Still, the expressions for the settling velocity and suspension vis-
cosity in the introduction account for the probability distribution of sphere locations
in a suspension. Thus, interparticle hydrodynamic interactions have very little im-
pact on these parameters for an aerosol consisting of spherical particles at typical
volume fractions.
The situation is more complicated for fractal aggregates due to the difference
between the fractal dimension and the spatial dimension. To understand why, let us
consider an aerosol consisting of a large number of identical N -sphere aggregates.
The average distance between the centers of mass of the aggregates is simply L̄/a =
(4πN/3π)1/3, regardless of the fractal dimension of the aggregates. Now let us
rewrite the average spacing in terms of the radius of gyration:
( )
3/d 1/3
L̄ (4πN/3φ)1/3 4πk f df−3
= = 0 N 3df (8.15)
Rg (N/k 1/df0) 3φ
220
Here, k0 and df are the prefactor and fractal dimension of the aggregates. For
DLCA aggregates (k −40 ≈ 1.3, df ≈ 1.78), and for a volume fraction of 10 , the
average spacing between aggregates is 24 radii of gyration for N = 10 and 8.3 radii
of gyration for N = 1000. Since the radius of gyration is approximately equal to the
effective hydrodynamic radius of a particle in continuum flow, we can consult the
near-continuum results in Figure 8.3 to estimate the hydrodynamic force between the
aggregates in the parallel flow configuration at these average separation distances.
For N = 1000, the drag force for the parallel configuration is approximately 85% of
the drag at infinite separation distance for Kn = 0.1. Compare this to the result for
a system of spheres in continuum flow, where the average separation is 34 radii, and
the force at this separation is 96% of the force at infinite separation for Kn = 0.1.
8.4.1 Aerosol clouds
One interesting problem involving hydrodynamic interactions among particles
is the settling of an unbounded aerosol cloud in a gravitational field. This prob-
lem has been studied extensively for spherical particles in the continuum. (For a
small sample, refer to the works of Burgers [131] or to Fuchs [1].) The behavior
of the particle cloud depends on the volume fraction of particles in the cloud. For
sufficiently low volume fractions, the particles behave as if they are isolated, as one
would expect. In this case, the velocity of each particle is obtained by balancing the
gravitational force Fg with the drag force,
Fg
vs1 = (8.16)
ζ0
221
For high volume fractions, the particles entrain the surrounding fluid, and the cloud
behaves as a gas bubble with the same viscosity as the fluid surrounding the cloud
[1, 131]. In this case, the velocity of the cloud is given by
4πR3nF 2
v = 3
g 4R nFg
s2 = (8.17)
5πµR 15µ
where R is the radius of the cloud and n is the number concentration of particles.
In the first equality, I explicitly write the velocity as the ratio between the total
gravitational force on the cloud and the drag on a spherical bubble with equal
viscosity in the inner and outer fluids. The particles behave as if they are isolated
when vs2/vs1  1 and as a cloud with velocity given by Eq. (8.17) when vs2/vs1  1
[1].
The above analysis is applicable for aggregates in continuum flow, with a few
important caveats. First, one must use appropriate values for the gravitational and
drag forces on the aggregate. Second, the effects of the aggregates on the effective
viscosity of the cloud must be negligible; of course, this restriction also applies to
the aforementioned case of spherical particles. Additionally, I will ignore the effects
of hydrodynamic interactions on the orientation of the aggregates. I will address
this restriction in more detail in the following section. With the above caveats, we
can write the ratio vs2/vs1 for a cloud consisting of identical N -sphere aggregates as
vs2 6φζ
?R2
= 0 (8.18)
vs1 5Na2
222
where ζ?0 ≡ 6πµaζ0 is the non-dimensional friction coefficient for the aggregate when
it is alone in an infinite fluid. Note that the volume fraction is simply the number
density times the volume of one aggregate, i.e. φ = 4πa3Nn.
3
Now, I will consider whether or not the above analysis applies to particles
– whether spheres or aggregates – in the transition regime. (Again, if one wishes
to consider aggregates, the analysis is subject to the caveats mentioned above for
particles in continuum flow.) To do so, I will consider an idealized situation: a cloud
of radius R that consists of identical particles in a cubic lattice. For this problem,
each particle experiences the same external force, so we can directly solve Eq. (8.13)
for the velocity of each particle in the cloud, Ui. If Fuchs analysis is applicable for
particles in the transition regime, then the calculated velocity should approach that
given by Eq. (8.17) when the ratio vs2/vs1 [Eq. (8.18)] is much greater than unity
and should approach Fg/ζ0 when the result of Eq. (8.18) is much less than unity.
Figure 8.5 shows the average particle velocity in the aerosol cloud as a function
of the aerosol volume fraction. The calculations are performed for 25-nm-radius
spheres (Kn = 2.7) and a 25-m-radius cloud. Average velocities are normalized by
the settling velocity of an isolated sphere (i.e. by vs1). The figure shows that for
vs2/vs1  1, the average velocity of particles in the cloud approaches vs2, and for
vs2/vs1  1, the particles behave as if they are isolated.
These results show that a cloud of aerosol particles can behave as a gas droplet
in continuum flow even when the individual particles are small relative to the mean
free path of the gas. This occurs for sufficiently large cloud sizes and particle volume
fractions.
223
Figure 8.5: Average velocity for a cloud of spheres, normalized by the settling
velocity of a single particle, vs1. The velocity of a gas bubble having the same
volume fraction and cloud radius is also shown. For vs2/vs1  1, the average velocity
approaches the gas bubble velocity, and the two curves coincide. For vs2/vs1  1,
the particles behave as if they are isolated.
224
Of course, in real-world situations the cloud will not be perfectly spherical, nor
will it be composed of monodisperse particles arranged in a regular grid. Neverthe-
less, we can reach some qualitative conclusions about the behavior of a real-world
cloud of aerosol particles from the results of the ideal case. First, the deposition
velocity of the particles in the cloud may be significantly greater than the deposition
velocity of an isolated particle. This enhanced settling velocity is due to the fact
that the particles entrain the surrounding fluid. Second, a collection of particles
that behave as if they are in the transition flow regime when they are isolated (or
when the volume fraction is very small) can behave like a cloud of continuum par-
ticles when the particle volume fraction increases. This is because the long-range
hydrodynamic interactions between particles decay as 1/r, regardless of the size of
the particle relative to the gas mean free path.
8.4.2 Additional considerations
Before concluding, I will address a number of assumptions that I have made for
my analysis. First, I have ignored rotational effects when considering interactions
between particles. The complete description of two particles has the same general
form as Eq. (8.1), but one replaces the 3-element velocity and force vectors with
6-element vectors that include the angular velocity of and torque exerted by the
particle, respectively. Likewise, one would need to incorporate rotational effects in
the generalized mobility tensors Mij, which now have size 6-by-6. See Brenner [29]
or Carrasco and Garcıa de la Torre [53] for the effects of rotation on a single particle
in the continuum and my previous work [94] for how one might consider rotational
225
effects in the kinetic regime. The two-particle principles follow from the application
of KR-based theories to single-particle systems.
Second, I have mostly ignored Brownian effects in my calculations, except to
note that Brownian forces would likely randomize the orientations of aggregates in
an aerosol. Strictly speaking, one must account for hydrodynamic interactions when
determining the probability distribution for the particle orientation. For example,
Mackaplow and Shaqfeh [138] have performed dynamic simulations showing non-
Brownian fibers aligning with gravity due to hydrodynamic effects. Therefore, the
orientation of aerosol aggregates will depend on the competition between hydrody-
namic effects that tend to align particles and Brownian effects that tend to random-
ize their orientation. This is analogous to the behavior of a perfectly-conducting
aerosol particle in an external electric field [96, 105].
Additionally, Brownian forces affect the spatial distribution of particles in a
cloud or suspension. Diffusion is partially responsible (along with shear) for breakup
of clouds [1], so the analysis in the previous section applies only for some initial pe-
riod while the cloud remains intact. For statistically homogeneous systems, one
must account for the particle probability distribution when determining suspension
behavior. In principle, one can apply the methods used in Stokesian dynamics sim-
ulations (see, e.g., Ermak and McCammon [135] and Brady and Bossis [136]) for
non-continuum flows, provided one modifies the hydrodynamic interaction terms
in the algorithms. I have provided here the translational hydrodynamic interac-
tion tensors, while I describe how to account for rotational and translation-rotation
coupling effects elsewhere (Chapter 5).
226
Third, my calculations have been performed for isothermal conditions. As
Fuchs notes, cloud behavior is often driven by temperature differences between the
cloud interior and the surrounding gas [1]. Because aerosols are often generated in
non-isothermal systems (e.g. flames), one must take care in applying my analysis to
real-world systems.
Finally, my calculations assume the particles are in an unbounded, non-periodic
system. Note that in periodic, statistically homogeneous suspensions, the sedimen-
tation velocity decreases with increasing particle volume fraction [128]. In contrast,
unbounded clouds of particles settle at a faster rate as volume fraction increases, as
is clear from Eq. (8.18). The difference is due to the gas behavior: for an unbounded
cloud, the gas can flow around the cloud to occupy the space vacated by the settling
cloud, whereas the gas must flow between the particles in the opposite direction of
the sedimentation velocity. This counter flow serves to reduce the sedimentation
velocity of the periodic suspension.
8.5 Conclusions
I have described how one would apply my extended Kirkwood-Riseman theory
to interactions between spheres and aggregate particles as a function of the distance
between particles and the particle size and shape (number of spheres, primary sphere
Knudsen number, fractal dimension). I have provided sample results for the effects
of hydrodynamic interactions between two spheres and two aggregates. In both
cases, the interactions are weaker than for particles in continuum flow, and the
227
interaction strength decreases with increasing Knudsen number. Note that in many
aerosols of engineering interest, the particle volume fraction is very small, and on
average one can rightly neglect these interactions. However, for aerosols near the
gel point, such interactions may be important.
I have applied my method to an unbounded cloud of spheres with non-negligible
Knudsen numbers and shown that it behaves similarly to a cloud of larger particles.
This analysis applies to an isothermal, spherical cloud of particles arranged in a reg-
ular grid, but the conclusion that hydrodynamic interactions among a large group
of particles can significantly affect settling behavior also applies in less restrictive
conditions.
The above work provides some of the hydrodynamic interaction terms neces-
sary to perform a Brownian dynamics simulation of an aerosol consisting of parti-
cles with non-negligible Knudsen numbers. One could either apply the point force
method described here or my more complicated EKR method to account for Brow-
nian (through the generalized Stokes-Einstein relation; see Ermak and McCammon
[135]) and hydrodynamic effects on the aerosol behavior. One can also account for
rotational effects using the EKR method, as discussed elsewhere [94]. Taken to-
gether, this work can form the basis for a dynamic simulation of a dense aerosol
system.
228
Chapter 9: NGDE: A MATLAB-based Code for Solving the Aerosol
General Dynamic Equation
The previous chapters in this dissertation have largely concentrated on single-
particle behavior (the exception being the study of interparticle hydrodynamic inter-
actions in Chapter 8). For the present chapter, I will shift my focus to the dynamic
behavior of an aerosol system undergoing nucleation, surface growth, and coagula-
tion. Note that this chapter provides the technical background for the NGDE code.
The User Manual for the code is included as Appendix F. One can obtain the NGDE
code on the Zachariah Group website.
9.1 Introduction
The behavior of an aerosol system is governed by the general dynamic equation
[2, 3, 45], which can be written as
∫ ∫
∂n 1 v ∞
= β(v − v′, v′)n(v′, t)n(v − v′, t)dv′ − β(v, v′)n(v, t)n(v′, t)dv′
∂t 2 0 0
− ∂ [n(v, t)G(v, t)] + S(v, t)−R(v, t) (9.1)
∂v
229
Here, n(v, t) is the particle size distribution (PSD) as a function of time t and particle
volume v, where n(v, t)dv is the number of aerosol particles per unit volume of gas
with particle volume between v and v+dv; β(v, v′) is the collision kernel for particles
with volumes v and v′; G(v, t) is the growth rate of particles with volume v due to
condensation/evaporation; S(v, t) is the rate at which particles are added to the
system (e.g. by homogeneous nucleation); and R(v, t) is the rate at which particles
are removed from the system (e.g. by deposition on surfaces). The integrals represent
how the size distribution changes due to coagulation.1
Eq. (9.1) is non-linear integro-differential equation that can be solved ana-
lytically only for a small number of cases. (See, for example, Refs. [140, 141]).
Thus, one must typically resort to numerical methods to solve for the evolution of
the PSD with time. However, solving the GDE numerically is complicated by the
orders-of-magnitude difference in volume between the smallest and largest particles
in an aerosol. As suggested by the bounds of the integrals in Eq. (9.1), one must
determine the size distribution for an infinite range of particle sizes. Fortunately,
the physical characteristics of the system allow one to establish upper and lower
bounds for the size distribution. For example, large particles may deposit quickly
due to gravitational forces, so one can assume particles greater than some upper
bound have an infinite removal rate. On the other hand, no particle can be smaller
than a single molecule (ignoring of course the subatomic domain), thus establishing
a natural lower bound for the size distribution. In addition, thermodynamic con-
1The integrals in Eq. (9.1) only account for growth due to particle-particle collisions. Of course,
it is possible that two particles could collide with sufficient energy to break them into smaller
particles, but such high-energy collisions are rare in situations of interest to the aerosol scientist
and are therefore ignored in Eq. (9.1).
230
siderations limit the smallest stable size of molecular clusters, such that one can
ignore particles consisting of fewer molecules than some critical size. Even with
these simplifications, solving the GDE numerically is hardly a trivial endeavor.
This chapter describes a fairly simple method for solving the general dynamic
equation: a MATLAB-based code using a nodal method that is similar in principle
to the sectional method pioneered by Gelbard and colleagues [26, 142, 143]. The
MATLAB version of the NGDE code is based on an earlier version of the code [27]
written in C. A number of new features have added to the MATLAB version of the
code. The most significant additions are the implementation of a dynamic time-step
algorithm that significantly speeds up the calculation run-time and the introduction
of a post-processing tool for visualizing the code results.
Before the NGDE code is described in detail, a brief overview of various tech-
niques used to solve the GDE numerically is provided. Next, the constituent models
for nucleation, coagulation, and surface growth used in NGDE; the dynamic time-
step algorithm; and the available calculation types and general solution procedure
are described. Results are presented for a few sample problems. These results are
compared to the results of other numerical solutions of the problems published in the
literature. The chapter closes with a discussion about the limitations of NGDE and
the ways in which one may improve the code, as well as a brief introduction to the
post-processing tool NGDEplot. More details about running NGDE, code input and
output, and code structure can be found in the NGDE User Manual (Appendix F).
231
9.2 Overview of Numerical Methods for Solving the GDE
Numerical methods for solving the GDE can be divided into three broad
classes: J-space methods, moment methods, and sectional methods [45]. J-space
methods [144] involve transforming the continuous PSD to J-space, integrating the
J-space distribution function with respect to time, and taking the inverse transform
to obtain the PSD as a function of time. Moment methods [145] start by assuming
some form for the size distribution and its moments (i.e. the product of the distri-
bution and powers of the particle volume, integrated over all volumes) and involve
solving for the evolution of the moments of the size distribution. In addition to the
integration schemes needed to solve for the time-dependent evolution of the PSD
(e.g. the Euler method or Kunge-Kutta method), both J-space and moment meth-
ods require the use of quadrature formulas to integrate the distribution or moment
equations with respect to volume. For this reason, these methods are somewhat
mathematically complex and can prove a hindrance for the novice aerosol scientist.
On the other hand, sectional methods [26, 142, 143] are more intuitive ways of
solving the general dynamic equation that make fewer assumptions about the shape
of the size distribution. These methods divide the size distribution into discrete size
bins and solve the GDE for each size bin as a system of coupled equations. The
number concentration of particles in bin k at time t is defined as
∫ vk
Nk(t) = n(v, t)dv (9.2)
vk−1
232
where vk−1 and vk are the lower and upper bounds (in terms of volume) of bin k.
The evolution of the number concentration of particles in bin k can be written as
∣∣∣∣ ∣∣∣∣ ∣ ∣dNk dNk dNk dNk ∣∣∣ dNk ∣= + + + ∣∣ (9.3)dt dt coag dt nucl dt growth dt dep
where the first, second, and third terms on the right-hand-side represent the changes
in the number concentration in bin k due to coagulation, nucleation, and surface
growth. These terms will be discussed later in this chapter. The final term in the
equation represents losses due to deposition. Deposition is not considered in NGDE;
nevertheless, this term is included in the above equation for completeness, in case
one wishes to include deposition in NGDE in the future. (See Section 9.5.) For the
sectional method, one must also track the total volume (or mass) of particles within
each size bin, ∫ vk
Vk(t) = vn(v, t)dv (9.4)
vk−1
using an equation similar to the equation above for the number concentration.
One complication inherent in sectional methods is that one must use average
properties within each bin to calculate the coagulation and growth rates. For ex-
ample, consider the increase in the number concentration of particles in bin k due
to coagulation of particles in bins i and j, β̄ijkNiNj. Here, β̄ijk must represent the
collision kernel for all particles with volumes within bins i and j that combine to
233
form particles with volumes within bin k. This collision kernel is defined as
∫ v ∫i vj θ(v′, v′ , v )β(v′, v′ )n(v′, t)n(v′k , t)dv′dv′
≡ vi−1 vβ̄ j−1
i j i j i j j i
ijk (9.5)
Ni(t)Nj(t)
where θ(v′i, v
′ ′ ′
j, vk) is a function such that θ = 1 when vk−1 < vi + vj < vk and θ = 0
otherwise [142]. In other words, θ ensures that only collisions between particles
whose combined volume is within bin k are included in the average collision kernel.
To evaluate β̄ijk, one must properly define the function θ, which is difficult to do
in practice. Furthermore, one must make some assumption about the form of the
particle size distribution within each bin, i.e. n(v, t) for vi−1 < v < vi and vj−1 <
v < vj. Similar considerations apply when determining the rates of condensation
and evaporation from particles in bin k.
These issues can be avoided by replacing the finite bins with discrete particle
size nodes, such that particles can only exist at the nodes. Thus, there is no need to
determine average properties within each size bin; instead, one need only determine
the collision kernel for discrete particle sizes. This nodal method is markedly simpler
than standard sectional methods, making it an ideal solution technique for anyone
first learning about the general dynamic equation (or for anyone who wants to
avoid the mathematical complexities of more rigorous techniques). While it has its
limitations – which will be discussed later in this chapter – the nodal method yields
results that are sufficiently accurate for many applications.
234
Figure 9.1: NGDE volume nodes are equally spaced on a logarithmic scale covering
12 orders of magnitude. (This figure original appears as Figure 1 in Prakash et al.
[27].)
9.3 NGDE Code Description
NGDE solves the general dynamic equation – specifically, the form given by
Eq. (9.3) – for the particle size distribution. The PSD is divided into discrete size
nodes; by default, the size nodes cover 12 orders of magnitude in particle volume,
3
from the size of a monomer (∼ 10−29 m ) to particles with diameters of a few microns
(∼ 10−17 3m ). The user chooses the number of volume nodes for determining the
PSD; the default is 41. The nodes are equally-spaced on a logarithmic scale in terms
of particle volume, as shown in Figure 9.1.
NGDE can perform four different calculations: (1) coagulation only; (2) co-
agulation plus nucleation; (3) coagulation, nucleation, and surface growth (i.e. the
full GDE); and (4) pure surface growth. The user input – including specification
of the calculation type and the number of nodes for the PSD – takes the form of
a MATLAB data structure. For details about the input options, please see the
235
NGDE User Manual (Appendix F). The pure coagulation calculation begins with
a monodisperse aerosol and continues until some fixed end-time. Currently, this
end time is 1000 times the estimated time to reach the self-preserving PSD. (See
Section 9.4.1 for information about the self-preserving size distribution.) The other
three calculation types begin with a slightly supersaturated (S = 1.001, where S
is the saturation ratio) vapor at some user-specified temperature. The calculation
proceeds as the system cools from the initial temperature to some end temperature
(300 K by default) at the specified cooldown rate. In all cases, NGDE determines
the coagulation, nucleation, and/or surface growth rates at each time step based on
the current conditions in the system. The details of how the code calculates these
rates and about the integration scheme are given below.
9.3.1 Coagulation
Coagulation – which occurs when two particles collide and combine to form a
larger particle – is governed by the Smoluchowski equation,
∣∣∣∣ ∑M ∑M ∑MdNk 1= χijkβi,jNiNj −Nk βi,kNi (9.6)dt coag 2 i=2 j=2 i=2
where M is the number of volume nodes; χijk is a size-splitting operator for nodes
i, j, and k; and βi,j ≡ β(vi, vj) is the collision kernel between nodes i and j. The
summations begin at node 2 because node 1 represents monomers (i.e. the vapor
form of the particulate species).2 Here, particles with volumes vi and vj collide to
2Note that for pure coagulation, node 1 represents particles, so the summations in Eq. (9.6)
begin at i = 1, j = 1.
236
form a larger particle with volume vi + vj.
The size-splitting operator is defined as follows [27]:
v −(v +v ) k+1 i j− if v ≤ v + v ≤ vv v k i j k+1k+1 k
χijk =  (vi+vj)−vk−1 − if v
(9.7)
v v − k−1
≤ vi + vj ≤ vk
 k k 10 otherwise
Since newly-formed particles with volume vi + vj likely fall between two volume
nodes, χijk divides the number of coagulated particles into nodes with volumes just
larger and just smaller than vi + vj. To do so, χijk simply uses linear interpolation
based on particle volume. The operator also ensures that collisions between particles
with volumes vi and vj only contribute to the growth of particles with volume vk if
vk−1 ≤ vi + vj < vk+1. The NGDE code assumes perfect coalescence upon collision,
which means that all particles are spherical.
Currently, users have two choices for calculating the collision kernel βi,j: a
form based on kinetic theory in the free molecule regime (i.e. Kn  1) and Fuchs’
form of the collision kernel for the transition regime (i.e. Kn ∼ 1), where Kn ≡ λ/a
is the Knudsen number for particles with radius a in a background gas with mean
free path λ. The free molecule form of the collision kernel is [2]
( )1/6( )1/2( )1/2
≡ 3 6kBT 1 1 1/3 1/3βi,j β(vi, vj) = + (v + v 2
4π ρ v v i j
) (9.8)
p i j
where kB is the Boltzmann constant, T is the gas temperature, and ρp is the particle
237
density. Fuchs’ form [1] is given by [146]
[ ]−1
dpi + dpj 8(Di +Dj)
βi,j = 2πDiDj(dpi + dpj) +
d + d + 2(g2 + g2)1/2 (c̄2 + c̄2)1/2pi pj i j i j (dpi + dpj)
(9.9)
where
2λ
Kni = (9.10a)
( dp)i 1/2
8kBT
c̄i = (9.10b)
πmi
8Di
`i = (9.10c)
πc̄i
1
g = [(d + ` 3 2 2 3/2i pi i) − (dpi + `i ) ]− dpi (9.10d)3dpi`i ( )
kBT 5 + 4Kni + 6Kn
2
i + 18Kn
3
D = ii (9.10e)
3πµdpi 5−Kni + (8 + π)Kn2i
and dpi, is the diameter, mi the mass, c̄i the mean thermal speed, and Di the
diffusion coefficient of particles with volume vi.
9.3.2 Nucleation
Nucleation occurs when a non-equilibrium, supersaturated vapor condenses
to return to equilibrium; condensation on existing particles is known as heteroge-
neous nucleation, while condensation to form new particles is called homogeneous
nucleation [2]. In the context of the NGDE code, nucleation strictly refers to ho-
mogeneous nucleation; condensation on existing particles falls under the purview of
surface growth, as discussed in the next subsection.
238
New particles form when monomers collide to form stable molecular clusters.
Clusters with fewer monomers than the critical cluster size k? are unstable and
quickly dissociate, while clusters consisting of k? monomers are stable and tend to
grow rapidly. The critical cluster size is a function of the vapor properties and the
thermodynamic conditions (e.g. saturation ratio, temperature, pressure) of the sys-
tem. There are a number of models for calculating the critical cluster size and the
nucleation rate; NGDE uses classical nucleation theory with the self-consistent cor-
rection (SCC) proposed by Girshick and Chiu [147]. For this model, the nucleation
rate Jk and critical cluster size k
? are
( )0.5 ( )
2 2σ 4θ
3
Jk = nsSv1 exp θ − (9.11)πm1 27 ln2 S
( )3
? 2 θk = (9.12)
3 lnS
This corresponds to a critical particle volume of
( )3
v? = k?
π 4σv1
v1 = (9.13)
6 kBT lnS
In these equations, ns is the monomer concentration at saturation; σ is the sur-
face tension of the condensed species; θ = s1σ/kBT is the non-dimensional surface
tension; and s1, v1, and m1 are the surface area, volume. and mass of a monomer.
Since the critical volume v? is unlikely to be exactly equal to any of the volume
nodes, NGDE places nucleated particles in the node just larger than v? and adjusts
239
the nucleation rate using the parameter ξk, where
v
?
if v ?k−1 < v ≤ vv kk
ξk = v? if v
? ≤ v (9.14)1

v2
0 otherwise
Thus, the nucleation rate for node k is given by [27]
dNk ∣∣∣∣ = ξkJk (9.15)dt nucl
and the rate of change in the monomer concentration is
dN1 ∣∣∣∣ = −k?Jk (9.16)dt nucl
9.3.3 Surface Growth
Condensation on or evaporation from the surface of a spherical particle with
volume vk is driven by the difference between the actual monomer concentration,
N1, and the monomer concentration over the particle at saturation, N
s
1,k, which is
given by ( ? ) ( )dp lnSs 4σv1N1,k = ns exp = ns exp (9.17)dpk kBTdpk
The increase in the saturation monomer concentration over a curved surface versus
the saturation concentration over a flat surface (ns in the above equation) is known
as the Kelvin effect. Note that this effect is usually negligible for the larger volume
240
nodes used in NGDE, but it plays a major role in the behavior of smaller parti-
cles. Condensation and evaporation rates also depend upon the collision frequency
between monomers and particles.
Incorporating the above effects, the condensation/evaporation rate for node k
is ∣
dNk ∣∣∣ = α1 + α2 − α3 − α4 (9.18)dt growth
where
 v1 s sv − β (N −Nv − 1,k−1 1 1,k−1)Nk−1 if N1 > N k k 1 1,k−1α1 =  (9.19a)0 otherwise
− v1 β (N −N s − 1,k+1 1 1,k+1)Nk+1 if N1 < N
s
 vk+1 vk 1,k+1α2 =  (9.19b)0 otherwise
 v1 − β1,k(N1 −N
s
1,k)Nk if N
s
v v 1
> N
 k+1 k 1,kα3 =  (9.19c)0 otherwise
−
v1
− β1,k(N −N
s
1 )Nk if N < N
s
1
 vk vk−1 1,k 1,kα4 =  (9.19d)0 otherwise
Here, α1 and α2 represent an increase in Nk due to condensation on particles with
volume vk−1 and evaporation from particles with volume vk+1, respectively, while
α3 and α4 represent a decrease in Nk due to condensation on and evaporation from
241
particles with volume vk, respectively. The leading factor in the above α’s represents
size-splitting to deal with the fact that particles with volume vk do not grow or shrink
into particles with volumes vk+1 or vk−1 within a single time step.
Eq. (9.18) applies for k ≥ 2; the monomer balance (k = 1) is given by
∣ ∑MdN1 ∣∣∣ = (γ2k − γ1k) (9.20)dt growth k=2
where  s s
β1,k(N1 −N 1,k
)Nk if N1 > N1,k
γ1k =  (9.21a)0 otherwise
−β (N −N s )N if N < N s 1,k 1 k 1 1,k 1,kγ2k =  (9.21b)0 otherwise
Here, γ1k represents the loss of monomers due to condensation on particles in node
k, and γ2k represents the gain of monomers due to evaporation from particles in
node k.
9.3.4 Solution Strategy
NGDE uses an explicit Eulerian method to integrate Eq. (9.3) with respect to
time. Specifically, the particle size distribution at time t+ ∆t is given by
[ ∣ ∣ ∣ ]
dNk(t) ∣∣∣ dNk(t) ∣∣∣ dNk(t) ∣N (t+ ∆t) = N ∣k k(t) + ∆t + + ∣ (9.22)dt coag dt nucl dt growth
242
Figure 9.2: NGDE solves the general dynamic equation according to the illustration
above. (This figure original appears as Figure 2 in Prakash et al. [27].)
where the coagulation, nucleation, and growth rates are given by Eqs. (9.6), (9.15),
and (9.18) for k ≥ 2, and the nucleation and growth rates are given by Eqs. (9.16)
and (9.20) for monomers (k = 1). These rates are based on the conditions at time
t. Figure 9.2 summarizes the calculations performed by NGDE.
Because the code uses an explicit solver, the time step must be small enough
to ensure that the calculated coagulation, nucleation, and surface growth rates do
not change significantly by the end of the time step. At the same time, the time
step must be large enough for the code to complete its simulation in a reasonable
CPU time. For these reasons, the MATLAB version of the code uses a dynamic
time-step algorithm to choose ∆t based on the conditions at time t:
∆t = min(∆tcoag, 0.5∆tneg, ∆tmon, ∆tsat, ∆tuser) (9.23)
243
The ∆t’s are defined as follows:
0.001
∆tcoag = (9.24a)
βminNtot
is 0.1% of the characteristic coagulation time, where βmin = min(βi,j) for i, j ≥ 2
and Ntot is the total number concentration of particles (excluding monomers);
(∣∣∣ ∣Nk ∣∆t ∣)neg = min ∣ (9.24b)k dN ∣k/dt
is the minimum time at which the number concentration in any node would become
negative based on current coagulation, nucleation, and growth rates;
0.001N1
∆tmon = (9.24c)|dN1/dt|
is the time in which the monomer concentration changes by 0.1%;
∣∣∣
∆tsat = ∣∣ 0.01S(Ps )∣∣
∣∣
(9.24d)
dTN1k 1− D ∣dt B T
is the time in which the saturation ratio changes by 1%, where Ps is the saturation
vapor pressure, dT/dt is the cooldown rate and D is a constant used to determine
the saturation vapor pressure; and
∆tuser = 10
−4 (9.24e)
244
is a user-specified maximum time step. The time-step algorithm has a number of
hard-wired coefficients, including those used to specify the allowable fraction of the
characteristic coagulation time and the allowable changes in the number concentra-
tion of particles, the number concentration of monomers, and the saturation ratio.
The chosen values of these parameters reflect the desire to minimize the required
calculation time while ensuring the calculation results are reasonable.
9.4 Sample Results
Sample problems have been developed to test the NGDE code. Test problems
for pure coagulation and pure surface growth are used to validate the relevant models
in the code. Additional sample problems representing combined nucleation and
coagulation and the full GDE are presented to show that the code is producing
reasonable results. All of the sample problems are for aluminum in argon gas.
The sample problems are described in the following subsections.
9.4.1 Pure Coagulation
An interesting feature of coagulating systems is that after a sufficiently long
time, the non-dimensional aerosol size distribution ψ = φn(v, t)/N2tot becomes in-
dependent of time and of the initial conditions of the system [2, 148, 149]. This
time-independent PSD is known as the self-preserving distribution (SPD). Here, the
non-dimensional particle volume is η = Ntotv/φ, Ntot and φ are the total number
concentration and volume fraction of particles in the system, and n(v, t)dv is the
245
number concentration of particles with volume between v and v + dv at time t.
The first test problem for NGDE involves coagulation of an initially monodis-
−3
perse aerosol with a particle diameter of 1 nm and number concentration of 1024 m .
The calculation is performed at T = 1773 K for aluminum particles with density
ρp = 2700kg/m
3. The calculated PSD should collapse to the self-preserving distri-
bution after time [149]
τf
tSPD = (9.25)
βfN0
where ( )1/6( )1/2
3 6kBT 1/6
βf = v0 (9.26)4π ρp
is the free molecule coagulation coefficient for particles with volume v0, N0 and v0
are the initial number concentration and volume of particles in the aerosol, and τf
is a time constant for coagulation in the free molecule regime. Vemury et al. [149]
found τf ≈ 5 when the bins in their sectional code are spaced such that vk = 2vk−1.
Thus, the self-preserving distribution should be reached after tSPD = 30 ns for the
initial conditions of this problem.
To test the effects of NGDE node spacing on the results, the pure coagulation
calculation has been performed with 21, 41, and 101 nodes, corresponding to node
spacings of vk = 4.0vk−1, vk = 2.0vk−1, and vk = 1.3vk−1. For all cases, the non-
dimensional size distribution ψ becomes independent of time and is approximately
equal to the self-preserving distribution ψf as calculated by Vemury et al. [149].
The results are shown in Figure 9.3. For all three cases, the calculated size dis-
tribution reaches a constant distribution by approximately 30 ns, which is in good
246
Figure 9.3: Non-dimensional size distribution for the pure coagulation problem after
1 µs, compared to the SPD for the free molecule regime, as determined by Vemury
et al. [149] using an accurate sectional method. The calculated size distribution
reaches a self-preserving distribution by 30 ns and remains constant until the end
of the calculation at 30 µs (i.e. 1000 times the estimated tSPD). Note that the
difference between the PSD calculated by NGDE and the self-preserving distribution
calculated by Vemury et al. [149] is due in part to ambiguities in defining a “bin
width” for a size distribution with “zero-width” nodes.
agreement with tSPD found by Vemury et al. [149].
3 For the case with 101 nodes,
the distribution from NGDE is in excellent agreement with the SPD, but there are
noticeable differences between the distribution from NDGE with 21 and 41 nodes
and the SPD. These differences are due in part to ambiguity in defining a bin width
for calculating n: for a sectional code, the bin width is simply the difference between
the largest and smallest particle volume in each bin, but for a nodal code, the “bins”
are zero-width nodes. In Figure 9.3 and in the NGDEplot post-processing tool, the
bin widths are set to ∆vk = (vk+1 − vk−1)/2, i.e. “bin” k has upper bound at the
midpoint between nodes k and k + 1 and lower bound at the midpoint between
nodes k − 1 and k.
An additional way to evaluate the NGDE results is to compare the moments of
3This is based on the time at which the moments of the particle size distribution are approxi-
mately equal to the steady-state values of the moments given in Table 9.1.
247
the calculated distribution to the moments of the self-preserving distribution. The
ith moment of the non-dimensional particle size distribution is defined as
∫ ∞
M = ηii φdη (9.27)
0
where i is any real number. Many of these moments are related to important proper-
ties of the aerosol size distribution: i = 0 represents the total number concentration
of particles; i = 1/3, 2/3, and 1 are proportional to the mean diameter, surface area,
and volume of particles in the system; and i = 2 is proportional to the intensity of
light scattered by the particles when they are much smaller than the wavelength of
incident light.
As shown in Table 9.1, the NGDE results with 101 nodes are in excellent
agreement with the SPD results, with differences of approximately 7% for the second
moment and less than 2% for all other moments. The NGDE results for 41 nodes
are in good agreement (less than 9% difference from the SPD) for all but the second
moment of the distribution (42% difference). These results suggest that using 41
nodes to represent 12 orders of magnitude in volume yields sufficiently accurate
results for pure coagulation.
9.4.2 Pure Surface Growth
The second test problem involves condensation of aluminum vapor on a mono-
disperse aerosol as the system cools from 1773 K to 300 K at 1000 K/s. Particles
are placed in node 25 out of 41, corresponding to a particle diameter of 79 nm. The
248
Table 9.1: Moments, Mi, of the particle size distribution for the pure coagulation
calculation. Results are shown for NGDE calculations with 21, 41, and 101 nodes.
The SPD results are from Vemury et al. [149].
i 21 Nodes 41 Nodes 101 Nodes SPD
-1/2 2.0303 1.7047 1.5877 1.5641
-1/3 1.5233 1.3649 1.3056 1.2937
-1/6 1.2025 1.1431 1.1201 1.1155
0 1.0000 1.0000 1.0000 1.0001
1/6 0.8766 0.9122 0.9266 0.9296
1/3 0.8103 0.8657 0.8884 0.8929
1/2 0.7896 0.8530 0.8789 0.8836
2/3 0.8111 0.8707 0.8947 0.8984
5/6 0.8777 0.9186 0.9347 0.9360
1 1.0000 1.0000 1.0000 0.9998
2 6.2769 2.9543 2.2399 2.0873
−3
initial number concentration is 1010 m .
For pure surface growth, the aerosol should remain monodisperse with constant
number concentration. Only the volume of each particle (and thus the total mass
or volume of particles) changes. Thus, one can easily solve the following system
of first-order, ordinary differential equations for pure surface growth with changing
temperature:
dT
= −dTdt (9.28a)
dt
dN1
= −β (N −N s1,p 1 1,p)Np (9.28b)dt
dvp − m1 dN1 m1= = β1,p(N1 −N s1,p) (9.28c)dt ρpNp dt ρp
Here, dTdt is the cooldown rate, β1,p is the collision kernel between monomers
and particles with volume vp given by Eq. (9.8), N
s
1,p is the saturation monomer
concentration over particles with volume vp (or diameter dp) given by Eq. (9.17),
249
Figure 9.4: Size distribution calculated by NGDE for the pure surface growth sample
problem. The distribution should remain monodisperse, but it spreads out over time
because of numerical diffusion introduced by the size-splitting algorithm.
m1 is the mass of a monomer, and ρp is the particle density. One can solve for the
system temperature T , monomer concentration N1, and particle volume vp using
any ODE solver, such as the MATLAB function ode45.
Figure 9.4 shows the size distribution calculated by NGDE for the pure sur-
face growth problem. The PSD is shown at several different times. While the size
distribution should remain monodisperse, the PSD calculated by NGDE spreads out
with time due to numerical diffusion. Note that sectional methods also suffer from
the same issue for pure surface growth. Fortunately, NGDE correctly determines
the volume mean particle size (Figure 9.5) and the (constant) total number concen-
tration. This shows that NGDE can provide correct results for the zeroth (number
concentration) and first (average volume) moments of the distribution for surface
growth problems, even though the distribution is much broader than expected.
250
Figure 9.5: Volume-mean particle diameter calculated by NGDE for the pure surface
growth problem. NGDE results are in excellent agreement with the mean diameter
from the “exact” solution of the problem [i.e. Eq. (9.28) solved using the MATLAB
function ode45, which integrates the system of ODEs using a 4th/5th order Runge-
Kutta method].
9.4.3 Nucleation and Coagulation
The third sample problem tests nucleation and coagulation of aluminum as
the system cools from 1773 K to 300 K at 1000 K/s. Initially, there are no particles
in the system, and aluminum vapor is very slightly super-saturated (S = 1.001).
This calculation is performed with 41 volume nodes.
Figure 9.6 shows the evolution of the size distribution with time, and Figure 9.7
shows the critical particle volume (i.e. the volume of newly-formed particles) and
the nucleation rate. Particles begin to form between t = 0.1 s and t = 0.18 s. At this
point, coagulation is negligible, so the particle distribution is very narrow, with most
particles existing at nodes near the critical volume for nucleation (v? ∼ 1 nm3 at this
point in the calculation). As the particles begin to coagulate, the size distribution
broadens while the average volume increases. At longer times, the distribution is
bimodal, with the mode at lower volumes representing newly formed particles and
251
Figure 9.6: Particle size distribution at select times for nucleation and coagulation
of aluminum. The distribution becomes bimodal due to the combined influence of
new particle formation at lower volume nodes and coagulation to populate the larger
volume nodes.
the mode at higher volumes representing particles formed earlier in the calculation
that have since coagulated. This behavior is common in atmospheric systems, where
sources continuously produce small particles that coagulate to form larger particles
[2]. The nucleation rate decreases from t = 0.2 s until the end of the calculation,
resulting in the decay of the lower mode of the bimodal PSD.
9.4.4 Full GDE for Condensation of Aluminum
The final sample problem involves solving the full GDE as a system of slightly
super-saturated aluminum vapor cools from 1773 K to 300 K at a rate of 1000 K/s.
Once again, the calculation is performed using 41 nodes.
Figure 9.8 shows the particle size distribution at select times. The particle
number concentration remains near zero for the first 0.1 seconds of the calculation.
At this point, the saturation ratio has reached a large enough value to allow for signif-
icant nucleation rates. Nucleation rates are very large between approximately 0.1 s
252
Figure 9.7: Critical volume and nucleation rate for nucleation and coagulation of
aluminum. Initially, the critical volume is very large (near one micron) due to the
low saturation ratio, but the critical volume quickly decreases as the temperature
drops and the saturation ratio increases. [Refer to Eq. (9.13) for the relationship
between critical volume and saturation ratio.]
and 0.15 s (Figure 9.9); after this brief burst of particle formation, nucleation rates
are negligible for the remainder of the calculation. These nucleated particles grow
quickly and form a nearly-lognormal peak centered near 106 nm3 by approximately
0.15 s. The smaller peak at this time is due to nucleation. After the saturation ratio
drops to nearly unity, larger particles grow slowly, while smaller particles evaporate
due to the Kelvin effect. These processes continue for the remainder of the calcula-
tion, resulting in a final lognormal distribution with a single peak at 6.1× 106 nm3.
The number concentration remains nearly constant after nucleation ceases, which
shows that coagulation is negligible for these number concentrations.
This sample problem shows the important role of surface growth in the aerosol
dynamics. For the same conditions as the nucleation plus coagulation problem, the
calculated number concentration is orders of magnitude lower because monomers
that condensed on existing particles are unavailable to nucleate to form new parti-
253
Figure 9.8: Particle size distribution at select times for nucleation, coagulation, and
surface growth of aluminum. Particle formation begins around t = 0.1 s and effectly
ends by t = 0.16 s. After this time, the size distribution changes due primarily to
surface growth. Coagulation plays a minor role due to the low number density and
short time frame of the problem.
Figure 9.9: Nucleation rate and saturation ratio early in the simulation for nucle-
ation, coagulation, and surface growth of aluminum by NGDE. Nucleation rates are
significant for the short period of time when the saturation ratio is greater than ∼ 3.
After this period, very few new particles form as condensation on existing particles
dominates the behavior of the system.
254
Figure 9.10: Monomer and total particle concentration for nucleation, coagulation,
and surface growth of aluminum. The particle concentration is nearly constant
(decrease of less than 2%) after particle nucleation stops around t = 0.16 s. Particles
grow primarily by surface growth, as evidenced by the steady decline in the monomer
concentration.
cles. At the same time, the volume-mean particle diameter is larger at the end of
the full GDE calculation than the mean diameter when surface growth is neglected
(230 nm versus 59 nm). Note that the total particle volume is nearly the same in
both calculations.
9.5 Limitations of NGDE
The sample problems discussed in the previous section demonstrate that NGDE
yields results in good agreement with other available numeric solutions of the general
dynamic equation. However, there are a number of limitations of the code. Some of
these limitations – particularly the numerical diffusion observed in the pure surface
growth problem – are intrinsic to the nodal method specifically and sectional meth-
ods in general. Other limitations can be addressed by modifying the code. Some of
these limitations, and suggestions for improvement, are discussed in this section.
255
One limitation is that NGDE does not include an energy balance. This means
that the code ignores the latent heat involved in nucleation, condensation, and
evaporation. Ignoring the latent heat is reasonable because NGDE requires a user-
specified cooldown rate for calculation types involving nucleation and growth; one
can assume that temperature changes due to particle formation and growth are
included implicitly in this cooldown rate. However, if one is concerned with a system
where the latent heat plays a significant role in the dynamics, then one would need
to add the energy balance to the code. This would also require the user to specify
data about the latent heat as a function of thermodynamic conditions. Such changes
could be made with only a modest effort.
A second limitation is that NGDE does not account for particle sources and
sinks. In aerosol systems, deposition often plays an important role in affecting the
particle size distribution. Some deposition mechanisms could be included fairly
easily in NGDE. For example, the settling velocity of a particle with diameter dk in
a gravitational field is given by
(ρ 2p − ρf )gd
u k
Cc(Knk)
g,k = (9.29)
18µ
where ρp−ρf is the difference between the particle and fluid densities, µ is the fluid
viscosity, and Cc is the Cunningham slip correction factor for spheres with Knudsen
number Knk = 2λ/dk. Assuming a well-mixed aerosol, the decrease in number
concentration of particles in node k due to settling in time ∆t in a box with height
L is Nkug,k∆t/L; one could simply add a subroutine to perform this calculation and
256
include it as a loss term in the particle mass balance. Similarly, one could add a
subroutine to add particles to a given volume node to represent an aerosol source
term.
Another limitation is that NGDE is written with specific situations in mind:
coagulation of an initially monodisperse aerosol; nucleation, coagulation, and sur-
face growth for an initially saturated system experiencing a constant decrease in
temperature; and surface growth on a monodisperse aerosol for a constant change
in temperature. It may be desirable to allow for some flexibility in the calculation
types, such as allowing the user to specify a polydisperse size distribution or non-
constant cooldown rate. Fortunately, many such changes would require only minor
changes to the code and code input. Even less effort is required to modify some of
the hard-wired parameters (e.g. initial saturation ratio, maximum allowable change
in saturation ratio in the dynamic time-step algorithm) in the code. For guidance
in changing the code, please refer to the NGDE User Manual (Appendix F).
9.6 NGDEplot
As part of the conversion of NGDE to MATLAB, a new post-processing tool
has been developed to display how certain parameters evolve with time. Currently,
NGDEplot can create movies showing the evolution of the particle size distribu-
tion (in various forms) and/or the scattering, absorption, and extinction coefficients
for the distribution. The user can save the frames for later playback using MAT-
LAB’s movie command. In addition, NGDEplot can display static plots showing
257
the nucleation rate, saturation ratio, and mean particle diameter.
NGDEplot works as follows. First, the code unpacks the simulation results
contained in the NGDE output parameter results2 and reads user-specified input.
The remainder of NGDEplot is fairly straightforward: other than simple manipula-
tion of the NGDE results, the code mostly consists of commands that control the
appearance of the plots. With that said, the light scattering calculations warrant a
brief explanation.
Before showing the light-scattering movie, NGDEplot determines the scatter-
ing, absorption, and extinction efficiencies for each volume node by calling a Mie
scattering code [89]. The Bohren Mie code was converted from a Fortran code
provided by Professors Eugene Clothiaux and Craig Bohren from the Pennsylva-
nia State University; the conversion was performed using the automated MATLAB
function f2matlab available through MATLAB File Exchange on the MathWorks
website. Extensive testing was performed to ensure that the converted code returns
correct values for the optical efficiencies.
Using the efficiencies for each volume node obtained from the Bohren Mie
code, NGDEplot calculates the scattering, absorption, and extinction coefficients
for the distribution, where the scattering coefficient is given by
∑M π
Ksca(λ, t) = d
2
4 pk
Qsca,k(λ)Nk(t) (9.30)
k=2
Here, dpk is the diameter of particles in node k, M is the total number of nodes,
and Qsca,k is the scattering efficiency for particles in node k for incident light with
258
wavelength λ. The extinction coefficient is calculated by replacing Qsca in the above
equation with Qext, while the absorption coefficient is simply Kabs = Kext − Ksca.
(Note that the optical cross sections are also functions of the refractive index of
the particles.) The optical coefficients are calculated at each time t for a range
wavelengths that includes the visible spectrum, as well as portions of the ultraviolet
and infrared spectra. Note that the summation begins at k = 2 unless the NGDE
results are from a pure coagulation calculation, in which case the summation begins
at k = 1.
More information about running NGDEplot can be found in the NGDE User
Manual (Appendix F). Sample screenshots of the size distribution and light scatter-
ing movies are shown in Figures 9.11 and 9.12.
9.7 Conclusions
The NGDE code uses a nodal method to solve the general dynamic equation
for the evolution of an aerosol particle size distribution with time. The method is
similar to sectional methods, but the discrete nature of the nodes makes the resulting
code much simpler than sectional codes. Even with this simplified approach, the
NGDE code gives results in good agreement with available results obtained using
other methods. Because of its simplicity and accuracy, NGDE is well-suited to serve
as a teaching tool in college courses on aerosol physics and dynamics.
The MATLAB version of the code includes a new dynamic time-step algorithm
that decreases the code execution time by orders of magnitude. For example, the
259
Figure 9.11: Screenshot of the particle size distribution movie from NGDEplot. This
still is from the end of full GDE sample problem described in Section 9.4.4.
260
Figure 9.12: Screenshot of the light scattering movie from NGDEplot. This still
is from the end of full GDE sample problem described in Section 9.4.4. The Mie
calculations are performed for a refractive index of n = 1 + 6.4i, which is an average
value for aluminum in the visible spectrum [150].
261
full GDE sample problem described in Section 9.4.4 takes approximately 10 minutes
to run with the dynamic time-step algorithm in MATLAB but more than a day to
run in the original C version of the code. Furthermore, the MATLAB version of
NGDE comes with post-processing tool NGDEplot that utilizes MATLAB’s built-
in plotting features to display results for the particle size distribution and the light
scattering, absorption, and extinction coefficients.
262
Chapter 10: Conclusions and Recommendations for Future Work
In this dissertation, I have described my method for determining the force and
torque on aggregates in the transition regime and discussed its use to study various
problems related to aerosol particle physics. Now, I will summarize the important
conclusions of my work and suggest areas for further study.
10.1 Conclusions
I have developed a method for calculating the translational, rotational, and
coupling friction tensors for aggregates consisting of spheres in point contact when
the primary sphere Knudsen number is in the transition regime (i.e. 0.01 < Kn <
100). My method addresses an important practical problem because a significant
fraction of terrestrial aerosols is generated as aggregates of very small primary
spheres [4, 7]. It synthesizes two previous approaches for computing the transport
properties of particles: the Kirkwood-Riseman method originally developed in the
late 1940s to determine the intrinsic viscosity and translational diffusion of polymers
[28], and the efforts of Loyalka and others to determine the velocity around a sphere
in rarefied flow [75–78]. Since that time, researchers have made significant improve-
ments to KR theory for describing hydrodynamic interactions between spheres in
263
the continuum [30, 51, 57, 58, 97]; however, no effort had been made to extend
this approach to the transition regime using the results of Loyalka. This gap in
the literature is likely due to the analytical and numerical complexities involved in
solving the Boltzmann transport equation for each primary sphere Knudsen number
of interest, even when the equation is posed in a simplified form like the Bhatnagar-
Gross-Krook equation [71]. My method plugs this gap and extends KR theory to
the transition regime in order to improve predictions of the transport behavior of
aerosol aggregates consisting of nano-scale primary spheres.
From the friction tensors determined using my method, one can obtain the
translational and rotational friction coefficients by averaging over all particle orienta-
tions; one can also obtain the diffusion tensors and coefficients through a generalized
Stokes-Einstein relation. Results for the translational friction coefficient compare
well to Direct Simulation Monte Carlo results and experimental data published
in the literature. The calculated translational friction coefficients also approach
the continuum and free molecule results from the Zeno algorithm and a Monte
Carlo algorithm in the limit of very small and very large primary sphere Knudsen
numbers, respectively. This suggests that the friction and diffusion coefficients ob-
tained using my method can be used in various relations describing aerosol transport
(e.g. to determine gravitational settling rates, electrical mobilities, or coagulation
rates for non-spherical particles), while the friction and diffusion tensors can be used
in aerosol dynamics simulations (e.g. for coagulation or Brownian motion).
Based on this success in calculating the friction tensors and coefficients, I
have applied the method to study a number of problems related to aerosol physics.
264
First, I looked at the effects of primary sphere size (characterized by the Knudsen
number) and the number of spheres (N) on the translational and rotational friction
coefficients of particles formed by diffusion-limited cluster aggregation. In both
cases, I found that the friction coefficients approach the continuum limit as the
Knudsen number decreases or as the number of spheres increases. For example,
DLCA aggregates consisting of 1000 spheres are in the continuum limit even when
the primary sphere Knudsen number is unity. This supports the general premise
of the Adjusted Sphere Method [41, 67], that one can determine the translational
friction coefficient of an aggregate using an aggregate Knudsen number based on its
hydrodynamic radius (i.e. a continuum measure of particle size) and its orientation-
averaged projected area (i.e. a free molecular measure of particle size). In response
to this finding, I showed that one can determine the rotational friction coefficient
using an aggregate Knudsen number for rotational motion that is proportional to
the ratio of continuum and free molecule rotational friction coefficients.
As part of this effort to determine the translational and rotational friction coef-
ficients, I showed that the ratio of translational to rotational characteristic diffusion
times is near unity for DLCA aggregates, regardless of primary sphere or aggregate
size. In other words, these particles rotate significantly during the average time
required to diffuse one radius of gyration. This finding is significant because most
aerosol studies ignore the effects of rotation on the particle dynamics. My results
suggest the effect of rotation on particle dynamics requires further study.
One major advantage of my method compared to the DSMC method is that
it is very fast: one can determine the friction tensors for aggregates with 100 pri-
265
mary spheres within seconds, while aggregates with 2000 spheres take minutes on a
single processor. (Compare this to DSMC, which takes days on a single processor
to determine the friction coefficient for a 20-sphere aggregate [41]). However, the
required computational time is still too excessive to incorporate this method in an
aerosol dynamics code. Therefore, I have used my results for the translational and
rotational friction coefficients of DLCA aggregates to develop an analytical expres-
sion that returns these coefficients as a function of the primary sphere radius, the
number of spheres in the aggregate, and the gas properties. My analytical expression
for the translational friction coefficient is more accurate than previous correlations
for DLCA aggregates [37–39] when compared to my EKR results. The rotational
friction coefficient expression appears to be the first of its kind published in the
literature. Researchers can use these expressions to quickly estimate the friction
coefficients for soot-like aggregates.
Next, I applied my EKR method to determine the orientation-averaged mo-
bility of a particle in an electric field. The mobility is a function of particle size and
field strength due to the interaction of the induced dipole in the particle with the
electric field. My orientation-averaged mobility results as a function of field strength
are in good agreement with experimental data published in the literature. In gen-
eral, for DLCA aggregates the mobility is less than 10% greater when the particle
is aligned with the field than when all orientations are equally probable (i.e. at very
low field strengths). Thus, one could in theory use the relationship between mobility
and field strength to obtain size information or to separate particles with similar
mobility at one field strength but different shapes (and hence different mobilities
266
from each other at a different field strength). However, my results suggest there are
several practical issues related to the experimental setup and to the accuracy of the
methods used to relate the data to size and shape information (such as my EKR
method). It is especially difficult to obtain shape information for either very large
or very small soot-like aggregates because it is difficult to operate a DMA at low
enough voltages to ensure that large aggregates have a fully random orientation,
and small aggregates require very high field strengths to align. In these limits, the
measured mobility at low and high field strengths would be nearly equal, which
would suggest – incorrectly – that these fractal aggregates are actually spherical.
Finally, I showed that one can use my method to determine the hydrodynamic
forces between spheres or aggregates in the transition regime, where the centers of
mass of the particles are separated by a distance r. This effort is a natural ex-
tension of the single-particle work discussed in the earlier chapters of my work: it
includes the interactions among the primary spheres in multiple aggregates as well
as within one aggregate. I have demonstrated that while the strength of interactions
between particles weakens as the primary sphere Knudsen number decreases, such
interactions follow the characteristic 1/r behavior observed for continuum particles.
From this result, I described how one can adopt the point force method for widely
separated particles in the continuum to particles in the transition regime; the only
difference is the leading coefficient that appears in the Oseen hydrodynamic inter-
action tensor. I have used this point force approach to show that a spherical cloud
of particles with non-negligible Knudsen numbers behaves just like a cloud in the
continuum, provided certain conditions (e.g. cloud radius, particle volume fraction)
267
are satisfied.
In addition to my work in calculating the transport behavior of aggregates, I
have made significant improvements to the NGDE code, which solves the general
dynamic equation for the change in the aerosol size distribution due to nucleation,
coagulation, and surface growth. In particular, I converted the code from C to
MATLAB to make use of MATLAB’s built-in plotting features, I added a dynamic
time-step algorithm to significantly reduce code execution time and improve code
stability, and I developed a post-processing tool to generate movies showing the
evolution of the size distribution and the optical properties of the aerosol. These
improvements will enhance the code’s intended use in teaching students about the
effects of nucleation, coagulation, and surface growth on aerosol dynamics.
10.2 Recommendations for Future Work
10.2.1 Friction Coefficient Expressions for non-DLCA Aggregates
Much of my work has focused on DLCA aggregates because such particles
are found in many areas of interest to the aerosol science. However, it is possible
to form particles with other fractal dimensions. (See, for example, Figure 8.3 of
Friedlander [2].) One could use my EKR method to determine the functional rela-
tionship between the primary sphere size and number of spheres in the aggregate
and its translational and rotational friction coefficients in order to develop analytic
expressions similar to my expressions for DLCA aggregates [Eqs. (4.38) and (6.26)].
One could include these expressions in an aerosol dynamics code to account for the
268
transport behavior of aggregates as a function of the primary sphere size, number
of primary spheres, and fractal dimension.
Similarly, one could compare the properties of aggregates generated with the
Mackowski algorithm – which I have used to create the particles used in this study
– with those generated by direct numerical simulation of the aggregate trajecto-
ries. This is of interest to determine how well the properties of particles generated
by a simplified algorithm match the properties of particles from a more realistic,
Langevin-style simulation (as described below).
10.2.2 Aggregates with Polydisperse Primary Spheres
All of my calculations involve aggregates that consist of equally-sized primary
spheres. This is an idealized situation; in reality, aggregates typically consist of
primary spheres with some distribution of radii. Bernal et al. [51] have demonstrated
how one can account for polydispersity in the continuum, and Spyrogianni et al.
[151] have applied the method to determine the effect of polydispersity on aggregate
settling rates. Now, I will describe how to address this issue in the transition regime.
To account for polydispersity, we must make a slight change to my extended
Kirkwood-Riseman method. In the continuum, the force on the ith sphere with
radius ai moving with velocity ui is given by
∑N
F c ci = −ζt,0(ai)ui − ζt,0(ai) Tij(ai, aj, rij) ·Fj (10.1)
i=6 j
where aj is the radius of the jth sphere and the hydrodynamic interaction tensor is a
269
function of ai, aj, and the distance between the spheres. My EKR method replaces
the hydrodynamic interaction tensor with the quotient of the velocity tensor and
the monomer friction coefficient for sphere j. Thus, to account for polydisperse
primaries, my EKR method becomes
∑N
Fi = −ζt,0(Kni)ui − ζt,0(Kn −1i) ζt,0 (Knj)Vij(Knj, rij) ·Fj (10.2)
i 6=j
The only difference between this expression and the expression for monodisperse
primaries is the appearance of the friction coefficient ratio ζt,0(Kni)/ζt,0(Knj) for
primary spheres with Knudsen numbers of Kni ≡ λ/ai and Knj ≡ λ/aj. For mono-
disperse primaries (Kni = Knj ≡ Kn), the ratio cancels, leaving the expression that
has appeared throughout the earlier chapters of this disseration,
∑N
Fi = −ζt,0(Kn)ui − Vij(Kn) ·Fj
i=6 j
There are a number of practical challenges that must be addressed in order
to determine the friction coefficient of aggregates with polydisperse primaries. The
first challenge is to generate the coordinates for the spheres in the aggregates. The
Mackowski algorithm I have been using creates aggregates with unit primary sphere
size. One would need to modify the algorithm to account for polydisperse primaries.
One could do so by sampling the primary sphere sizes from a specified distribution
(such as a lognormal distribution a geometric standard deviation σg = 1.2, in agree-
ment with experiments of soot formation in premixed diffusion flames Köylü and
270
Faeth [42]). Alternatively, one could obtain particles from a dynamic simulation of
particle aggregation [151].
Also, one would need to make significant modifications to my MATLAB pro-
gram for calculating the friction tensors (bgk tensors; see Appendix G) to account
for polydispersity. Currently, my program loads the velocity and monomer friction
coefficient results obtained by solving the BGK equation. These results are stored
in individual MATLAB data files indexed by the non-dimensional monomer radius
r0 [defined by Eq. (2.5d)]. To account for polydisperse primaries, one would need to
access multiple data files, each one associated with the Knudsen number of one of
the monomers in the aggregate. This would require there to be a file of BGK results
for each primary sphere Knudsen number. Alternatively, one could compile all of
the BGK results in a table and interpolate based on the primary sphere Knudsen
number and the distance between spheres. Likewise, one would need to interpolate
to determine the coefficients c1 and c2 [see Eq. (2.59)] for the velocity far from the
sphere. Such changes would not be difficult to implement.
With that said, it is unlikely that accounting for polydispersity would have
a significant impact on aggregate transport properties. For example, Spyrogianni
et al. [151] found that aggregates in the continuum with polydisperse primaries
settle faster than aggregates with monodisperse aggregates with equal mean primary
diameters, but only because the polydisperse aggregates have greater mass than the
monodisperse aggregates. In other words, the settling velocity of aggregates in the
continuum is unaffected by monomer polydispersity, once one corrects for differences
in particle mass. Nevertheless, this does not guarantee that the same behavior would
271
be observed in the transition regime, especially if there is a broad distribution of
primary sizes. (See, for example, the particles described in Ref. [152].) One could
test whether or not this is the case using my EKR method, with Eq. (10.2) to
account for polydispersity.
10.2.3 Rotational and Coupling Interactions
My EKR method ignores rotational and coupling hydrodynamic interactions
between spheres in an aggregate. Such interactions are weaker than the translational
interactions included in my method, so one can safely neglect these effects when
the average distance between spheres in the aggregate increases. However, when
the average distance between spheres is small, rotational and coupling interactions
become significant, and the error in the rotational friction coefficient calculated
using the EKR method can be large (up to 40% for the cases I have studied).
Expressions for the rotational and coupling hydrodynamic interaction tensors
in the continuum are available in the literature, as summarized by Carrasco and
Garcıa de la Torre [53]. I have shown that to order r−3ij , the rotational and coupling
hydrodynamic interaction tensors are related to the vorticity and velocity fields
around a rotating sphere. (See Appendix D.) This suggests that one could obtain
the rotational and coupling interaction tensors as a function of Knudsen number by
solving the BGK equation for the flow field around a rotating sphere. Loyalka [77]
has already performed this calculation; unfortunately, the paper does not provide
detailed results for the velocity around the rotating sphere, nor does it go into much
detail about the solution procedure. Still, the problem is not significantly different
272
from the translating sphere problem; the main difference is in the far-field velocity
distribution function, where instead of a uniform translational velocity U there is a
position-specific velocity ω×r. The rotational problem is slightly more complicated
due to this difference, but in principal one can use the same approach developed by
Lea [74] that I have described in Chapter 2.
After obtaining the velocity and vorticity fields around a rotating sphere – and
thus the coupling and rotational hydrodynamic interaction tensors as a function of
Knudsen number – the next step would be to incorporate these tensors in the EKR
method. The approach is analogous to the method described by Carrasco and Garcıa
de la Torre [53] for calculating the force and torque on a particle in the continuum;
the only difference is that one replaces the continuum translational, rotational, and
coupling interaction tensors with the velocity field around a translating sphere, the
vorticity field around a rotating sphere, and the velocity field around a rotating
sphere obtained by solving the BGK equation as a function of Knudsen number.
This effort should significantly improve the accuracy of the calculated rotational
friction coefficient, especially for small aggregates near the continuum regime.
10.2.4 Brownian Dynamics
The final suggested extension of my research is to incorporate my EKR method
into Brownian dynamics simulations to study the effects of particle rotation and
hydrodynamic interactions on various parameters of interest to the aerosol scien-
tist (e.g. coagulation rates, sedimentation rates, the size and shape of coagulated
aerosols). This would involve solving the Langevin equations for each particle in an
273
N -particle system. The Langevin equations for particle i describe its translational
and rotational velocities (ui and ωi) and position in space-fixed coordinates and
orientation in particle-fixed coordinates (xi and (φi, ωi, ψi)):
N
dmiu ∑i
= −Ξ †t,i ·ui −Ξc,i ·ωi + FH,ij + FE,i + FB,i (10.3)dt
j=1
dω ∑N
Ii · i + ωi × (Ii ·ωi) = −Ξc,i ·ui −Ξr,i ·ωi + TH,ij + TE,i + TB,i (10.4)
dt
j=1
dxi
= ui (10.5)
 dt d 1idt 
 
  ηiωx′,i − 3iωy′,i + 2iωz′,i  

d2i
  

 ω ′ 

dt 1 3i x ,i
+ ηiωy′,i − 1iωz′,i 
 =  (10.6) 2d   3i dt  −2iωx′,i + 1iωy′,i + ηiωz′,i
dηi −
dt 1i
ωx′,i − 2iωy′,i − 3iωz′,i
In these equations, mi is the mass, Ii is moment of inertia tensor, FE,i and TE,i
are the external forces and torques (e.g. from an electric field), and FB,i and TB,i
are fluctuating Brownian forces and torques on particle i, while FH,ij and TH,ij
are the hydrodynamic force and torque on particle i due to particle j. Note that
the rotational velocity and orientation are written in terms of the particle-fixed
axes (x′i, y
′
i, z
′
i), which are related to the co-moving axes (i.e. the axes parallel to the
space-fixed axes that translate with the particle) by the Euler angles (φi, θi, ψi). The
friction tensors represent the friction on the particle when it is alone in an infinite
fluid (i.e. ignoring the effects of the other particles) and are written in terms of the
274
co-moving axes. Eq. (10.6) is written in terms of the Euler quaternion (1i, 2i, 3i, ηi),
which is related to the Euler angles. (See Ref. [153].) Note that in principal one
must consider the effects of all of the particles when determining the fluctuating
Brownian force and torque on particle i [136].
Previous studies of Brownian dynamics are restricted to the continuum or free
molecule limits [121, 127, 154, 155]. Many of these studies use simplified meth-
ods to calculation the aggregate friction coefficient, though Fernandes and Garćıa
de la Torre [127] apply methods based on Kirkwood-Riseman theory to determine
the translational friction tensor Ξt. Furthermore, the rotational dynamics are of-
ten neglected for fractal aggregates [121, 127, 154, 155], though they have been
considered in studies of simple shapes shapes such as ellipsoids [153, 156]. Hydro-
dynamic interactions are included in Stokesian dynamics simulations for particles in
the continuum [135, 136] but are typically excluded for non-continuum particles.
Eqs. (10.3)–(10.6) can be solved for a system of particles to determine the
effects of the rotation and/or hydrodynamic interactions on the dynamic behavior
of the system. To do so, one would have to repeat the simulation multiple times,
average the results, and compare various figures of merit (e.g. coagulation rates,
sedimentation rates, particle size and fractal dimension) to the same parameters for
simulations of the same system that neglect rotational and hydrodynamic effects.
During the calculation, one would need to perform the EKR calculation at each
time step to determine the friction tensors, hydrodynamic forces and torques, and
probability distribution for the Brownian forces and torques for each particle.
One could also perform a Brownian simulation for a single aggregate in an
275
external field; in this case, one need only perform the EKR calculation for the
friction tensors of the particle at the start of the simulation. Calculations could be
performed for a range of external field strengths. Results of Brownian trajectories
for zero field strength can be used to evaluate the results for the translational and
rotational diffusion coefficients computed using rigid body hydrodynamic theory
(i.e. the results described in Chapters 3–6). Brownian dynamics results for non-zero
field strength could be used to determine the electrical mobility of a particle and
can be compared to the results of the particle alignment calculations described in
Chapter 7.
Ultimately, these Brownian simulations can be used to evaluate the validity of
various assumptions (e.g. treating aggregates as equivalent spheres, ignoring rotation
effects, neglecting hydrodynamic interactions) typically made in studies of cluster
aggregation kinetics and other dynamic aerosol processes.
Clearly, this suggested project would be a substantial undertaking, both in set-
ting up the problem (i.e. writing the code to solve the Langevin equations above) and
in ensemble-averaging and interpreting the results. Additionally, one needs access
to powerful computing resources to tackle this problem, given the large number of
simulations that would be required to account for the statistical fluctuations inher-
ent in solving the problem and the computational time required to perform a single
Brownian trajectory. Given these issues, as well as the likelihood that rotational
and hydrodynamic effects may only have minor effects on the dynamic behavior of
aerosol systems, it is hardly surprising that this problem has received little atten-
tion in the literature. Nevertheless, my method for calculating the friction tensors of
276
aerosol aggregates in the transition regime provides one of the missing pieces needed
to tackle this difficult problem.
277
Appendix A: Derivation of Expressions in Chapter 2
In this Appendix, I present derivations for several expressions that appear in
the solution of the BGK equation for flow around a sphere, as presented in Chapter 2.
A.1 Derivation of the Expression for g [Eq. (2.42)]
In this section, I will present the derivation of the constant g in Eq. (2.42) for
flow around a sphere. This derivation follows Appendix D of Lea [74], but I have
included more of the intermediate steps for clarity. I have also included information
from Law and Loyalka [76] because that study accounts for non-isothermal condi-
tions around the sphere. For this derivation, I will refer to the angles defined in
Fig. 2.4.
We start by writing the velocity perturbation vector in terms of coordinates
(x′, y′, z′):
√ √
ε2(r) = 2 ρ3 sinα
′ ê ′x′ + 2 ρ2 cosα êz′ (A.1)
Note that the z′-direction is simply êr; the y
′ direction is chosen to be perpendicular
to the plane containing U∞ and êz′ ; and the x
′ direction is of course mutually
278
orthogonal to the y′- and z′-directions:
êz′ × êU
êz′ = êr, êy′ = ′ , êx′ = êy′ × êz′ (A.2)sinα
We must now write out the local coordinates (x′, y′, z′), in terms of coordinates
(x, y, z). First, we write êz′ in terms of (x, y, z):
êz′ = sin θ
′ cosφ′ êx + sin θ
′ sinφ′ ê + cos θ′y êz (A.3)
Next, we write êU in terms of (x, y, z):
êU = sinα êx + cosα êz (A.4)
Next, we evaluate the cross product to determine êy′ :
1 [
êy′ = ′ sin θ
′ sinφ′ cosα êx
sinα ]
+ (sinα′ cos θ′ − sin θ′ cosφ′ cosα) êy − sin θ′ sinφ′ sinα êz (A.5)
Finally, we evaluate the cross product to determine êx′ :
1 [ (
ê 2 ′ ′ ′ ′ 2 ′ 2 ′
)
x′ = sinα cos θ − cosα sin θ cos θ cosφ + sinα sin θ sin φ êx
sinα′ (
− cosα sin θ′ cos θ′
)
( sinφ
′ + sinα sin2 θ′ sinφ′ cosφ′ êy
+ cosα sin2 θ′ − sinα sin θ′ cos θ′
) ]
cosφ′ êz (A.6)
279
Now that we have defined the coordinates (x′, y′, z′) in terms of coordinates
(x, y, z), we can write our velocity perturbation vector in terms of coordinates
(x, y, z):
√ {[ (
ε2(r) = 2 ρ2 cosα
′ sin θ′ cosφ′ + 2 ′ ′ ′ ′
′ ′)]
ρ3 sin[α cos θ − cosα sin θ cos θ cosφ
+ sin 2 2 ′ ′ ′(α sin θ sin φ êx + ρ2 cosα sin θ sinφ
− ρ cosα sin θ′ cos θ′ sinφ′ ′ ′ ′
)]
[3 ( + sinα sin
2 θ sinφ cosφ êy )] }
+ ρ cosα′ cos θ′2 + ρ3 cosα sin
2 θ′ − sinα sin θ′ cos θ′ cosφ′ êz
(A.7)
We can now consider our equation for the source term, Eq. (2.33):
√
A(r0) =− π∫U cosα [ ( ) ]
− 2 dr · 3T2 ε1(r) + T3 Ω̂ ε2(r) + T4− T2 ε3(r) (Ω̂ · n̂)
π V |r − r |20 2
Here, we have simply written out the triple integral in terms of Cartesian coordi-
nates. Again, the argument of the Tn functions is |r−r0|. Before proceeding further,
we will introduce a new coordinate system, (r, t, φ′), where r is the distance from
the origin to point r, t = |r − r0|, and φ′ is the angle of rotation about the z-axis,
with φ′ = 0 corresponding to the positive x-axis. These coordinates are related to
the (x, y, z) coordinates by the following expressions:
x = r sin θ′ cosφ′ (A.8a)
280
y = r sin θ′ sinφ′ (A.8b)
z = r cos θ′ (A.8c)
where the angles θ and θ′ are
r2 + r2 − t2
cos θ′ = 0 (A.9a)
2r0r
t2 + r20 − r2cos θ = (A.9b)
√ 2r0t
(2rr )2 − (r2 + r2 − t2)20
r sin θ′ = t sin θ = 0 (A.9c)
2r0
Note that θ and θ′ are independent of φ′.
The Jacobian determinant of coordinates (r, t, φ′) is tr/r0. In these coordi-
nates, our equation for A(r0) becomes
√ ∫ ∞ ∫ √r22 r −r2 ∫0 dt 2π [
A(r0) = − πU cosα− dr ( dφ
′ T2(t)ε)1(r)π r r0 0 r−r t0 0 ]
3
+ T3(t)Ω̂ · ε2(r) + T4(t)− T2(r) ε3(r) (cos θ)
2
Here, θ is the angle between n̂ and Ω̂, ε2 is given by Eq. (A.7), and
r0 − r
Ω̂ = = − sin θ cosφ′êx − sin θ sinφ′êy + cos θê| − | zr0 r
We can integrate over φ′ analytically. We start with the term involving ε1(r).
We use Eq. (2.48a) to write the density perturbation in terms of its radial and
281
angular components, write cosα′ in terms of the angles (α, θ′, φ′), and integrate over
φ′:
∫ 2π ∫ 2π
dφ′ T2(t)ε1(r) = T2(t)ρ1(r)∫ dφ
′ cosα′
0 0
2π
= T2(t)ρ1(r) dφ
′ [cosα cos θ′ + sinα sin θ′ cosφ′]
0
=2πT2(t)ρ1(r) cosα cos θ
′
We get a very similar result for ε3, with ρ1 replaced by ρ4 and the appropriate Tn
functions.
For the term involving ε2(r), we must first evaluate the dot product Ω̂ · ε2(r):
√ [
Ω̂ · ε2(r) = 2 ρ2 cosα′(cos θ cos θ′ − sin θ sin θ′ cos2 φ′ − sin θ sin θ′ sin2 φ′)
+ ρ3(cosα sin θ sin θ
′ cos θ′ cos2 φ′ − sinα sin θ cos2 θ′ cosφ′
− sinα sin θ sin2 θ′ sin2 φ′ cosφ′ + cosα sin θ sin θ′ cos θ′ sin2 φ′
+ sinα sin θ sin2 θ′ sin2 φ′ cosφ′]+ cosα cos θ sin
2 θ′
[− sinα cos θ sin θ
′ cos θ′ cosφ′)
√
= 2 ρ2(cos θ cos θ
′ − sin θ sin θ′)(cosα cos θ′ + sinα sin θ′ cosφ′)
+ ρ3(cosα sin θ sin θ
′ cos θ′ − sinα sin θ cos2 θ′ cosφ′]
+ cosα cos θ sin2 θ′ − sinα cos θ sin θ′ cos θ′ cosφ′)
282
When we separate the terms involving sinα from those involving cosα,
√ [
Ω̂ · ε2(r) = 2 cosα ρ2(cos θ cos2 θ′ − sin θ sin θ′]cos θ
′)
+ ρ3(sin θ sin θ
′
[ cos θ
′ + cos θ sin2 θ′)
√
+ 2 sinα ρ (cos θ sin θ′ cos θ′ cosφ′2 − sin θ sin2 θ′] cosφ
′)
− ρ3(sin θ cos2 θ′ cosφ′ + cos θ sin θ′ cos θ′ cosφ′)
we can see that the term involving sinα are odd functions of φ′. This means that
these terms will disappear when we integrate over φ′. On the other hand, the term
involving cosα does not depend on φ′, so the integral is simply the cosα term
multiplied by 2π:
∫ 2π √
dφ′
[
T (t)Ω̂ · ε (r) = 2 2πT (t) ρ (cos θ cos2 θ′3 2 3 2 − sin θ sin θ′ cos θ′)
0 ]
+ ρ3(sin θ sin θ
′ cos θ′ + cos θ sin2 θ′)
Thus, we can write the function A(r0) as
A(r0) = gU cosα (A.10)
where the constant g is defined by
∫ ∞
g = −π1/2 r+ 2 dr [q(r) ·a(r)]
r r0 0
283
∫ √r2−r20
− dta1(r) = 2 ∫ T2(t) cos θ cos θ
′
r−r t0 √
√ r2−r20 dt
a2(r) =− 2 2∫ T3(t) cos θ(cos θ cos
2 θ′ − sin θ sin θ′ cos θ′)
r−√r
t
0
√ r2−r20 dt
a3(r) =− 2(2 ) ∫ T3(t)(cos θ(sin θ sin θ
′ c)os θ
′ + cos θ sin2 θ′)
r−r t0 √
1/2
2 r
2−r20 dt 3
a4(r) =− 2 T4(t)− T2(t) cos θ cos θ′
3 r−r t 20
Finally, we plug in our expressions for cos θ, cos θ′, sin θ, and sin θ′ to get our
expression for g: ∫ ∞ r
g = −π1/2 + 2 dr [q(r) ·a(r)] (A.11)
r r0 0
∫ √
1 r
2−r20 [ ]
a 41(r) = ∫ t − 2r
2t2 + (r4 − 4 T2(t)r0) dt (A.12)2r2r 20 r−r0 √
− r21 −r20 [ t
a2(r) = √ t6 − t4(r2 + r20)− t2(r2 − r2)20
2 2r20r
2
r−r0 ]
+ (r2∫ − r
2
0)(r
4 − T3(t)r40) dt (A.13)3
√ t
1 r
2−r20 [
a3(r) = √ t6 − t4(3r2 + r2) + t2(r20 − r2 2 22 2 0)(3r + r0)2 2r0r r−r0 ]
− 2 − 2 2 T3(t)( ) ∫ (r r0) dt3√ t1/2 r2−r2 [ ] [ ]
(A.14)
2 1 0 4 − 2 2 4 − 2 3 dta4(r) = t 2r t + (r r0) T4(t)− T2(t) (A.15)3 2r2 20r r−r 2 t0
A.2 Derivation of the Source Term Expressions (Eqns. 2.40–2.41)
In this section, I will present the derivation of the W terms that appear in the
source term expressions for flow around a sphere. This derivation follows Appendix
284
C of Lea [74], but I have included more of the intermediate steps for clarity. I have
also included information from Law and Loyalka [76] to account for non-isothermal
conditions. For this derivation, I will refer to the angles defined in Fig. 2.3.
We start by writing the source terms SA(r) and SU(r),
∫
SA1(r) =π
−3/2 A∫(r0) T2(|r − r0|)dΩ̂ω√
SA2(r) = 2π
−3/2
∫ A(r0) T3(|r − r0|)Ω̂zdΩ̂ω√
SA3(r) =√2π−3/2 ∫ A(r0) T(3(|r − r0|)Ω̂xdΩ̂ω )
2 −3/2 | − | − 3SA4(r) = π ∫ A(r0) T4( r r0 ) T2(|r − r0|) dΩ̂3 ω 2
S (r) =− 2π−3/2U1 Ω̂∫ ·U∞T3(|r − r0|)dΩ̂ω√
SU2(r) =− 2 2π−3/2 ∫ Ω̂ ·U∞T4(|r − r0|)Ω̂zdΩ̂ω√
SU3(r) =− 2√2π−3/2 ∫ Ω̂ ·U∞T(4(|r − r0|)Ω̂xdΩ̂ω )
2 3
SU4(r) =− 2 π−3/2 Ω̂ ·U∞ T5(|r − r0|)− T3(|r − r0|) dΩ̂
3 ω 2
where dΩ̂ = sin θdθdφ, Ω̂x = sin θ cosφ, and Ω̂x = cos θ. Introducing the variable t,
where ( )
t2 + r2| − | − r
2
t = r r0 = arccos
0
2rt
r2 − r20 − t2sin θdθ = dt
2rt2
285
the source term equations become
∫ √r2−r2 [ ]2 2 2 ∫0 r − r − t 2π
S (r) =π−3/2A1 ∫ dt T2(t)
0
[ dφA(r0)r−r 2rt20 √ 0√ r2−r2 ] ∫0 r2 − r2 − t2 2π
SA2(r) = 2π
−3/2 dt T (t) cos θ 0∫ 3 dφA(r0)r−r 2rt2√0√ r2−r2 [ ] ∫ 00 r2 − r2 − t2 2π
S −3/2A3(r) =√2π ∫ dt T(3(t) sin θ
0
)[ dφA(r0) cosφr−r 2rt2√0 0r2−r2 ]2 0 3 r2 ∫− r2 − t2 2π
S (r) = π−3/2A4 ∫ dt T4(t[)− T2(t)
0
] ∫ dφA(r2 0)3 r−r 2 2rt√0 0r2−r20 r2 − r2 − t2 2π
S (r) =− 2π−3/2U1 ∫ dt T
0
3(t) dφ (Ω̂ ·U2 ∞)
r−r 2rt0 √ 0
√ r2−r2 [ ]2 2 2 ∫0 2π
SU2(r) =−
r − r − t
2 2π−3/2 ∫ dt T4(t) cos θ
0 dφ (Ω̂ ·U∞)
r−r [ 2rt2√0 0√ r2−r2 ] ∫0 r2 − r2 − t2 2π
SU3(r) =− 2 2π−3/2√ ∫ dt T4(t) sin θ
0 dφ (Ω̂ ·U∞) cosφ
2rt2r−√r0 0
r22 −r
2 ( )[ ]
0 3 r2 2− −3/2 − r0 − t
2
SU4(r) = ∫2 π dt T5(t)− T3(t)3 r−r 2 2rt202π
× dφ (Ω̂ ·U∞)
0
We can integrate analytically over φ, but first we must write A(r0) and Ω̂ ·U∞
in terms of the angles α, θ, and φ. The dot product is
( ) ( )
Ω̂ ·U∞ = sinα, 0, cosα · sin θ cosφ, sin θ sinφ, cos θ
= sinα sin θ cosφ+ cosα cos θ
As we saw in Section A.1,
A(r0) = gU cosα
′ = gU(cosα cos θ′ − sinα sin θ′ cosφ)
286
We can substitute these expressions into our integrals over φ:
∫ 2π ∫ 2π
dφA(r0) =gU (cosα cos θ
′ − sinα sin θ′ cosφ)dφ
0 0
∫ =2πg∫U cosα cos θ′2π 2π
dφA(r0) cosφ =gU cosφ(cosα cos θ
′ − sinα sin θ′ cosφ)dφ
0 0
∫ =∫− πgU sinα sin θ′2π 2π
dφ Ω̂ ·U∞ = (sinα sin θ cosφ+ cosα cos θ)dφ
0 0
∫ =2∫π cosα cos θ2π 2π
dφ Ω̂ ·U∞ cosφ = cosφ(sinα sin θ cosφ+ cosα cos θ)dφ
0 0
=π sinα sin θ
Thus, our source terms are
∫ √r2−r2 [ ]0 2 2 2
S (r) =2π−1/2
r − r − t
A1 gU cosα ∫ dt T (t)
0 cos θ′2
r−r 2rt
2
0 √
√ r2−r2 [ ]0 r2 − r2 − t2
SA2(r) =2 2π
−1/2gU cosα dt T (t) 03 cos θ cos θ
′
r−r 2rt
2
√0
√ ∫ r2−r2 [ ]0 r2 − r2 2
S −1/2 0
− t
A3(r) =−( 2π) gU sinα ∫ dt [T3(t) ] [ sin θ sin θ
′
r−r 2rt
2
√0
1/2 r2−r2 ]2 0 3 r2 − r2 − t2
SA4(r) =2 gU c∫osα dt T4(t)− T2(t)
0 cos θ′
3π r−r [ 2 2rt2√ 0r2−r2 ]0 2 2 2
S −1/2
r − r0 − t
U1(r) =− 4π cosα ∫ dt T3(t) [ cos θ2rt2r−r0 √√ r2−r2 ]0 r2 − r2 2
S (r) =− 4 2π−1/2 cosα dt T (t) 0 − t 2U2 ∫ 4 cos θr−r 2rt2√0√ r2−r2 [ ]0 r2 − r2 − t2
SU3(r) =− 2 2π−1/2 sinα dt T (t) 04 sin2 θ
r−r 2rt
2
0
287
( )1/2 ∫ √r2−r2 [ ] [ ]2 0 3 r2 − r2 − t2
SU4(r) =− 4 cosα dt T5(t)− T3(t) 0 cos θ
3π 2r−r 2 2rt0
Finally, we must write cos θ, cos θ′, sin θ, and sin θ′ in terms of r, t, and r0 and
substitute these expressions into our source term equations:
r2 + r2 − t2
cos θ′ = 0 (A.16a)
2r0r
t2 + r2 − r2
cos θ = 0 (A.16b)
√ 2rt
′ (2rr
2
0) − (r2 + r2 − t2)2
r0 sin θ = t sin θ =
0 (A.16c)
2r
This gives us the final expression for the source terms:
 
WA1(r) cosα

 WA2(r) cosα
SA(r) = gU   (A.17)WA3(r) sinα
WA4(r) cosα
 WU1(r) cosα 

WU2(r) cosαSU(r) = U   (A.18)WU3(r) sinα

WU4(r) cosα
288
∫
√ 1
[ ]
WA1(r) = ∫ t
4 − 2r2 T2(t)t2 + (r4 − r40) dt (A.19)2 πr2r0 [ t21
W (r) = √ t6 − (r2 + r2)t4 − (r2 − r2 2A2 0 0) t2 (A.20)
2 2πr3r0 ]
2 2 2 − 2 2 T3(t)+(r + r0)∫(r[ r0) dt (A.21)t3−1
WA3(r) = √ t6 − (3r2 + r2)t4 + (r2 − r20 0)(3r2 + r2)t20 (A.22)
4 2πr3r0 ] T (t)
(− 3(r)2 − r2 30) dtt31/2 ∫
2 1 [ ] [ ]
(A.23)
W (r) = √ t4A4 ∫ − 2r
2t2 + (r4 − 4 − 3 dtr0) T4(t) T2(t) (A.24)3 2
1 [ πr
2r0 ] 2 t2
W (r) =√ t4 2U1 ∫ − (r −
2 2 T3(t)r0) dt (A.25)πr2 t3
√ 1
[ ]
W (r) = t6 + t4(r2U2 ∫ − r
2)− t2(r2 − 2 2 2 2 3 T4(t)0 r0) − (r − r0) dt (A.26)
2πr3 t4
−1 [
W 6 4U3(r) = √ t − t (3r2 + r2) + t20 (r2 − r2)(3r2 + r20 0) (A.27)
2 2πr3 ]
− 2 − 2 3 T4(t)( (r) r0) ∫t41/2
2 1 [
dt ] [ ] (A.28)
√ 4 − 2 − 2 2 − 3 dtWU4(r) = t (r r0) T5(t) T3(t) (A.29)3 πr2 2 t3
√
The integration limits are r − r0 and r2 − r20.
A.3 Derivation of H
In this section, I will present the derivation of the H terms that appear in the
equations for flow around a sphere. This derivation follows Appendix E of Lea [74],
but I have included more of the intermediate steps for clarity. I have also included
information from Law and Loyalka [76] to account for non-isothermal conditions.
For this derivation, I will refer to the angles defined in Fig. 2.2.
289
We start by writing Lψ(r′) from Eq. (2.39b):
∫ ∫ ∫
Lψ(r′ 1) = π−3/2 r′ 2dr′ sin θ′dθ′ dφ′ Λψ(r′)
|r − r′|2
 ( ) 
3
 T ′ ′ ′[ 1 ε1(r ) + T2 Ω̂ · ε2(r ) + T3− T1 ε3(r )2   √
] 
2 T ε (r′
( )
 2 1 ) + T3 Ω̂ · ε (r
′
2 ) + T −3 T ′ 4 2 2 ε3(r ) Ωz
Λψ(r′) ≡ 


√ [ ( ) ] 
√ [ 2 T2 ε1(r
′) + T Ω̂ · ε (r′) + T −3 T ε (r′3 2 4 2 3 ) Ωx 
( ) ( ) 2 ]
2 T −3 T ε (r′) + T −3 T Ω̂ · ε (r′
( )
9 ′
3 3 2 1 1 4 2 2 2
) + T5−3 T3 + T ε (r )4 1 3
Note that the argument of ε1, ε2, and ε3 is r
′. Here, we have explicitly written the
volume integral in polar coordinates, though we have not written the integration
bounds because the bounds are difficult to formulate in these coordinates. Instead,
it makes sense to transform the integral over θ′ to an integral over t ≡ |r − r′|. We
relate t to θ′ by
√
t = r2 + r′ 2 − 2rr′ cos θ′
Taking the derivative of both sides, we get
rr′ sin θ′√ ′ rr
′ sin θ′
dt = dθ = dθ′
r2 + r′ 2 − 2rr′ cos θ t
Thus, our differential volume is now
r′
dr′ = r′ 2 sin θ′dr′dθ′dφ′ = tdr′dtdφ′ (A.30)
r
290
and Eq. (2.39b) becomes
∫ ∫ √ √∞ ′ r2−r2+ r′ 2−r2 ∫0 0 2π
Lψ(r′) = π−3/2 r ′ dtdr dφ′ Λψ(r′)
r r0 |r−r′| t 0
Next, we must write the expression Λψ(r′) in terms of the angles defined in
Fig. 2.2. The density and temperature perturbations are simply
ε ′1(r ) = ρ1(r
′) cosα′ = ρ ′ ′ ′1(r)[cosα cos θ + sinα sin θ cosφ ]
( )1/2 ( )1/2
′ 2 ′ ′ 2ε (r ) = ρ (r ) cosα = ρ (r)[cosα cos θ′3 4 4 + sinα sin θ
′ cosφ′]
3 3
This is the same expression that we used in Section A.1. Similarly, we have
r − r′
Ω̂ = ′ = − sin θ cosφ
′êx − sin θ sinφ′êy + cos θê|r − zr |
and
√ [
Ω̂ · ε2(r′) = 2 cosα ρ2(cos θ cos2 θ′ − sin θ sin θ′]cos θ
′)
+ ρ3(sin θ s[in θ
′ cos θ′ + cos θ sin2 θ′)
√
+ 2 sinα ρ2(cos θ sin θ
′ cos θ′ cosφ′ − sin θ sin2 θ′] cosφ
′)
− ρ3(sin θ cos2 θ′ cosφ′ + cos θ sin θ′ cos θ′ cosφ′)
291
Integrating over φ′, we have
∫ 2π
ε (r′)dφ′ =2πρ (r′) cos θ′∫ 1 1 cosα02π √ [
Ω̂ · ε (r′2 )dφ′ =2 2π ρ (cos θ cos2 θ′2 − sin θ sin θ′ cos θ′)
0 ]
∫ + ρ3(sin θ sin θ′ cos θ′ + cos θ sin2 θ′) cosα2π
∫ ε1(r
′)Ω̂zdφ
′ =2πρ1(r
′) cos θ cos θ′ cosα
0
2π √ [
Ω̂ · ε2(r′)Ω̂zdφ′ =2 2π ρ2(cos2 θ cos2 θ′ − sin θ cos θ sin θ′ cos θ′)
0 ]
∫ + ρ3(sin θ cos θ sin θ′ cos θ′ + cos2 θ sin2 θ′) cosα2π
∫ ε1(r
′)Ω̂xdφ
′ =− πρ1(r′) sin θ sin θ′ sinα
0
2π √ [
Ω̂ · ε2(r′)Ω̂ dφ′x = 2π ρ2(sin2 θ sin2 θ′ − sin θ cos θ sin θ′ cos θ′)
0 ]
+ ρ3(sin
2 θ cos2 θ′ + sin θ cos θ sin θ′ cos θ′) sinα
292
We can now write Lψ(r′) as
   cosα 0 0 0  H11 H   12
H13 H14
∫    

∞ r′  0 cosα 0 0 

 H21 H22 H 23 H24Lψ(r′) = π−1/2 · 
r r0  0 0 sinα 0  H31 H32 H H 33 34
0 0 0 cosα H41 H42 H43 H44
 ρ1(r′) 

ρ2(r′)·  
 dr′
ρ (r′3 )
ρ4(r
′)
where
∫
′ ′T1(t)H11(r, r ) =2 c∫os θ dt[ t√ ] T2(t)
H ′ 2 ′ ′ ′12(r, r ) =2 2∫ cos θ cos θ − sin θ sin θ cos θ dt√ [ ] tT2(t)
H13(r, r
′) =2(2 ) sin θ sin θ
′ cos θ′ + cos θ sin2∫ [ ]θ
′ dt
t
1/2
′ 2 3 dtH14(r, r ) =2
′
∫ cos θ T3(t)− T1(t)3 2 t√
′ ′T2(t)H21(r, r ) =2∫2 cos θ cos θ dt[ t ] T3(t)
H22(r, r
′) =4∫ cos
2 θ cos2 θ′ − sin θ cos θ sin θ′ cos θ′ dt
t
H (r, r′
[ ] T3(t)
23 ) =4
′ ′ 2 2 ′
√ (
sin)θ cos∫θ sin θ cos θ + cos θ sin θ dtt1/2 [ ]
2 3 dt
H24(r, r
′) =2 2 ∫ cos θ cos θ
′ T4(t)− T2(t)
3 2 t
√
H31(r, r
′) =− ′T2(t)2 sin θ sin θ dt
t
293
∫
′ [ ] T3(t)H32(r, r ) =2∫ sin
2 θ sin2 θ′ − sin θ cos θ sin θ′ cos θ′ dt
[ ] tT3(t)
H33(r, r
′) =2 si 2 2 ′(n θ)cos ∫θ + sin θ cos[θ sin θ
′ cos θ′ dt
t
√ 1/2 ]2 3 dt
H34(r, r
′) =− ′( 2) sin θ sin θ T4(t)− T2(t)3 2 t1/2 ∫ [ ]
2
H ′41(r, r ) =2 ( ) c
′
∫os θ T3(t)−
3 dt
T1(t)
3 2 t
√ 1/2 [ ]
H (r, r′
2 [ 2 ′ − ′ ′] − 3 dt42 ) =2 2( cos θ cos θ sin θ sin θ cos θ T4(t) T2(t)3√ ) 2 t1/2 ∫ [ ]2 [
H (r, r′) =2 2 sin θ sin θ′ cos θ′ + cos θ sin2 θ′
]
− 3 dt43 ( )∫ [ ] T4(t) T2(t)3 2 t
2 9 dt
H44(r, r
′) =2 cos θ′ T5(t)− 3 T3(t) + T1(t)
3 4 t
Finally, we will write cos θ, cos θ′, sin θ, and sin θ′ in terms of r′, r, and t:
2 ′ 2 2
cos θ′
r + r − t
= ′ (A.31a)2rr
t2 + r2 − r′ 2
cos θ = (A.31b)
√ 2rt
(2rr′)2 − (r2 + r′ 2 − t2)2
r′ sin θ′ = t sin θ = (A.31c)
2r
Substituting these expressions into the above equations for H(r, r′), we get
∫
H (r, r′
−1 [ ]
) = t211 −
T1(t)
(r2 + r′ 2) dt (A.32)
rr′ ∫ t
H (r, r′
[ ]
) =√ 1 t4 − 2r2 2 − T2(t)12 ∫ t (r
′ 4 − r4) dt (A.33)
2rr′ 2 [ t2−1 ] T2(t)
H ′ 413(r, r ) =√ ∫ t − 2(r
′ 2 + r2)[t
2 + (r′ 2 − r2)2 ] dt (A.34)2rr′ 2√ [ ] t2
H14(r, r
′) =√− 2 2 − 2 ′ 2 − 3 dtt (r + r ) T (t) T (t) (A.35)
3rr′
3 1
2 t
294
∫
H (r, r′
−1 [ ]
21 ) =√ 4∫ t − 2r
′ 2t2 − (r4 − T2(t)r′ 4) dt (A.36)
2r2r′
′ 1 [ t
2
H (r, r ) = t6 4 ′ 2 222 ′ − t (r + r )− t
2(r2 − r′ 2)2
2r2r 2 ]
+(r′ 2 − T3(t)∫ r2[ )(r
′ 4 − r4) dt (A.37)
t3
H (r, r′
−1
) = t623 ′ − t
4(3r′ 2 + r2) + t2(r′ 2 − r2)(3r′ 2 + r2)
2r2r 2
−(r′
]
2 − 2 3 T3(t)∫r [) dt [ ] (A.38)t3′ ]√−1 4 − ′ 2 2 − 4 − ′ 4 − 3 dtH24(r, r ) = ∫ t 2r t (r r ) T4(t) T2(t) (A.39)3r2r′ [ ] 2 t′ √1 4 − ′ 2 2 2 T2(t)H31(r, r ) = ∫ t 2(r + r )t + (r
2 − r′ 2)2 dt (A.40)
2 2r2r t2
H (r, r′
−1 [ 6
32 ) = ′ t − t
4(3r2 + r′ 2) + t2(r2 − r′ 2)(3r2 + r′ 2)
4r2r 2
−(r2 −∫r′
]
2 3 T3(t)
[ ) dt (A.41)t31
H33(r, r
′) = t6′ − t
4(3r2 + 3r′ 2)
4r2r 2 ]
2 4 2 ′ 2 ′ 4 − 4 − ′ 4 2 − ′ 2 T3(t)+t (3r ∫+ [2r r + 3r ) (r r )(r r ) dt (A.42)t3
H (r, r′34 ) = √
1
t4 [− 2(r
′ 2 + r2)t2
2 3r2r ] ]
+(r2 − r′ 2)2 − 3 dt∫ T4(t) T[2(t)√ [ 2] t ]
(A.43)
H41(r, r
′) =√− 2 3 dt∫ t
2 − (r2 + r′ 2) T3(t)−[ T1(t) ] (A.44)3rr′ 2 t
H (r, r′
1 [ ] 3 dt
42 ) =√ t4 − 2r2∫ t
2 − (r′ 4 − r4) T
′ 4
(t)− T2(t) (A.45)
3rr 2 [ 2 t−1
H43(r, r
′) =√ t4 − 2(r′ 2 + r2)t2
3rr′ 2 ] [ ]
+(r′ 2 − r2 2 3 dt∫ ) T4(t)− [T2(t) ] (A.46)
−2 [ ]2 t 9 dt
H44(r, r
′) = t2 − (r2 + r′ 2′ ) T5(t)− 3 T3(t) + T1(t) (A.47)3rr 4 t
295
√ √
where the integration extends from |r − r′| to r2 + r20 + r′ 2 − r20.
A.4 Derivation of the Drag Expression (Eq. (2.54))
In this section, I will present the derivation of the expression for the drag,
Eq. (2.54). For this section, I will be using the same coordinates that I used in
Section A.1.
We will start by writing our expression for the shear stress [Eq. (2.52)] in
compressed form:
τij(r0) = ρ∞ [Bij + Cij −Dij] (A.48)
∫ [
B ≡ π−3/2ij T3((|r − r0|)ε1(r) + T4(|r − r0|)Ω̂ · ε2(r)V ) ]
| 3 dr+ T5( r − r0|)− T3(|r − r0|) ε3(r) ΩiΩj (A.49a)
2 |r − r0|
where ∫
C ≡ π−3/2A(r ) c c e−c2ij 0 i j dc (A.49b)
∫ c · n̂>0
−3/2 2Dij ≡ π 2U∞ · c c −cicj e dc (A.49c)
c · n̂>0
In terms of this new notation, the drag force is
( ) ∫ ∫
2kBT
2π π
∞ [
F̃ = ρ r̃2D,Z ∞ 0 dφ dα sinα (Bzx + Czx −Dzx) sinαm 0 0 ]
+ (Bzz + Czz −Dzz) cosα (A.50)
296
where α is the angle between U∞ and n̂.
We will first focus on the expression Bij. In terms of the coordinates (r, t, φ
′)
from Section A.1, we have
∫ ∞ ∫ √r2−r2 ∫0 2π [ ]
B = π−3/2
r dt
dr dφ′ij T3(t)ε1(r)− T4(t)Ω̂ · ε2(r) ΩiΩj
r r0 0 r−r t0 0
Next, we substitute the expressions for ε1(r) and U∞ · ε2(r) from Section A.1 into
the integral:
∫ ∫ ∫ [
− r dt ( )Bij =π 3/2 dr dφ′ T3(t)ρ1(r) cosα cos θ′ + sinα sin θ′ cosφ′
r0 {t√ ( ) (
+ 2 T4(t) co)s}α ρ (r) cos θ cos
2 θ′2 { − s(in θ sin θ
′ cos θ′ + ρ sin θ sin θ′3 cos θ
′
√
+ cos θ sin2 θ′ + ) 2 T4(t) s(inα ρ2(r) cos θ sin θ
′ cos θ′ cosφ′ )}
−√sin θ(sin2 θ′ cosφ′ − ρ)3(r) sin θ cos2 θ′( cosφ
′ + cos θ sin θ′ cos θ]′ cosφ′
2 3 )
+ T5(t)− T3(t) ρ4(r) cosα cos θ′ + sinα sin θ′ cosφ′ ΩiΩj
3 2
We can integrate analytically over φ′ for each i, j. As we saw in the derivation
of the drag force, we need only consider two components of the shear stress tensor,
τzx and τzz. Thus, we will only perform the integration for Bzx and Bzz. In these
coordinates, Ωx = − sin θ cosφ′ and Ωz = cos θ. The integral will be non-zero only
for even functions of φ′, meaning terms containing cosφ′ will be zero while terms
containing cos2 φ′ are multiplied by π and terms independent of φ′ are multiplied by
297
2π. Thus, the terms Bzx and Bzz are
∫ ∫ [
B =π−1/2
r dt
zx U sinα dr − T3(t)q1(r) sin θ cos θ sin θ′
( r0 t√ )
+ 2 T4(t)q2(r)( sin
2 θ cos θ sin2 θ′ − sin θ cos2 θ sin θ′ cos θ′
√ )
+√2 T(4(t)q3(r) sin2 θ c)os θ cos2 θ′ + sin θ cos2]θ sin θ′ cos θ′
− 2 3T5(t)∫− T3(∫t) ρ4[(r) sin θ cos θ sin θ
′ (A.51)
3 2
B =2π−1/2
r dt
zz U cosα dr T3(t)q (r) cos
2
1 θ cos θ
′
( r0 t√ )
+ 2 T (t)q 3 2 ′ 2 ′ ′4 2(r)( cos θ cos θ − sin θ cos θ sin θ cos θ√ )
+√2 T((t)q (r) cos3 2 ′ 2 ′ ′4 3 θ s)in θ + sin θ cos θ]sin θ cos θ
2
+ T5(t)−
3
T3(t) ρ4(r) cos
2 θ cos θ′ (A.52)
3 2
We will eventually substitute Eq. (A.9) for cos θ, cos θ′, sin θ, and sin θ′, but we will
defer that substitution until later in this section.
Let us now turn our attention to Cij and Dij. In terms of the (x, y, z) coordi-
nates, with z normal to the sphere at r0, these expressions are
∫ ∞ ∫ ∞ ∫ ∞
2 2 2
C −3/2ij = π gU cosα dcz dcy dcx cicj e
−cx−cy−cz
0 −∞ −∞
∫ ∞ ∫ ∞ ∫ ∞
2 2 2
Dij = 2π
−3/2U dcz dcy dcx (cx sinα + cz cosα)cic e
−cx−cy−cz
j
0 −∞ −∞
Here, we have substituted A(r0) = gU cosα and U∞ · c = Ucz cosα into our ex-
pressions for Cij and Dij, respectively. The above integrals are straightforward to
298
compute for any ij, giving
1
Czx = 0 Czz = gU cosα
4
U
Dzx = sinα D = π
−1/2U cosα
2π1/2
zz
We can now consider the drag force given by Eq. (A.50). Since θ and θ′ are
independent of α and φ, the integration in the drag expression is straightforward,
giving the following expression for the drag:
( ) ∫
2k T 2π
∫ π
B ∞ [
F̃ =ρ r̃2D,Z ∞ 0 dφ dα sinα (Bzx −Dzx) sinαm 0 0 ]
+ ((Bzz + Cz)z −D∫ zz) cos∫α
2k T 2π π
[( )
B ∞ 2 Bzx 1=ρ U r̃ dφ dα − sin3∞( 0 ) αm 0 0 U]sinα 2π1/2
Bzz g
+ ( + − π
−1/2
) [ sinα cos
2 α
U cosα 4
1/2 ]
2kBT∞ 4π 2Bzx Bzz g
=ρ Ũ r̃2 + + − 2π−1/2∞ ( m ) 0 3 U sinα U cosα 41/2 [ ( ) ]
ρ∞Ũ 2πkBT∞ 2 1/2 2Bzx Bzz= 1/2
3 ( )r̃0 4π[ + + gπ − 8m U sinα ∫U cosα1/2 ∫
− ρ∞Ũ 2πkBT∞ r dt= { r̃
2 8− gπ1/20 + 8 dr3 m ( r0) t
× T3(t)q1(r) sin θ cos θ sin θ′ − cos2 θ cos θ′
√ (
− 2 T (t)q (r) cos3 θ cos2 θ′ − 2 sin θ cos2 θ sin θ′ ′4 2
′)
cos θ
√ (
+ sin2 θ cos θ sin2 θ − 2 T (t)q (r) cos3 θ sin2 θ′4 3 )
+ 2√sin(θ cos2 θ sin θ′ cos θ)′ + sin2(θ cos θ cos
2 θ′ }]
2 3
+ T5(t)− T3(t) q4(r) sin θ cos θ sin θ′ − cos2 θ cos θ′
3 2
299
We next substitute Eq. (A.9) for cos θ, cos θ′, sin θ, and sin θ′:
( )1/2 [ ∫ 1/2 2
F̃ =− ρ∞Ũ 2πkBT∞ r̃2 1/2 2π rD,Z {3 ∫ 0 8− gπ + drm r20
× √q1(r) T3(t)
[ ]
dt t4 − (r2 − r2)2
πr2∫ t3 0
− √q2(r) T4(t)
[ ]
∫ dt t
6 + t4(r2 − r2)− t2(r2 − r2)2 − (r2 − r20 0 0)3
2πr3 t4 [ ]
+ √q3(r) T4(t)dt t6 − t4(3r2∫ [ + r
2 2
] 0) + t (3r
2 + r2 2 2 2 2 3
3 t4 0
)}(r] − r0)− (r − r0)√2πr
√2q4(r) dt 3
[ ]
+ T (t)− T (t) t45 3 − (r2 − r20)2
3πr2 t3 2
The integrals over t in the above expressions are the WU expressions from
Section A.2. After replacing the integrals over t with Eqns. (A.25–A.28), we arrive
at our final expression for the drag in terms of q(r) (Eq. (2.54)):
( )1/2 [ ∫ {
−ρ∞Ũ 2πk T
1/2 2
B ∞
F̃ = r̃2
2π r
D,Z 0 8− gπ1/2 + dr q1(r)WU1(r)3 m r20 }]
− q2(r)WU2(r)− 2q3(r)WU3(r) + q4(r)WU4(r)
300
Appendix B: BGK Results
My method for calculating the drag on an aerosol fractal aggregate in the
transition regime requires knowledge of the velocity field around a sphere. I de-
termined the velocity field using the Bhatnagar-Gross-Krook model [71] in the lin-
earized Boltzmann equation, following the procedure of Lea and Loyalka [75] and
Law and Loyalka [76]. I am providing results here in case anyone wishes to apply
my method for calculating the drag force on an aggregate.
I have listed the calculated values of q2(r) and q3(r) for a range of Knudsen
numbers in Tables B.1–B.24. I have included the values for c1 and c2 in Table B.25.
These variables have been defined Chapter 3. Note that when I compute the drag
force on an aggregate, I use the asymptotic solutions for q2 and q3 when the distance
rij between primary spheres is greater than the maximum r in the tables below.
301
Table B.1: Results for a = 0.01 (Kn = 88.8)
r q2(r) q3(r) r q2(r) q3(r)
0.0135 -8.6764E-01 -6.9165E-02 5.1318 -5.3060E-05 -3.4932E-05
0.0283 -2.0298E-01 9.3393E-05 5.3750 -5.1404E-05 -3.4073E-05
0.0549 -5.3819E-02 -2.6852E-04 5.6173 -4.9893E-05 -3.3251E-05
0.0933 -1.8869E-02 -3.8681E-04 5.8582 -4.8509E-05 -3.2461E-05
0.1434 -8.1915E-03 -3.2227E-04 6.0971 -4.7236E-05 -3.1700E-05
0.2050 -4.1498E-03 -2.5401E-04 6.3334 -4.6062E-05 -3.0965E-05
0.2779 -2.3575E-03 -2.0311E-04 6.5666 -4.4975E-05 -3.0253E-05
0.3622 -1.4627E-03 -1.6673E-04 6.7961 -4.3967E-05 -2.9565E-05
0.4574 -9.7309E-04 -1.4044E-04 7.0214 -4.3029E-05 -2.8899E-05
0.5634 -6.8506E-04 -1.2099E-04 7.2418 -4.2157E-05 -2.8250E-05
0.6800 -5.0536E-04 -1.0622E-04 7.4570 -4.1342E-05 -2.7623E-05
0.8069 -3.8766E-04 -9.4762E-05 7.6664 -4.0582E-05 -2.7016E-05
0.9437 -3.0736E-04 -8.5675E-05 7.8695 -3.9871E-05 -2.6429E-05
1.0901 -2.5064E-04 -7.8335E-05 8.0658 -3.9208E-05 -2.5861E-05
1.2459 -2.0936E-04 -7.2310E-05 8.2548 -3.8588E-05 -2.5314E-05
1.4106 -1.7853E-04 -6.7291E-05 8.4362 -3.8009E-05 -2.4788E-05
1.5838 -1.5496E-04 -6.3055E-05 8.6094 -3.7470E-05 -2.4283E-05
1.7652 -1.3660E-04 -5.9438E-05 8.7741 -3.6969E-05 -2.3800E-05
1.9542 -1.2202E-04 -5.6316E-05 8.9299 -3.6504E-05 -2.3340E-05
2.1505 -1.1026E-04 -5.3594E-05 9.0763 -3.6075E-05 -2.2904E-05
2.3536 -1.0064E-04 -5.1200E-05 9.2131 -3.5679E-05 -2.2492E-05
2.5630 -9.2671E-05 -4.9076E-05 9.3400 -3.5317E-05 -2.2105E-05
2.7782 -8.5989E-05 -4.7176E-05 9.4566 -3.4989E-05 -2.1745E-05
2.9986 -8.0328E-05 -4.5462E-05 9.5626 -3.4693E-05 -2.1413E-05
3.2239 -7.5484E-05 -4.3907E-05 9.6578 -3.4429E-05 -2.1108E-05
3.4534 -7.1301E-05 -4.2485E-05 9.7421 -3.4197E-05 -2.0832E-05
3.6866 -6.7661E-05 -4.1175E-05 9.8150 -3.3997E-05 -2.0587E-05
3.9229 -6.4468E-05 -3.9963E-05 9.8766 -3.3829E-05 -2.0374E-05
4.1618 -6.1649E-05 -3.8832E-05 9.9267 -3.3693E-05 -2.0192E-05
4.4027 -5.9142E-05 -3.7773E-05 9.9651 -3.3588E-05 -2.0046E-05
4.6450 -5.6900E-05 -3.6775E-05 9.9917 -3.3516E-05 -1.9936E-05
4.8882 -5.4883E-05 -3.5830E-05 10.0065 -3.3475E-05 -1.9868E-05
302
Table B.2: Results for a = 0.025 (Kn = 35.5)
r q2(r) q3(r) r q2(r) q3(r)
0.0285 -1.1687E+00 -2.0677E-01 5.1468 -2.2857E-04 -1.2281E-04
0.0433 -5.4136E-01 -1.8587E-02 5.3900 -2.1898E-04 -1.1877E-04
0.0699 -2.1113E-01 -2.5890E-03 5.6323 -2.1031E-04 -1.1500E-04
0.1083 -8.9083E-02 -1.7876E-03 5.8732 -2.0244E-04 -1.1148E-04
0.1584 -4.2528E-02 -1.5542E-03 6.1121 -1.9528E-04 -1.0819E-04
0.2200 -2.2709E-02 -1.2898E-03 6.3484 -1.8875E-04 -1.0510E-04
0.2929 -1.3300E-02 -1.0571E-03 6.5816 -1.8278E-04 -1.0219E-04
0.3772 -8.3971E-03 -8.7321E-04 6.8111 -1.7731E-04 -9.9451E-05
0.4724 -5.6372E-03 -7.3198E-04 7.0364 -1.7228E-04 -9.6876E-05
0.5784 -3.9811E-03 -6.2338E-04 7.2568 -1.6767E-04 -9.4437E-05
0.6950 -2.9330E-03 -5.3888E-04 7.4720 -1.6342E-04 -9.2138E-05
0.8219 -2.2391E-03 -4.7214E-04 7.6814 -1.5950E-04 -8.9968E-05
0.9587 -1.7619E-03 -4.1859E-04 7.8845 -1.5589E-04 -8.7920E-05
1.1051 -1.4227E-03 -3.7500E-04 8.0808 -1.5256E-04 -8.5987E-05
1.2609 -1.1745E-03 -3.3902E-04 8.2698 -1.4950E-04 -8.4166E-05
1.4256 -9.8848E-04 -3.0896E-04 8.4512 -1.4667E-04 -8.2450E-05
1.5988 -8.4587E-04 -2.8355E-04 8.6244 -1.4407E-04 -8.0837E-05
1.7802 -7.3443E-04 -2.6186E-04 8.7891 -1.4169E-04 -7.9325E-05
1.9692 -6.4582E-04 -2.4317E-04 8.9449 -1.3951E-04 -7.7911E-05
2.1655 -5.7428E-04 -2.2694E-04 9.0913 -1.3751E-04 -7.6594E-05
2.3686 -5.1573E-04 -2.1274E-04 9.2281 -1.3569E-04 -7.5372E-05
2.5780 -4.6720E-04 -2.0022E-04 9.3550 -1.3405E-04 -7.4243E-05
2.7932 -4.2653E-04 -1.8912E-04 9.4716 -1.3257E-04 -7.3209E-05
3.0136 -3.9211E-04 -1.7922E-04 9.5776 -1.3125E-04 -7.2268E-05
3.2389 -3.6271E-04 -1.7034E-04 9.6728 -1.3009E-04 -7.1419E-05
3.4684 -3.3738E-04 -1.6233E-04 9.7571 -1.2907E-04 -7.0664E-05
3.7016 -3.1540E-04 -1.5508E-04 9.8300 -1.2820E-04 -7.0003E-05
3.9379 -2.9619E-04 -1.4849E-04 9.8916 -1.2748E-04 -6.9438E-05
4.1768 -2.7930E-04 -1.4247E-04 9.9417 -1.2689E-04 -6.8966E-05
4.4177 -2.6436E-04 -1.3694E-04 9.9801 -1.2644E-04 -6.8594E-05
4.6600 -2.5108E-04 -1.3186E-04 10.0067 -1.2614E-04 -6.8324E-05
4.9032 -2.3921E-04 -1.2716E-04 10.0215 -1.2596E-04 -6.8163E-05
303
Table B.3: Results for a = 0.05 (Kn = 17.8)
r q2(r) q3(r) r q2(r) q3(r)
0.0535 -1.2925E+00 -3.3188E-01 5.1718 -7.7134E-04 -3.6368E-04
0.0683 -8.5319E-01 -7.8173E-02 5.4150 -7.3460E-04 -3.4943E-04
0.0949 -4.5889E-01 -1.8149E-02 5.6573 -7.0147E-04 -3.3632E-04
0.1333 -2.3791E-01 -7.6369E-03 5.8982 -6.7151E-04 -3.2423E-04
0.1834 -1.2860E-01 -5.3797E-03 6.1371 -6.4435E-04 -3.1309E-04
0.2450 -7.4124E-02 -4.3695E-03 6.3734 -6.1966E-04 -3.0279E-04
0.3179 -4.5581E-02 -3.6359E-03 6.6066 -5.9718E-04 -2.9326E-04
0.4022 -2.9714E-02 -3.0511E-03 6.8361 -5.7667E-04 -2.8443E-04
0.4974 -2.0380E-02 -2.5840E-03 7.0614 -5.5792E-04 -2.7626E-04
0.6034 -1.4601E-02 -2.2116E-03 7.2818 -5.4078E-04 -2.6867E-04
0.7200 -1.0858E-02 -1.9135E-03 7.4970 -5.2508E-04 -2.6164E-04
0.8469 -8.3375E-03 -1.6730E-03 7.7064 -5.1069E-04 -2.5512E-04
0.9837 -6.5807E-03 -1.4768E-03 7.9095 -4.9750E-04 -2.4908E-04
1.1301 -5.3188E-03 -1.3151E-03 8.1058 -4.8541E-04 -2.4347E-04
1.2859 -4.3882E-03 -1.1805E-03 8.2948 -4.7434E-04 -2.3829E-04
1.4506 -3.6857E-03 -1.0671E-03 8.4762 -4.6419E-04 -2.3349E-04
1.6238 -3.1443E-03 -9.7086E-04 8.6494 -4.5491E-04 -2.2906E-04
1.8052 -2.7194E-03 -8.8836E-04 8.8141 -4.4644E-04 -2.2497E-04
1.9942 -2.3803E-03 -8.1711E-04 8.9699 -4.3873E-04 -2.2122E-04
2.1905 -2.1057E-03 -7.5513E-04 9.1163 -4.3172E-04 -2.1779E-04
2.3936 -1.8803E-03 -7.0086E-04 9.2531 -4.2538E-04 -2.1465E-04
2.6030 -1.6932E-03 -6.5307E-04 9.3800 -4.1966E-04 -2.1181E-04
2.8182 -1.5362E-03 -6.1074E-04 9.4966 -4.1455E-04 -2.0924E-04
3.0386 -1.4032E-03 -5.7307E-04 9.6026 -4.1001E-04 -2.0695E-04
3.2639 -1.2894E-03 -5.3939E-04 9.6978 -4.0602E-04 -2.0492E-04
3.4934 -1.1914E-03 -5.0916E-04 9.7821 -4.0255E-04 -2.0314E-04
3.7266 -1.1064E-03 -4.8193E-04 9.8550 -3.9960E-04 -2.0162E-04
3.9629 -1.0321E-03 -4.5731E-04 9.9166 -3.9714E-04 -2.0035E-04
4.2018 -9.6678E-04 -4.3497E-04 9.9667 -3.9516E-04 -1.9931E-04
4.4427 -9.0910E-04 -4.1466E-04 10.0051 -3.9366E-04 -1.9852E-04
4.6850 -8.5790E-04 -3.9614E-04 10.0317 -3.9262E-04 -1.9797E-04
4.9282 -8.1222E-04 -3.7920E-04 10.0465 -3.9204E-04 -1.9766E-04
304
Table B.4: Results for a = 0.075 (Kn = 11.8)
r q2(r) q3(r) r q2(r) q3(r)
0.0785 -1.3351E+00 -4.0357E-01 5.1968 -1.6182E-03 -7.2215E-04
0.0933 -1.0118E+00 -1.3874E-01 5.4400 -1.5378E-03 -6.9182E-04
0.1199 -6.4227E-01 -4.4412E-02 5.6823 -1.4654E-03 -6.6404E-04
0.1583 -3.8015E-01 -1.8857E-02 5.9232 -1.4000E-03 -6.3856E-04
0.2084 -2.2541E-01 -1.1779E-02 6.1621 -1.3407E-03 -6.1516E-04
0.2700 -1.3832E-01 -9.0517E-03 6.3984 -1.2869E-03 -5.9366E-04
0.3429 -8.8756E-02 -7.4545E-03 6.6316 -1.2379E-03 -5.7390E-04
0.4272 -5.9595E-02 -6.2821E-03 6.8611 -1.1933E-03 -5.5570E-04
0.5224 -4.1735E-02 -5.3573E-03 7.0864 -1.1526E-03 -5.3896E-04
0.6284 -3.0351E-02 -4.6129E-03 7.3068 -1.1154E-03 -5.2354E-04
0.7450 -2.2817E-02 -4.0085E-03 7.5220 -1.0814E-03 -5.0934E-04
0.8719 -1.7659E-02 -3.5140E-03 7.7314 -1.0503E-03 -4.9628E-04
1.0087 -1.4017E-02 -3.1061E-03 7.9345 -1.0218E-03 -4.8426E-04
1.1551 -1.1375E-02 -2.7666E-03 8.1308 -9.9574E-04 -4.7321E-04
1.3109 -9.4102E-03 -2.4816E-03 8.3198 -9.7191E-04 -4.6307E-04
1.4756 -7.9174E-03 -2.2402E-03 8.5012 -9.5013E-04 -4.5376E-04
1.6488 -6.7607E-03 -2.0341E-03 8.6744 -9.3024E-04 -4.4524E-04
1.8302 -5.8488E-03 -1.8568E-03 8.8391 -9.1212E-04 -4.3746E-04
2.0192 -5.1183E-03 -1.7032E-03 8.9949 -8.9565E-04 -4.3037E-04
2.2155 -4.5249E-03 -1.5693E-03 9.1413 -8.8071E-04 -4.2394E-04
2.4186 -4.0368E-03 -1.4517E-03 9.2781 -8.6721E-04 -4.1812E-04
2.6280 -3.6306E-03 -1.3481E-03 9.4050 -8.5508E-04 -4.1289E-04
2.8432 -3.2890E-03 -1.2563E-03 9.5216 -8.4424E-04 -4.0822E-04
3.0636 -2.9991E-03 -1.1745E-03 9.6276 -8.3463E-04 -4.0409E-04
3.2889 -2.7510E-03 -1.1014E-03 9.7228 -8.2620E-04 -4.0046E-04
3.5184 -2.5370E-03 -1.0358E-03 9.8071 -8.1888E-04 -3.9733E-04
3.7516 -2.3511E-03 -9.7680E-04 9.8800 -8.1266E-04 -3.9467E-04
3.9879 -2.1886E-03 -9.2350E-04 9.9416 -8.0748E-04 -3.9248E-04
4.2268 -2.0458E-03 -8.7525E-04 9.9917 -8.0332E-04 -3.9073E-04
4.4677 -1.9196E-03 -8.3146E-04 10.0301 -8.0016E-04 -3.8941E-04
4.7100 -1.8075E-03 -7.9162E-04 10.0567 -7.9799E-04 -3.8853E-04
4.9532 -1.7076E-03 -7.5531E-04 10.0715 -7.9678E-04 -3.8805E-04
305
Table B.5: Results for a = 0.1 (Kn = 8.88)
r q2(r) q3(r) r q2(r) q3(r)
0.1035 -1.3563E+00 -4.5258E-01 5.2218 -2.7589E-03 -1.1976E-03
0.1183 -1.1049E+00 -1.9204E-01 5.4650 -2.6195E-03 -1.1456E-03
0.1449 -7.7518E-01 -7.5273E-02 5.7073 -2.4940E-03 -1.0980E-03
0.1833 -5.0281E-01 -3.4494E-02 5.9482 -2.3805E-03 -1.0545E-03
0.2334 -3.2005E-01 -2.0792E-02 6.1871 -2.2777E-03 -1.0146E-03
0.2950 -2.0679E-01 -1.5336E-02 6.4234 -2.1844E-03 -9.7808E-04
0.3679 -1.3770E-01 -1.2420E-02 6.6566 -2.0995E-03 -9.4456E-04
0.4522 -9.4958E-02 -1.0442E-02 6.8861 -2.0222E-03 -9.1380E-04
0.5474 -6.7806E-02 -8.9315E-03 7.1114 -1.9517E-03 -8.8557E-04
0.6534 -5.0023E-02 -7.7233E-03 7.3318 -1.8873E-03 -8.5966E-04
0.7700 -3.8012E-02 -6.7380E-03 7.5470 -1.8285E-03 -8.3589E-04
0.8969 -2.9657E-02 -5.9256E-03 7.7564 -1.7747E-03 -8.1409E-04
1.0337 -2.3685E-02 -5.2501E-03 7.9595 -1.7255E-03 -7.9411E-04
1.1801 -1.9310E-02 -4.6837E-03 8.1558 -1.6805E-03 -7.7581E-04
1.3359 -1.6031E-02 -4.2052E-03 8.3448 -1.6394E-03 -7.5909E-04
1.5006 -1.3524E-02 -3.7977E-03 8.5262 -1.6019E-03 -7.4381E-04
1.6738 -1.1571E-02 -3.4483E-03 8.6994 -1.5676E-03 -7.2988E-04
1.8552 -1.0024E-02 -3.1465E-03 8.8641 -1.5364E-03 -7.1721E-04
2.0442 -8.7802E-03 -2.8842E-03 9.0199 -1.5081E-03 -7.0576E-04
2.2405 -7.7669E-03 -2.6549E-03 9.1663 -1.4824E-03 -6.9536E-04
2.4436 -6.9310E-03 -2.4534E-03 9.3031 -1.4592E-03 -6.8602E-04
2.6530 -6.2339E-03 -2.2754E-03 9.4300 -1.4384E-03 -6.7767E-04
2.8682 -5.6466E-03 -2.1174E-03 9.5466 -1.4198E-03 -6.7025E-04
3.0886 -5.1473E-03 -1.9766E-03 9.6526 -1.4033E-03 -6.6372E-04
3.3139 -4.7193E-03 -1.8506E-03 9.7478 -1.3889E-03 -6.5802E-04
3.5434 -4.3498E-03 -1.7376E-03 9.8321 -1.3763E-03 -6.5313E-04
3.7766 -4.0285E-03 -1.6359E-03 9.9050 -1.3657E-03 -6.4901E-04
4.0129 -3.7474E-03 -1.5440E-03 9.9666 -1.3568E-03 -6.4565E-04
4.2518 -3.5001E-03 -1.4609E-03 10.0167 -1.3497E-03 -6.4297E-04
4.4927 -3.2814E-03 -1.3854E-03 10.0551 -1.3443E-03 -6.4099E-04
4.7350 -3.0872E-03 -1.3169E-03 10.0817 -1.3406E-03 -6.3968E-04
4.9782 -2.9140E-03 -1.2545E-03 10.0965 -1.3386E-03 -6.3899E-04
306
Table B.6: Results for a = 0.25 (Kn = 3.55)
r q2(r) q3(r) r q2(r) q3(r)
0.2535 -1.3934E+00 -6.0267E-01 5.3718 -1.5604E-02 -6.7970E-03
0.2683 -1.2953E+00 -3.9984E-01 5.6150 -1.4839E-02 -6.5073E-03
0.2949 -1.1258E+00 -2.5248E-01 5.8573 -1.4147E-02 -6.2415E-03
0.3333 -9.2440E-01 -1.6101E-01 6.0982 -1.3520E-02 -5.9971E-03
0.3834 -7.3047E-01 -1.0879E-01 6.3371 -1.2951E-02 -5.7725E-03
0.4450 -5.6573E-01 -7.9479E-02 6.5734 -1.2432E-02 -5.5658E-03
0.5179 -4.3563E-01 -6.2340E-02 6.8066 -1.1960E-02 -5.3755E-03
0.6022 -3.3681E-01 -5.1482E-02 7.0361 -1.1528E-02 -5.2003E-03
0.6974 -2.6303E-01 -4.3952E-02 7.2614 -1.1134E-02 -5.0389E-03
0.8034 -2.0816E-01 -3.8309E-02 7.4818 -1.0773E-02 -4.8903E-03
0.9200 -1.6719E-01 -3.3840E-02 7.6970 -1.0443E-02 -4.7533E-03
1.0469 -1.3631E-01 -3.0173E-02 7.9064 -1.0140E-02 -4.6273E-03
1.1837 -1.1277E-01 -2.7095E-02 8.1095 -9.8629E-03 -4.5113E-03
1.3301 -9.4596E-02 -2.4474E-02 8.3058 -9.6090E-03 -4.4047E-03
1.4859 -8.0380E-02 -2.2218E-02 8.4948 -9.3765E-03 -4.3068E-03
1.6506 -6.9114E-02 -2.0262E-02 8.6762 -9.1640E-03 -4.2171E-03
1.8238 -6.0075E-02 -1.8555E-02 8.8494 -8.9697E-03 -4.1349E-03
2.0052 -5.2735E-02 -1.7058E-02 9.0141 -8.7926E-03 -4.0599E-03
2.1942 -4.6708E-02 -1.5738E-02 9.1699 -8.6315E-03 -3.9917E-03
2.3905 -4.1708E-02 -1.4569E-02 9.3163 -8.4853E-03 -3.9295E-03
2.5936 -3.7519E-02 -1.3531E-02 9.4531 -8.3532E-03 -3.8733E-03
2.8030 -3.3979E-02 -1.2604E-02 9.5800 -8.2344E-03 -3.8229E-03
3.0182 -3.0961E-02 -1.1774E-02 9.6966 -8.1282E-03 -3.7777E-03
3.2386 -2.8369E-02 -1.1028E-02 9.8026 -8.0340E-03 -3.7377E-03
3.4639 -2.6128E-02 -1.0357E-02 9.8978 -7.9514E-03 -3.7026E-03
3.6934 -2.4177E-02 -9.7503E-03 9.9821 -7.8797E-03 -3.6722E-03
3.9266 -2.2469E-02 -9.2009E-03 10.0550 -7.8186E-03 -3.6462E-03
4.1629 -2.0965E-02 -8.7023E-03 10.1166 -7.7678E-03 -3.6247E-03
4.4018 -1.9635E-02 -8.2487E-03 10.1667 -7.7270E-03 -3.6074E-03
4.6427 -1.8453E-02 -7.8354E-03 10.2051 -7.6960E-03 -3.5943E-03
4.8850 -1.7398E-02 -7.4580E-03 10.2317 -7.6746E-03 -3.5853E-03
5.1282 -1.6453E-02 -7.1130E-03 10.2465 -7.6628E-03 -3.5802E-03
307
Table B.7: Results for a = 0.5 (Kn = 1.78)
r q2(r) q3(r) r q2(r) q3(r)
0.5035 -1.4050E+00 -7.2269E-01 5.6218 -5.2901E-02 -2.3187E-02
0.5183 -1.3617E+00 -5.7935E-01 5.8650 -5.0428E-02 -2.2238E-02
0.5449 -1.2801E+00 -4.5249E-01 6.1073 -4.8182E-02 -2.1364E-02
0.5833 -1.1674E+00 -3.5102E-01 6.3482 -4.6138E-02 -2.0559E-02
0.6334 -1.0369E+00 -2.7496E-01 6.5871 -4.4274E-02 -1.9816E-02
0.6950 -9.0248E-01 -2.2000E-01 6.8234 -4.2572E-02 -1.9131E-02
0.7679 -7.7481E-01 -1.8073E-01 7.0566 -4.1016E-02 -1.8499E-02
0.8522 -6.6010E-01 -1.5238E-01 7.2861 -3.9591E-02 -1.7916E-02
0.9474 -5.6080E-01 -1.3140E-01 7.5114 -3.8285E-02 -1.7378E-02
1.0534 -4.7688E-01 -1.1534E-01 7.7318 -3.7087E-02 -1.6881E-02
1.1700 -4.0698E-01 -1.0264E-01 7.9470 -3.5988E-02 -1.6422E-02
1.2969 -3.4918E-01 -9.2270E-02 8.1564 -3.4979E-02 -1.6000E-02
1.4337 -3.0151E-01 -8.3598E-02 8.3595 -3.4053E-02 -1.5610E-02
1.5801 -2.6217E-01 -7.6204E-02 8.5558 -3.3203E-02 -1.5252E-02
1.7359 -2.2960E-01 -6.9808E-02 8.7448 -3.2424E-02 -1.4922E-02
1.9006 -2.0252E-01 -6.4215E-02 8.9262 -3.1711E-02 -1.4620E-02
2.0738 -1.7987E-01 -5.9286E-02 9.0994 -3.1058E-02 -1.4343E-02
2.2552 -1.6081E-01 -5.4914E-02 9.2641 -3.0462E-02 -1.4089E-02
2.4442 -1.4468E-01 -5.1015E-02 9.4199 -2.9920E-02 -1.3858E-02
2.6405 -1.3094E-01 -4.7525E-02 9.5663 -2.9427E-02 -1.3648E-02
2.8436 -1.1916E-01 -4.4390E-02 9.7031 -2.8981E-02 -1.3458E-02
3.0530 -1.0900E-01 -4.1564E-02 9.8300 -2.8580E-02 -1.3287E-02
3.2682 -1.0018E-01 -3.9010E-02 9.9466 -2.8221E-02 -1.3134E-02
3.4886 -9.2497E-02 -3.6697E-02 10.0526 -2.7903E-02 -1.2999E-02
3.7139 -8.5759E-02 -3.4596E-02 10.1478 -2.7623E-02 -1.2879E-02
3.9434 -7.9821E-02 -3.2685E-02 10.2321 -2.7380E-02 -1.2776E-02
4.1766 -7.4569E-02 -3.0943E-02 10.3050 -2.7173E-02 -1.2688E-02
4.4129 -6.9899E-02 -2.9352E-02 10.3666 -2.7001E-02 -1.2615E-02
4.6518 -6.5733E-02 -2.7896E-02 10.4167 -2.6863E-02 -1.2556E-02
4.8927 -6.2003E-02 -2.6564E-02 10.4551 -2.6758E-02 -1.2512E-02
5.1350 -5.8652E-02 -2.5342E-02 10.4817 -2.6685E-02 -1.2481E-02
5.3782 -5.5631E-02 -2.4219E-02 10.4965 -2.6645E-02 -1.2464E-02
308
Table B.8: Results for a = 0.75 (Kn = 1.18)
r q2(r) q3(r) r q2(r) q3(r)
0.7535 -1.4087E+00 -8.0092E-01 5.8718 -1.0172E-01 -4.4921E-02
0.7683 -1.3830E+00 -6.9036E-01 6.1150 -9.7188E-02 -4.3148E-02
0.7949 -1.3334E+00 -5.8475E-01 6.3573 -9.3051E-02 -4.1510E-02
0.8333 -1.2613E+00 -4.9151E-01 6.5982 -8.9269E-02 -3.9998E-02
0.8834 -1.1717E+00 -4.1337E-01 6.8371 -8.5810E-02 -3.8600E-02
0.9450 -1.0716E+00 -3.5010E-01 7.0734 -8.2640E-02 -3.7309E-02
1.0179 -9.6787E-01 -2.9982E-01 7.3066 -7.9732E-02 -3.6114E-02
1.1022 -8.6636E-01 -2.6012E-01 7.5361 -7.7062E-02 -3.5012E-02
1.1974 -7.7099E-01 -2.2859E-01 7.7614 -7.4609E-02 -3.3989E-02
1.3034 -6.8404E-01 -2.0327E-01 7.9818 -7.2354E-02 -3.3045E-02
1.4200 -6.0642E-01 -1.8257E-01 8.1970 -7.0281E-02 -3.2173E-02
1.5469 -5.3814E-01 -1.6536E-01 8.4064 -6.8374E-02 -3.1368E-02
1.6837 -4.7864E-01 -1.5080E-01 8.6095 -6.6621E-02 -3.0626E-02
1.8301 -4.2709E-01 -1.3830E-01 8.8058 -6.5010E-02 -2.9941E-02
1.9859 -3.8254E-01 -1.2741E-01 8.9948 -6.3530E-02 -2.9311E-02
2.1506 -3.4407E-01 -1.1785E-01 9.1762 -6.2160E-02 -2.8733E-02
2.3238 -3.1082E-01 -1.0937E-01 9.3494 -6.0931E-02 -2.8202E-02
2.5052 -2.8202E-01 -1.0180E-01 9.5141 -5.9794E-02 -2.7716E-02
2.6942 -2.5701E-01 -9.5018E-02 9.6699 -5.8758E-02 -2.7272E-02
2.8905 -2.3522E-01 -8.8905E-02 9.8163 -5.7817E-02 -2.6869E-02
3.0936 -2.1617E-01 -8.3378E-02 9.9531 -5.6965E-02 -2.6504E-02
3.3030 -1.9944E-01 -7.8367E-02 10.0800 -5.6197E-02 -2.6176E-02
3.5182 -1.8471E-01 -7.3811E-02 10.1966 -5.5510E-02 -2.5881E-02
3.7386 -1.7168E-01 -6.9660E-02 10.3026 -5.4899E-02 -2.5620E-02
3.9639 -1.6012E-01 -6.5871E-02 10.3978 -5.4363E-02 -2.5391E-02
4.1934 -1.4982E-01 -6.2406E-02 10.4821 -5.3897E-02 -2.5192E-02
4.4266 -1.4061E-01 -5.9232E-02 10.5550 -5.3500E-02 -2.5022E-02
4.6629 -1.3236E-01 -5.6321E-02 10.6166 -5.3170E-02 -2.4881E-02
4.9018 -1.2494E-01 -5.3648E-02 10.6667 -5.2904E-02 -2.4768E-02
5.1427 -1.1825E-01 -5.1190E-02 10.7051 -5.2702E-02 -2.4682E-02
5.3850 -1.1220E-01 -4.8927E-02 10.7317 -5.2563E-02 -2.4622E-02
5.6282 -1.0671E-01 -4.6843E-02 10.7465 -5.2486E-02 -2.4590E-02
309
Table B.9: Results for a = 1.0 (Kn = 0.888)
r q2(r) q3(r) r q2(r) q3(r)
1.0035 -1.4105E+00 -8.6031E-01 6.1218 -1.5598E-01 -6.9474E-02
1.0183 -1.3930E+00 -7.7068E-01 6.3650 -1.4934E-01 -6.6822E-02
1.0449 -1.3591E+00 -6.8131E-01 6.6073 -1.4325E-01 -6.4366E-02
1.0833 -1.3087E+00 -5.9800E-01 6.8482 -1.3767E-01 -6.2094E-02
1.1334 -1.2437E+00 -5.2370E-01 7.0871 -1.3255E-01 -5.9989E-02
1.1950 -1.1678E+00 -4.5947E-01 7.3234 -1.2784E-01 -5.8041E-02
1.2679 -1.0852E+00 -4.0506E-01 7.5566 -1.2351E-01 -5.6236E-02
1.3522 -1.0000E+00 -3.5946E-01 7.7861 -1.1953E-01 -5.4565E-02
1.4474 -9.1578E-01 -3.2139E-01 8.0114 -1.1585E-01 -5.3018E-02
1.5534 -8.3500E-01 -2.8949E-01 8.2318 -1.1247E-01 -5.1586E-02
1.6700 -7.5939E-01 -2.6259E-01 8.4470 -1.0935E-01 -5.0260E-02
1.7969 -6.8989E-01 -2.3968E-01 8.6564 -1.0648E-01 -4.9036E-02
1.9337 -6.2683E-01 -2.1996E-01 8.8595 -1.0384E-01 -4.7905E-02
2.0801 -5.7014E-01 -2.0281E-01 9.0558 -1.0140E-01 -4.6861E-02
2.2359 -5.1949E-01 -1.8776E-01 9.2448 -9.9164E-02 -4.5899E-02
2.4006 -4.7443E-01 -1.7443E-01 9.4262 -9.7108E-02 -4.5015E-02
2.5738 -4.3443E-01 -1.6254E-01 9.5994 -9.5222E-02 -4.4203E-02
2.7552 -3.9894E-01 -1.5188E-01 9.7641 -9.3496E-02 -4.3459E-02
2.9442 -3.6745E-01 -1.4227E-01 9.9199 -9.1920E-02 -4.2780E-02
3.1405 -3.3948E-01 -1.3357E-01 10.0663 -9.0488E-02 -4.2162E-02
3.3436 -3.1460E-01 -1.2567E-01 10.2031 -8.9189E-02 -4.1602E-02
3.5530 -2.9242E-01 -1.1848E-01 10.3300 -8.8018E-02 -4.1098E-02
3.7682 -2.7262E-01 -1.1190E-01 10.4466 -8.6970E-02 -4.0646E-02
3.9886 -2.5488E-01 -1.0589E-01 10.5526 -8.6039E-02 -4.0245E-02
4.2139 -2.3897E-01 -1.0038E-01 10.6478 -8.5219E-02 -3.9892E-02
4.4434 -2.2466E-01 -9.5319E-02 10.7321 -8.4507E-02 -3.9586E-02
4.6766 -2.1175E-01 -9.0668E-02 10.8050 -8.3900E-02 -3.9325E-02
4.9129 -2.0008E-01 -8.6386E-02 10.8666 -8.3395E-02 -3.9108E-02
5.1518 -1.8952E-01 -8.2442E-02 10.9167 -8.2989E-02 -3.8934E-02
5.3927 -1.7992E-01 -7.8803E-02 10.9551 -8.2681E-02 -3.8802E-02
5.6350 -1.7119E-01 -7.5445E-02 10.9817 -8.2468E-02 -3.8711E-02
5.8782 -1.6324E-01 -7.2342E-02 10.9965 -8.2350E-02 -3.8660E-02
310
Table B.10: Results for a = 1.25 (Kn = 0.710)
r q2(r) q3(r) r q2(r) q3(r)
1.2535 -1.4115E+00 -9.0799E-01 6.3718 -2.1200E-01 -9.5307E-02
1.2683 -1.3987E+00 -8.3292E-01 6.6150 -2.0338E-01 -9.1777E-02
1.2949 -1.3739E+00 -7.5591E-01 6.8573 -1.9545E-01 -8.8505E-02
1.3333 -1.3365E+00 -6.8152E-01 7.0982 -1.8815E-01 -8.5469E-02
1.3834 -1.2872E+00 -6.1247E-01 7.3371 -1.8142E-01 -8.2654E-02
1.4450 -1.2281E+00 -5.5016E-01 7.5734 -1.7523E-01 -8.0043E-02
1.5179 -1.1616E+00 -4.9505E-01 7.8066 -1.6951E-01 -7.7621E-02
1.6022 -1.0906E+00 -4.4694E-01 8.0361 -1.6423E-01 -7.5375E-02
1.6974 -1.0178E+00 -4.0524E-01 8.2614 -1.5936E-01 -7.3293E-02
1.8034 -9.4558E-01 -3.6915E-01 8.4818 -1.5486E-01 -7.1363E-02
1.9200 -8.7559E-01 -3.3787E-01 8.6970 -1.5071E-01 -6.9576E-02
2.0469 -8.0911E-01 -3.1064E-01 8.9064 -1.4688E-01 -6.7922E-02
2.1837 -7.4689E-01 -2.8679E-01 9.1095 -1.4334E-01 -6.6393E-02
2.3301 -6.8934E-01 -2.6578E-01 9.3058 -1.4008E-01 -6.4981E-02
2.4859 -6.3654E-01 -2.4714E-01 9.4948 -1.3708E-01 -6.3679E-02
2.6506 -5.8841E-01 -2.3050E-01 9.6762 -1.3432E-01 -6.2480E-02
2.8238 -5.4473E-01 -2.1556E-01 9.8494 -1.3178E-01 -6.1378E-02
3.0052 -5.0519E-01 -2.0208E-01 10.0141 -1.2946E-01 -6.0369E-02
3.1942 -4.6946E-01 -1.8987E-01 10.1699 -1.2733E-01 -5.9447E-02
3.3905 -4.3720E-01 -1.7877E-01 10.3163 -1.2540E-01 -5.8607E-02
3.5936 -4.0806E-01 -1.6865E-01 10.4531 -1.2365E-01 -5.7845E-02
3.8030 -3.8175E-01 -1.5939E-01 10.5800 -1.2207E-01 -5.7159E-02
4.0182 -3.5796E-01 -1.5091E-01 10.6966 -1.2065E-01 -5.6544E-02
4.2386 -3.3642E-01 -1.4312E-01 10.8026 -1.1939E-01 -5.5997E-02
4.4639 -3.1690E-01 -1.3595E-01 10.8978 -1.1828E-01 -5.5517E-02
4.6934 -2.9919E-01 -1.2935E-01 10.9821 -1.1732E-01 -5.5100E-02
4.9266 -2.8308E-01 -1.2327E-01 11.0550 -1.1650E-01 -5.4745E-02
5.1629 -2.6841E-01 -1.1765E-01 11.1166 -1.1582E-01 -5.4449E-02
5.4018 -2.5503E-01 -1.1247E-01 11.1667 -1.1527E-01 -5.4211E-02
5.6427 -2.4281E-01 -1.0767E-01 11.2051 -1.1485E-01 -5.4031E-02
5.8850 -2.3163E-01 -1.0323E-01 11.2317 -1.1456E-01 -5.3906E-02
6.1282 -2.2140E-01 -9.9116E-02 11.2465 -1.1440E-01 -5.3838E-02
311
Table B.11: Results for a = 1.5 (Kn = 0.592)
r q2(r) q3(r) r q2(r) q3(r)
1.5035 -1.4121E+00 -9.4744E-01 6.6218 -2.6772E-01 -1.2157E-01
1.5183 -1.4022E+00 -8.8310E-01 6.8650 -2.5731E-01 -1.1719E-01
1.5449 -1.3832E+00 -8.1567E-01 7.1073 -2.4770E-01 -1.1313E-01
1.5833 -1.3542E+00 -7.4889E-01 7.3482 -2.3882E-01 -1.0935E-01
1.6334 -1.3156E+00 -6.8510E-01 7.5871 -2.3062E-01 -1.0585E-01
1.6950 -1.2683E+00 -6.2576E-01 7.8234 -2.2304E-01 -1.0259E-01
1.7679 -1.2139E+00 -5.7164E-01 8.0566 -2.1602E-01 -9.9564E-02
1.8522 -1.1544E+00 -5.2293E-01 8.2861 -2.0953E-01 -9.6754E-02
1.9474 -1.0919E+00 -4.7950E-01 8.5114 -2.0353E-01 -9.4147E-02
2.0534 -1.0281E+00 -4.4094E-01 8.7318 -1.9798E-01 -9.1728E-02
2.1700 -9.6463E-01 -4.0677E-01 8.9470 -1.9284E-01 -8.9485E-02
2.2969 -9.0283E-01 -3.7644E-01 9.1564 -1.8809E-01 -8.7407E-02
2.4337 -8.4358E-01 -3.4947E-01 9.3595 -1.8370E-01 -8.5484E-02
2.5801 -7.8750E-01 -3.2539E-01 9.5558 -1.7965E-01 -8.3707E-02
2.7359 -7.3494E-01 -3.0381E-01 9.7448 -1.7591E-01 -8.2066E-02
2.9006 -6.8605E-01 -2.8437E-01 9.9262 -1.7247E-01 -8.0555E-02
3.0738 -6.4085E-01 -2.6680E-01 10.0994 -1.6930E-01 -7.9166E-02
3.2552 -5.9924E-01 -2.5086E-01 10.2641 -1.6640E-01 -7.7892E-02
3.4442 -5.6104E-01 -2.3635E-01 10.4199 -1.6375E-01 -7.6726E-02
3.6405 -5.2605E-01 -2.2309E-01 10.5663 -1.6133E-01 -7.5665E-02
3.8436 -4.9405E-01 -2.1095E-01 10.7031 -1.5913E-01 -7.4702E-02
4.0530 -4.6479E-01 -1.9981E-01 10.8300 -1.5716E-01 -7.3834E-02
4.2682 -4.3805E-01 -1.8957E-01 10.9466 -1.5538E-01 -7.3055E-02
4.4886 -4.1361E-01 -1.8014E-01 11.0526 -1.5380E-01 -7.2363E-02
4.7139 -3.9125E-01 -1.7144E-01 11.1478 -1.5241E-01 -7.1755E-02
4.9434 -3.7078E-01 -1.6340E-01 11.2321 -1.5120E-01 -7.1226E-02
5.1766 -3.5204E-01 -1.5597E-01 11.3050 -1.5016E-01 -7.0776E-02
5.4129 -3.3486E-01 -1.4910E-01 11.3666 -1.4930E-01 -7.0402E-02
5.6518 -3.1909E-01 -1.4273E-01 11.4167 -1.4861E-01 -7.0100E-02
5.8927 -3.0460E-01 -1.3683E-01 11.4551 -1.4809E-01 -6.9871E-02
6.1350 -2.9128E-01 -1.3136E-01 11.4817 -1.4772E-01 -6.9714E-02
6.3782 -2.7902E-01 -1.2628E-01 11.4965 -1.4752E-01 -6.9626E-02
312
Table B.12: Results for a = 1.75 (Kn = 0.5074)
r q2(r) q3(r) r q2(r) q3(r)
1.7535 -1.4125E+00 -9.8074E-01 6.8718 -3.2190E-01 -1.4768E-01
1.7683 -1.4047E+00 -9.2461E-01 7.1150 -3.0991E-01 -1.4250E-01
1.7949 -1.3896E+00 -8.6480E-01 7.3573 -2.9882E-01 -1.3769E-01
1.8333 -1.3664E+00 -8.0441E-01 7.5982 -2.8854E-01 -1.3321E-01
1.8834 -1.3352E+00 -7.4547E-01 7.8371 -2.7901E-01 -1.2904E-01
1.9450 -1.2966E+00 -6.8939E-01 8.0734 -2.7018E-01 -1.2517E-01
2.0179 -1.2514E+00 -6.3702E-01 8.3066 -2.6199E-01 -1.2157E-01
2.1022 -1.2011E+00 -5.8879E-01 8.5361 -2.5439E-01 -1.1822E-01
2.1974 -1.1471E+00 -5.4482E-01 8.7614 -2.4735E-01 -1.1511E-01
2.3034 -1.0909E+00 -5.0498E-01 8.9818 -2.4082E-01 -1.1222E-01
2.4200 -1.0339E+00 -4.6901E-01 9.1970 -2.3477E-01 -1.0954E-01
2.5469 -9.7720E-01 -4.3658E-01 9.4064 -2.2916E-01 -1.0705E-01
2.6837 -9.2179E-01 -4.0731E-01 9.6095 -2.2398E-01 -1.0475E-01
2.8301 -8.6835E-01 -3.8089E-01 9.8058 -2.1918E-01 -1.0262E-01
2.9859 -8.1736E-01 -3.5696E-01 9.9948 -2.1475E-01 -1.0065E-01
3.1506 -7.6913E-01 -3.3523E-01 10.1762 -2.1067E-01 -9.8834E-02
3.3238 -7.2384E-01 -3.1546E-01 10.3494 -2.0691E-01 -9.7164E-02
3.5052 -6.8152E-01 -2.9741E-01 10.5141 -2.0346E-01 -9.5637E-02
3.6942 -6.4216E-01 -2.8089E-01 10.6699 -2.0031E-01 -9.4231E-02
3.8905 -6.0565E-01 -2.6574E-01 10.8163 -1.9743E-01 -9.2953E-02
4.0936 -5.7186E-01 -2.5182E-01 10.9531 -1.9481E-01 -9.1794E-02
4.3030 -5.4065E-01 -2.3899E-01 11.0800 -1.9245E-01 -9.0746E-02
4.5182 -5.1184E-01 -2.2716E-01 11.1966 -1.9033E-01 -8.9808E-02
4.7386 -4.8527E-01 -2.1623E-01 11.3026 -1.8844E-01 -8.8974E-02
4.9639 -4.6077E-01 -2.0613E-01 11.3978 -1.8678E-01 -8.8239E-02
5.1934 -4.3817E-01 -1.9677E-01 11.4821 -1.8533E-01 -8.7602E-02
5.4266 -4.1734E-01 -1.8810E-01 11.5550 -1.8410E-01 -8.7058E-02
5.6629 -3.9812E-01 -1.8006E-01 11.6166 -1.8307E-01 -8.6605E-02
5.9018 -3.8037E-01 -1.7260E-01 11.6667 -1.8224E-01 -8.6242E-02
6.1427 -3.6399E-01 -1.6567E-01 11.7051 -1.8161E-01 -8.5966E-02
6.3850 -3.4885E-01 -1.5923E-01 11.7317 -1.8118E-01 -8.5775E-02
6.6282 -3.3485E-01 -1.5324E-01 11.7465 -1.8094E-01 -8.5669E-02
313
Table B.13: Results for a = 2.0 (Kn = 0.444)
r q2(r) q3(r) r q2(r) q3(r)
2.0035 -1.4128E+00 -1.0093E+00 7.1218 -3.7389E-01 -1.7335E-01
2.0183 -1.4064E+00 -9.5964E-01 7.3650 -3.6055E-01 -1.6742E-01
2.0449 -1.3941E+00 -9.0600E-01 7.6073 -3.4816E-01 -1.6191E-01
2.0833 -1.3751E+00 -8.5099E-01 7.8482 -3.3665E-01 -1.5677E-01
2.1334 -1.3494E+00 -7.9641E-01 8.0871 -3.2595E-01 -1.5198E-01
2.1950 -1.3172E+00 -7.4353E-01 8.3234 -3.1601E-01 -1.4752E-01
2.2679 -1.2792E+00 -6.9324E-01 8.5566 -3.0677E-01 -1.4338E-01
2.3522 -1.2361E+00 -6.4607E-01 8.7861 -2.9817E-01 -1.3952E-01
2.4474 -1.1893E+00 -6.0229E-01 9.0114 -2.9019E-01 -1.3592E-01
2.5534 -1.1397E+00 -5.6196E-01 9.2318 -2.8278E-01 -1.3259E-01
2.6700 -1.0886E+00 -5.2498E-01 9.4470 -2.7590E-01 -1.2949E-01
2.7969 -1.0369E+00 -4.9118E-01 9.6564 -2.6952E-01 -1.2661E-01
2.9337 -9.8554E-01 -4.6031E-01 9.8595 -2.6360E-01 -1.2394E-01
3.0801 -9.3524E-01 -4.3213E-01 10.0558 -2.5812E-01 -1.2147E-01
3.2359 -8.8653E-01 -4.0638E-01 10.2448 -2.5305E-01 -1.1919E-01
3.4006 -8.3980E-01 -3.8282E-01 10.4262 -2.4838E-01 -1.1708E-01
3.5738 -7.9531E-01 -3.6123E-01 10.5994 -2.4408E-01 -1.1514E-01
3.7552 -7.5322E-01 -3.4142E-01 10.7641 -2.4012E-01 -1.1336E-01
3.9442 -7.1360E-01 -3.2319E-01 10.9199 -2.3649E-01 -1.1174E-01
4.1405 -6.7645E-01 -3.0640E-01 11.0663 -2.3318E-01 -1.1025E-01
4.3436 -6.4172E-01 -2.9090E-01 11.2031 -2.3017E-01 -1.0890E-01
4.5530 -6.0932E-01 -2.7659E-01 11.3300 -2.2745E-01 -1.0768E-01
4.7682 -5.7917E-01 -2.6334E-01 11.4466 -2.2501E-01 -1.0659E-01
4.9886 -5.5113E-01 -2.5108E-01 11.5526 -2.2283E-01 -1.0562E-01
5.2139 -5.2508E-01 -2.3970E-01 11.6478 -2.2091E-01 -1.0476E-01
5.4434 -5.0089E-01 -2.2914E-01 11.7321 -2.1925E-01 -1.0402E-01
5.6766 -4.7845E-01 -2.1934E-01 11.8050 -2.1782E-01 -1.0338E-01
5.9129 -4.5762E-01 -2.1023E-01 11.8666 -2.1663E-01 -1.0286E-01
6.1518 -4.3829E-01 -2.0176E-01 11.9167 -2.1568E-01 -1.0243E-01
6.3927 -4.2035E-01 -1.9387E-01 11.9551 -2.1495E-01 -1.0211E-01
6.6350 -4.0370E-01 -1.8654E-01 11.9817 -2.1445E-01 -1.0189E-01
6.8782 -3.8824E-01 -1.7971E-01 11.9965 -2.1417E-01 -1.0177E-01
314
Table B.14: Results for a = 2.5 (Kn = 0.355)
r q2(r) q3(r) r q2(r) q3(r)
2.5035 -1.4132E+00 -1.0558E+00 7.6218 -4.7018E-01 -2.2274E-01
2.5183 -1.4087E+00 -1.0157E+00 7.8650 -4.5476E-01 -2.1548E-01
2.5449 -1.4000E+00 -9.7139E-01 8.1073 -4.4034E-01 -2.0870E-01
2.5833 -1.3866E+00 -9.2488E-01 8.3482 -4.2687E-01 -2.0237E-01
2.6334 -1.3682E+00 -8.7757E-01 8.5871 -4.1429E-01 -1.9646E-01
2.6950 -1.3449E+00 -8.3051E-01 8.8234 -4.0253E-01 -1.9095E-01
2.7679 -1.3168E+00 -7.8453E-01 9.0566 -3.9156E-01 -1.8581E-01
2.8522 -1.2844E+00 -7.4021E-01 9.2861 -3.8132E-01 -1.8101E-01
2.9474 -1.2483E+00 -6.9798E-01 9.5114 -3.7177E-01 -1.7654E-01
3.0534 -1.2093E+00 -6.5806E-01 9.7318 -3.6286E-01 -1.7237E-01
3.1700 -1.1680E+00 -6.2057E-01 9.9470 -3.5457E-01 -1.6850E-01
3.2969 -1.1251E+00 -5.8554E-01 10.1564 -3.4686E-01 -1.6490E-01
3.4337 -1.0815E+00 -5.5290E-01 10.3595 -3.3968E-01 -1.6156E-01
3.5801 -1.0378E+00 -5.2256E-01 10.5558 -3.3303E-01 -1.5846E-01
3.7359 -9.9433E-01 -4.9440E-01 10.7448 -3.2686E-01 -1.5560E-01
3.9006 -9.5171E-01 -4.6828E-01 10.9262 -3.2114E-01 -1.5295E-01
4.0738 -9.1024E-01 -4.4404E-01 11.0994 -3.1587E-01 -1.5051E-01
4.2552 -8.7018E-01 -4.2156E-01 11.2641 -3.1102E-01 -1.4827E-01
4.4442 -8.3172E-01 -4.0069E-01 11.4199 -3.0657E-01 -1.4622E-01
4.6405 -7.9499E-01 -3.8130E-01 11.5663 -3.0250E-01 -1.4434E-01
4.8436 -7.6005E-01 -3.6328E-01 11.7031 -2.9879E-01 -1.4264E-01
5.0530 -7.2694E-01 -3.4653E-01 11.8300 -2.9544E-01 -1.4110E-01
5.2682 -6.9564E-01 -3.3094E-01 11.9466 -2.9242E-01 -1.3972E-01
5.4886 -6.6614E-01 -3.1641E-01 12.0526 -2.8973E-01 -1.3849E-01
5.7139 -6.3837E-01 -3.0289E-01 12.1478 -2.8736E-01 -1.3740E-01
5.9434 -6.1228E-01 -2.9028E-01 12.2321 -2.8529E-01 -1.3646E-01
6.1766 -5.8779E-01 -2.7853E-01 12.3050 -2.8353E-01 -1.3566E-01
6.4129 -5.6482E-01 -2.6755E-01 12.3666 -2.8205E-01 -1.3499E-01
6.6518 -5.4331E-01 -2.5732E-01 12.4167 -2.8087E-01 -1.3445E-01
6.8927 -5.2317E-01 -2.4777E-01 12.4551 -2.7996E-01 -1.3404E-01
7.1350 -5.0431E-01 -2.3885E-01 12.4817 -2.7934E-01 -1.3376E-01
7.3782 -4.8667E-01 -2.3052E-01 12.4965 -2.7900E-01 -1.3360E-01
315
Table B.15: Results for a = 3.0 (Kn = 0.296)
r q2(r) q3(r) r q2(r) q3(r)
3.0035 -1.4134E+00 -1.0922E+00 8.1218 -5.5593E-01 -2.6910E-01
3.0183 -1.4101E+00 -1.0587E+00 8.3650 -5.3910E-01 -2.6070E-01
3.0449 -1.4036E+00 -1.0211E+00 8.6073 -5.2329E-01 -2.5283E-01
3.0833 -1.3935E+00 -9.8093E-01 8.8482 -5.0844E-01 -2.4547E-01
3.1334 -1.3797E+00 -9.3932E-01 9.0871 -4.9450E-01 -2.3859E-01
3.1950 -1.3621E+00 -8.9716E-01 9.3234 -4.8142E-01 -2.3216E-01
3.2679 -1.3405E+00 -8.5518E-01 9.5566 -4.6916E-01 -2.2615E-01
3.3522 -1.3153E+00 -8.1393E-01 9.7861 -4.5768E-01 -2.2053E-01
3.4474 -1.2868E+00 -7.7386E-01 10.0114 -4.4693E-01 -2.1530E-01
3.5534 -1.2554E+00 -7.3528E-01 10.2318 -4.3688E-01 -2.1041E-01
3.6700 -1.2216E+00 -6.9841E-01 10.4470 -4.2749E-01 -2.0586E-01
3.7969 -1.1859E+00 -6.6336E-01 10.6564 -4.1873E-01 -2.0162E-01
3.9337 -1.1489E+00 -6.3020E-01 10.8595 -4.1056E-01 -1.9768E-01
4.0801 -1.1111E+00 -5.9893E-01 11.0558 -4.0296E-01 -1.9402E-01
4.2359 -1.0728E+00 -5.6954E-01 11.2448 -3.9590E-01 -1.9064E-01
4.4006 -1.0347E+00 -5.4193E-01 11.4262 -3.8936E-01 -1.8751E-01
4.5738 -9.9688E-01 -5.1606E-01 11.5994 -3.8330E-01 -1.8462E-01
4.7552 -9.5978E-01 -4.9183E-01 11.7641 -3.7772E-01 -1.8196E-01
4.9442 -9.2361E-01 -4.6915E-01 11.9199 -3.7259E-01 -1.7952E-01
5.1405 -8.8855E-01 -4.4793E-01 12.0663 -3.6789E-01 -1.7730E-01
5.3436 -8.5472E-01 -4.2807E-01 12.2031 -3.6360E-01 -1.7527E-01
5.5530 -8.2224E-01 -4.0949E-01 12.3300 -3.5972E-01 -1.7344E-01
5.7682 -7.9115E-01 -3.9210E-01 12.4466 -3.5622E-01 -1.7180E-01
5.9886 -7.6150E-01 -3.7584E-01 12.5526 -3.5310E-01 -1.7033E-01
6.2139 -7.3329E-01 -3.6062E-01 12.6478 -3.5035E-01 -1.6904E-01
6.4434 -7.0650E-01 -3.4637E-01 12.7321 -3.4795E-01 -1.6792E-01
6.6766 -6.8112E-01 -3.3303E-01 12.8050 -3.4590E-01 -1.6696E-01
6.9129 -6.5710E-01 -3.2055E-01 12.8666 -3.4418E-01 -1.6616E-01
7.1518 -6.3442E-01 -3.0886E-01 12.9167 -3.4280E-01 -1.6552E-01
7.3927 -6.1301E-01 -2.9792E-01 12.9551 -3.4175E-01 -1.6503E-01
7.6350 -5.9283E-01 -2.8767E-01 12.9817 -3.4103E-01 -1.6469E-01
7.8782 -5.7382E-01 -2.7808E-01 12.9965 -3.4062E-01 -1.6451E-01
316
Table B.16: Results for a = 4.0 (Kn = 0.222)
r q2(r) q3(r) r q2(r) q3(r)
4.0035 -1.4138E+00 -1.1457E+00 9.1218 -6.9885E-01 -3.5284E-01
4.0183 -1.4117E+00 -1.1208E+00 9.3650 -6.8069E-01 -3.4266E-01
4.0449 -1.4076E+00 -1.0921E+00 9.6073 -6.6347E-01 -3.3309E-01
4.0833 -1.4014E+00 -1.0606E+00 9.8482 -6.4714E-01 -3.2411E-01
4.1334 -1.3928E+00 -1.0272E+00 10.0871 -6.3170E-01 -3.1567E-01
4.1950 -1.3816E+00 -9.9255E-01 10.3234 -6.1709E-01 -3.0777E-01
4.2679 -1.3677E+00 -9.5714E-01 10.5566 -6.0329E-01 -3.0035E-01
4.3522 -1.3512E+00 -9.2145E-01 10.7861 -5.9031E-01 -2.9341E-01
4.4474 -1.3322E+00 -8.8588E-01 11.0114 -5.7806E-01 -2.8691E-01
4.5534 -1.3108E+00 -8.5077E-01 11.2318 -5.6654E-01 -2.8084E-01
4.6700 -1.2871E+00 -8.1638E-01 11.4470 -5.5572E-01 -2.7516E-01
4.7969 -1.2616E+00 -7.8292E-01 11.6564 -5.4557E-01 -2.6987E-01
4.9337 -1.2344E+00 -7.5055E-01 11.8595 -5.3607E-01 -2.6493E-01
5.0801 -1.2059E+00 -7.1938E-01 12.0558 -5.2719E-01 -2.6034E-01
5.2359 -1.1764E+00 -6.8949E-01 12.2448 -5.1890E-01 -2.5608E-01
5.4006 -1.1461E+00 -6.6091E-01 12.4262 -5.1119E-01 -2.5214E-01
5.5738 -1.1154E+00 -6.3368E-01 12.5994 -5.0404E-01 -2.4849E-01
5.7552 -1.0846E+00 -6.0780E-01 12.7641 -4.9742E-01 -2.4513E-01
5.9442 -1.0538E+00 -5.8320E-01 12.9199 -4.9132E-01 -2.4205E-01
6.1405 -1.0232E+00 -5.5989E-01 13.0663 -4.8572E-01 -2.3922E-01
6.3436 -9.9314E-01 -5.3783E-01 13.2031 -4.8059E-01 -2.3665E-01
6.5530 -9.6364E-01 -5.1698E-01 13.3300 -4.7594E-01 -2.3433E-01
6.7682 -9.3486E-01 -4.9727E-01 13.4466 -4.7174E-01 -2.3224E-01
6.9886 -9.0690E-01 -4.7867E-01 13.5526 -4.6799E-01 -2.3037E-01
7.2139 -8.7982E-01 -4.6113E-01 13.6478 -4.6467E-01 -2.2872E-01
7.4434 -8.5369E-01 -4.4458E-01 13.7321 -4.6177E-01 -2.2729E-01
7.6766 -8.2854E-01 -4.2899E-01 13.8050 -4.5929E-01 -2.2607E-01
7.9129 -8.0438E-01 -4.1430E-01 13.8666 -4.5722E-01 -2.2505E-01
8.1518 -7.8126E-01 -4.0047E-01 13.9167 -4.5555E-01 -2.2423E-01
8.3927 -7.5916E-01 -3.8745E-01 13.9551 -4.5428E-01 -2.2360E-01
8.6350 -7.3806E-01 -3.7520E-01 13.9817 -4.5340E-01 -2.2317E-01
8.8782 -7.1796E-01 -3.6368E-01 13.9965 -4.5291E-01 -2.2293E-01
317
Table B.17: Results for a = 5.0 (Kn = 0.178)
r q2(r) q3(r) r q2(r) q3(r)
5.0035 -1.4139E+00 -1.1835E+00 10.1218 -8.1081E-01 -4.2599E-01
5.0183 -1.4125E+00 -1.1638E+00 10.3650 -7.9254E-01 -4.1455E-01
5.0449 -1.4097E+00 -1.1407E+00 10.6073 -7.7507E-01 -4.0376E-01
5.0833 -1.4054E+00 -1.1149E+00 10.8482 -7.5839E-01 -3.9363E-01
5.1334 -1.3995E+00 -1.0871E+00 11.0871 -7.4248E-01 -3.8403E-01
5.1950 -1.3918E+00 -1.0577E+00 11.3234 -7.2735E-01 -3.7502E-01
5.2679 -1.3821E+00 -1.0272E+00 11.5566 -7.1299E-01 -3.6656E-01
5.3522 -1.3705E+00 -9.9603E-01 11.7861 -6.9938E-01 -3.5861E-01
5.4474 -1.3569E+00 -9.6443E-01 12.0114 -6.8648E-01 -3.5116E-01
5.5534 -1.3414E+00 -9.3271E-01 12.2318 -6.7429E-01 -3.4417E-01
5.6700 -1.3241E+00 -9.0116E-01 12.4470 -6.6278E-01 -3.3763E-01
5.7969 -1.3050E+00 -8.6997E-01 12.6564 -6.5194E-01 -3.3151E-01
5.9337 -1.2844E+00 -8.3934E-01 12.8595 -6.4174E-01 -3.2580E-01
6.0801 -1.2624E+00 -8.0941E-01 13.0558 -6.3218E-01 -3.2048E-01
6.2359 -1.2392E+00 -7.8032E-01 13.2448 -6.2323E-01 -3.1553E-01
6.4006 -1.2151E+00 -7.5213E-01 13.4262 -6.1487E-01 -3.1094E-01
6.5738 -1.1902E+00 -7.2493E-01 13.5994 -6.0709E-01 -3.0669E-01
6.7552 -1.1647E+00 -6.9876E-01 13.7641 -5.9987E-01 -3.0277E-01
6.9442 -1.1388E+00 -6.7365E-01 13.9199 -5.9320E-01 -2.9916E-01
7.1405 -1.1128E+00 -6.4960E-01 14.0663 -5.8705E-01 -2.9586E-01
7.3436 -1.0867E+00 -6.2663E-01 14.2031 -5.8142E-01 -2.9285E-01
7.5530 -1.0608E+00 -6.0471E-01 14.3300 -5.7630E-01 -2.9011E-01
7.7682 -1.0351E+00 -5.8383E-01 14.4466 -5.7167E-01 -2.8766E-01
7.9886 -1.0098E+00 -5.6398E-01 14.5526 -5.6752E-01 -2.8546E-01
8.2139 -9.8498E-01 -5.4512E-01 14.6478 -5.6385E-01 -2.8353E-01
8.4434 -9.6072E-01 -5.2723E-01 14.7321 -5.6064E-01 -2.8184E-01
8.6766 -9.3707E-01 -5.1026E-01 14.8050 -5.5789E-01 -2.8040E-01
8.9129 -9.1414E-01 -4.9418E-01 14.8666 -5.5559E-01 -2.7920E-01
9.1518 -8.9191E-01 -4.7897E-01 14.9167 -5.5374E-01 -2.7823E-01
9.3927 -8.7045E-01 -4.6457E-01 14.9551 -5.5232E-01 -2.7749E-01
9.6350 -8.4977E-01 -4.5097E-01 14.9817 -5.5135E-01 -2.7699E-01
9.8782 -8.2989E-01 -4.3812E-01 14.9965 -5.5080E-01 -2.7670E-01
318
Table B.18: Results for a = 6.0 (Kn = 0.148)
r q2(r) q3(r) r q2(r) q3(r)
6.0035 -1.4140E+00 -1.2117E+00 11.1218 -8.9929E-01 -4.9006E-01
6.0183 -1.4129E+00 -1.1956E+00 11.3650 -8.8154E-01 -4.7774E-01
6.0449 -1.4109E+00 -1.1762E+00 11.6073 -8.6447E-01 -4.6609E-01
6.0833 -1.4078E+00 -1.1544E+00 11.8482 -8.4807E-01 -4.5508E-01
6.1334 -1.4035E+00 -1.1306E+00 12.0871 -8.3235E-01 -4.4468E-01
6.1950 -1.3978E+00 -1.1051E+00 12.3234 -8.1732E-01 -4.3488E-01
6.2679 -1.3907E+00 -1.0784E+00 12.5566 -8.0296E-01 -4.2564E-01
6.3522 -1.3821E+00 -1.0508E+00 12.7861 -7.8927E-01 -4.1695E-01
6.4474 -1.3719E+00 -1.0225E+00 13.0114 -7.7625E-01 -4.0877E-01
6.5534 -1.3602E+00 -9.9376E-01 13.2318 -7.6389E-01 -4.0109E-01
6.6700 -1.3469E+00 -9.6485E-01 13.4470 -7.5218E-01 -3.9389E-01
6.7969 -1.3322E+00 -9.3596E-01 13.6564 -7.4110E-01 -3.8714E-01
6.9337 -1.3161E+00 -9.0728E-01 13.8595 -7.3065E-01 -3.8083E-01
7.0801 -1.2987E+00 -8.7895E-01 14.0558 -7.2081E-01 -3.7494E-01
7.2359 -1.2802E+00 -8.5112E-01 14.2448 -7.1157E-01 -3.6946E-01
7.4006 -1.2606E+00 -8.2390E-01 14.4262 -7.0292E-01 -3.6438E-01
7.5738 -1.2401E+00 -7.9739E-01 14.5994 -6.9484E-01 -3.5964E-01
7.7552 -1.2189E+00 -7.7164E-01 14.7641 -6.8732E-01 -3.5527E-01
7.9442 -1.1972E+00 -7.4671E-01 14.9199 -6.8036E-01 -3.5125E-01
8.1405 -1.1750E+00 -7.2265E-01 15.0663 -6.7394E-01 -3.4756E-01
8.3436 -1.1525E+00 -6.9949E-01 15.2031 -6.6804E-01 -3.4419E-01
8.5530 -1.1299E+00 -6.7723E-01 15.3300 -6.6267E-01 -3.4115E-01
8.7682 -1.1073E+00 -6.5589E-01 15.4466 -6.5779E-01 -3.3838E-01
8.9886 -1.0848E+00 -6.3546E-01 15.5526 -6.5344E-01 -3.3593E-01
9.2139 -1.0624E+00 -6.1594E-01 15.6478 -6.4957E-01 -3.3376E-01
9.4434 -1.0403E+00 -5.9730E-01 15.7321 -6.4618E-01 -3.3186E-01
9.6766 -1.0187E+00 -5.7955E-01 15.8050 -6.4328E-01 -3.3024E-01
9.9129 -9.9739E-01 -5.6265E-01 15.8666 -6.4084E-01 -3.2889E-01
10.1518 -9.7661E-01 -5.4657E-01 15.9167 -6.3888E-01 -3.2780E-01
10.3927 -9.5638E-01 -5.3130E-01 15.9551 -6.3739E-01 -3.2697E-01
10.6350 -9.3673E-01 -5.1681E-01 15.9817 -6.3635E-01 -3.2640E-01
10.8782 -9.1769E-01 -5.0307E-01 15.9965 -6.3578E-01 -3.2609E-01
319
Table B.19: Results for a = 7.0 (Kn = 0.1269)
r q2(r) q3(r) r q2(r) q3(r)
7.0035 -1.4140E+00 -1.2337E+00 12.1218 -9.7017E-01 -5.4667E-01
7.0183 -1.4132E+00 -1.2200E+00 12.3650 -9.5328E-01 -5.3376E-01
7.0449 -1.4117E+00 -1.2035E+00 12.6073 -9.3693E-01 -5.2151E-01
7.0833 -1.4094E+00 -1.1845E+00 12.8482 -9.2114E-01 -5.0990E-01
7.1334 -1.4060E+00 -1.1637E+00 13.0871 -9.0592E-01 -4.9892E-01
7.1950 -1.4017E+00 -1.1413E+00 13.3234 -8.9132E-01 -4.8853E-01
7.2679 -1.3962E+00 -1.1176E+00 13.5566 -8.7731E-01 -4.7872E-01
7.3522 -1.3896E+00 -1.0928E+00 13.7861 -8.6389E-01 -4.6947E-01
7.4474 -1.3817E+00 -1.0672E+00 14.0114 -8.5107E-01 -4.6075E-01
7.5534 -1.3725E+00 -1.0410E+00 14.2318 -8.3886E-01 -4.5255E-01
7.6700 -1.3621E+00 -1.0145E+00 14.4470 -8.2725E-01 -4.4484E-01
7.7969 -1.3504E+00 -9.8772E-01 14.6564 -8.1622E-01 -4.3760E-01
7.9337 -1.3375E+00 -9.6093E-01 14.8595 -8.0579E-01 -4.3083E-01
8.0801 -1.3234E+00 -9.3426E-01 15.0558 -7.9593E-01 -4.2449E-01
8.2359 -1.3083E+00 -9.0785E-01 15.2448 -7.8665E-01 -4.1858E-01
8.4006 -1.2921E+00 -8.8182E-01 15.4262 -7.7794E-01 -4.1308E-01
8.5738 -1.2751E+00 -8.5627E-01 15.5994 -7.6979E-01 -4.0798E-01
8.7552 -1.2573E+00 -8.3128E-01 15.7641 -7.6218E-01 -4.0326E-01
8.9442 -1.2389E+00 -8.0692E-01 15.9199 -7.5514E-01 -3.9890E-01
9.1405 -1.2199E+00 -7.8325E-01 16.0663 -7.4861E-01 -3.9491E-01
9.3436 -1.2005E+00 -7.6031E-01 16.2031 -7.4262E-01 -3.9126E-01
9.5530 -1.1808E+00 -7.3814E-01 16.3300 -7.3714E-01 -3.8794E-01
9.7682 -1.1609E+00 -7.1676E-01 16.4466 -7.3218E-01 -3.8495E-01
9.9886 -1.1409E+00 -6.9620E-01 16.5526 -7.2772E-01 -3.8229E-01
10.2139 -1.1209E+00 -6.7643E-01 16.6478 -7.2376E-01 -3.7993E-01
10.4434 -1.1010E+00 -6.5748E-01 16.7321 -7.2029E-01 -3.7785E-01
10.6766 -1.0813E+00 -6.3933E-01 16.8050 -7.1732E-01 -3.7609E-01
10.9129 -1.0618E+00 -6.2199E-01 16.8666 -7.1482E-01 -3.7463E-01
11.1518 -1.0427E+00 -6.0542E-01 16.9167 -7.1281E-01 -3.7343E-01
11.3927 -1.0239E+00 -5.8963E-01 16.9551 -7.1127E-01 -3.7253E-01
11.6350 -1.0055E+00 -5.7459E-01 16.9817 -7.1021E-01 -3.7190E-01
11.8782 -9.8759E-01 -5.6027E-01 16.9965 -7.0962E-01 -3.7156E-01
320
Table B.20: Results for a = 8.0 (Kn = 0.111)
r q2(r) q3(r) r q2(r) q3(r)
8.0035 -1.4141E+00 -1.2514E+00 13.1218 -1.0278E+00 -5.9716E-01
8.0183 -1.4135E+00 -1.2396E+00 13.3650 -1.0119E+00 -5.8386E-01
8.0449 -1.4122E+00 -1.2251E+00 13.6073 -9.9639E-01 -5.7121E-01
8.0833 -1.4104E+00 -1.2084E+00 13.8482 -9.8139E-01 -5.5920E-01
8.1334 -1.4078E+00 -1.1899E+00 14.0871 -9.6688E-01 -5.4780E-01
8.1950 -1.4044E+00 -1.1699E+00 14.3234 -9.5287E-01 -5.3700E-01
8.2679 -1.4000E+00 -1.1486E+00 14.5566 -9.3938E-01 -5.2678E-01
8.3522 -1.3947E+00 -1.1262E+00 14.7861 -9.2642E-01 -5.1712E-01
8.4474 -1.3884E+00 -1.1029E+00 15.0114 -9.1400E-01 -5.0800E-01
8.5534 -1.3811E+00 -1.0789E+00 15.2318 -9.0212E-01 -4.9940E-01
8.6700 -1.3726E+00 -1.0544E+00 15.4470 -8.9079E-01 -4.9130E-01
8.7969 -1.3631E+00 -1.0295E+00 15.6564 -8.8000E-01 -4.8370E-01
8.9337 -1.3526E+00 -1.0045E+00 15.8595 -8.6977E-01 -4.7656E-01
9.0801 -1.3410E+00 -9.7939E-01 16.0558 -8.6008E-01 -4.6988E-01
9.2359 -1.3284E+00 -9.5441E-01 16.2448 -8.5093E-01 -4.6364E-01
9.4006 -1.3149E+00 -9.2962E-01 16.4262 -8.4231E-01 -4.5782E-01
9.5738 -1.3006E+00 -9.0515E-01 16.5994 -8.3424E-01 -4.5242E-01
9.7552 -1.2855E+00 -8.8107E-01 16.7641 -8.2669E-01 -4.4741E-01
9.9442 -1.2697E+00 -8.5747E-01 16.9199 -8.1968E-01 -4.4279E-01
10.1405 -1.2534E+00 -8.3441E-01 17.0663 -8.1317E-01 -4.3855E-01
10.3436 -1.2365E+00 -8.1194E-01 17.2031 -8.0719E-01 -4.3466E-01
10.5530 -1.2193E+00 -7.9012E-01 17.3300 -8.0171E-01 -4.3114E-01
10.7682 -1.2018E+00 -7.6897E-01 17.4466 -7.9675E-01 -4.2795E-01
10.9886 -1.1840E+00 -7.4853E-01 17.5526 -7.9228E-01 -4.2510E-01
11.2139 -1.1662E+00 -7.2881E-01 17.6478 -7.8831E-01 -4.2258E-01
11.4434 -1.1483E+00 -7.0981E-01 17.7321 -7.8482E-01 -4.2039E-01
11.6766 -1.1304E+00 -6.9156E-01 17.8050 -7.8182E-01 -4.1850E-01
11.9129 -1.1127E+00 -6.7401E-01 17.8666 -7.7931E-01 -4.1693E-01
12.1518 -1.0951E+00 -6.5722E-01 17.9167 -7.7729E-01 -4.1566E-01
12.3927 -1.0778E+00 -6.4114E-01 17.9551 -7.7574E-01 -4.1470E-01
12.6350 -1.0608E+00 -6.2578E-01 17.9817 -7.7467E-01 -4.1403E-01
12.8782 -1.0441E+00 -6.1113E-01 17.9965 -7.7408E-01 -4.1367E-01
321
Table B.21: Results for a = 9.0 (Kn = 0.0987)
r q2(r) q3(r) r q2(r) q3(r)
9.0035 -1.4141E+00 -1.2659E+00 14.1218 -1.0750E+00 -6.4219E-01
9.0183 -1.4136E+00 -1.2555E+00 14.3650 -1.0601E+00 -6.2867E-01
9.0449 -1.4126E+00 -1.2427E+00 14.6073 -1.0456E+00 -6.1577E-01
9.0833 -1.4111E+00 -1.2278E+00 14.8482 -1.0314E+00 -6.0349E-01
9.1334 -1.4090E+00 -1.2112E+00 15.0871 -1.0177E+00 -5.9182E-01
9.1950 -1.4062E+00 -1.1931E+00 15.3234 -1.0043E+00 -5.8074E-01
9.2679 -1.4027E+00 -1.1737E+00 15.5566 -9.9149E-01 -5.7023E-01
9.3522 -1.3984E+00 -1.1532E+00 15.7861 -9.7909E-01 -5.6028E-01
9.4474 -1.3932E+00 -1.1318E+00 16.0114 -9.6716E-01 -5.5087E-01
9.5534 -1.3872E+00 -1.1097E+00 16.2318 -9.5573E-01 -5.4198E-01
9.6700 -1.3803E+00 -1.0870E+00 16.4470 -9.4479E-01 -5.3361E-01
9.7969 -1.3724E+00 -1.0638E+00 16.6564 -9.3434E-01 -5.2572E-01
9.9337 -1.3636E+00 -1.0404E+00 16.8595 -9.2441E-01 -5.1831E-01
10.0801 -1.3539E+00 -1.0168E+00 17.0558 -9.1499E-01 -5.1136E-01
10.2359 -1.3433E+00 -9.9314E-01 17.2448 -9.0607E-01 -5.0490E-01
10.4006 -1.3319E+00 -9.6958E-01 17.4262 -8.9766E-01 -4.9881E-01
10.5738 -1.3196E+00 -9.4620E-01 17.5994 -8.8976E-01 -4.9316E-01
10.7552 -1.3067E+00 -9.2307E-01 17.7641 -8.8236E-01 -4.8794E-01
10.9442 -1.2931E+00 -9.0031E-01 17.9199 -8.7547E-01 -4.8310E-01
11.1405 -1.2789E+00 -8.7796E-01 18.0663 -8.6908E-01 -4.7866E-01
11.3436 -1.2642E+00 -8.5610E-01 18.2031 -8.6318E-01 -4.7459E-01
11.5530 -1.2490E+00 -8.3475E-01 18.3300 -8.5778E-01 -4.7089E-01
11.7682 -1.2335E+00 -8.1399E-01 18.4466 -8.5288E-01 -4.6756E-01
11.9886 -1.2177E+00 -7.9383E-01 18.5526 -8.4846E-01 -4.6457E-01
12.2139 -1.2018E+00 -7.7431E-01 18.6478 -8.4453E-01 -4.6192E-01
12.4434 -1.1857E+00 -7.5545E-01 18.7321 -8.4109E-01 -4.5961E-01
12.6766 -1.1695E+00 -7.3725E-01 18.8050 -8.3812E-01 -4.5764E-01
12.9129 -1.1534E+00 -7.1973E-01 18.8666 -8.3563E-01 -4.5597E-01
13.1518 -1.1373E+00 -7.0287E-01 18.9167 -8.3362E-01 -4.5464E-01
13.3927 -1.1214E+00 -6.8670E-01 18.9551 -8.3208E-01 -4.5363E-01
13.6350 -1.1057E+00 -6.7120E-01 18.9817 -8.3101E-01 -4.5292E-01
13.8782 -1.0902E+00 -6.5637E-01 18.9965 -8.3043E-01 -4.5254E-01
322
Table B.22: Results for a = 10.0 (Kn = 0.0888)
r q2(r) q3(r) r q2(r) q3(r)
10.0035 -1.4141E+00 -1.2781E+00 15.1218 -1.1142E+00 -6.8287E-01
10.0183 -1.4137E+00 -1.2689E+00 15.3650 -1.1004E+00 -6.6917E-01
10.0449 -1.4129E+00 -1.2573E+00 15.6073 -1.0868E+00 -6.5613E-01
10.0833 -1.4116E+00 -1.2439E+00 15.8482 -1.0735E+00 -6.4370E-01
10.1334 -1.4099E+00 -1.2288E+00 16.0871 -1.0605E+00 -6.3185E-01
10.1950 -1.4076E+00 -1.2123E+00 16.3234 -1.0479E+00 -6.2058E-01
10.2679 -1.4047E+00 -1.1945E+00 16.5566 -1.0357E+00 -6.0987E-01
10.3522 -1.4011E+00 -1.1757E+00 16.7861 -1.0239E+00 -5.9972E-01
10.4474 -1.3969E+00 -1.1559E+00 17.0114 -1.0126E+00 -5.9011E-01
10.5534 -1.3918E+00 -1.1354E+00 17.2318 -1.0016E+00 -5.8100E-01
10.6700 -1.3860E+00 -1.1143E+00 17.4470 -9.9113E-01 -5.7241E-01
10.7969 -1.3794E+00 -1.0926E+00 17.6564 -9.8110E-01 -5.6432E-01
10.9337 -1.3719E+00 -1.0706E+00 17.8595 -9.7153E-01 -5.5670E-01
11.0801 -1.3637E+00 -1.0484E+00 18.0558 -9.6243E-01 -5.4955E-01
11.2359 -1.3546E+00 -1.0260E+00 18.2448 -9.5380E-01 -5.4286E-01
11.4006 -1.3449E+00 -1.0036E+00 18.4262 -9.4563E-01 -5.3661E-01
11.5738 -1.3343E+00 -9.8128E-01 18.5994 -9.3798E-01 -5.3079E-01
11.7552 -1.3231E+00 -9.5912E-01 18.7641 -9.3079E-01 -5.2539E-01
11.9442 -1.3113E+00 -9.3721E-01 18.9199 -9.2407E-01 -5.2039E-01
12.1405 -1.2988E+00 -9.1562E-01 19.0663 -9.1784E-01 -5.1579E-01
12.3436 -1.2859E+00 -8.9441E-01 19.2031 -9.1208E-01 -5.1158E-01
12.5530 -1.2725E+00 -8.7365E-01 19.3300 -9.0680E-01 -5.0774E-01
12.7682 -1.2587E+00 -8.5336E-01 19.4466 -9.0200E-01 -5.0428E-01
12.9886 -1.2446E+00 -8.3360E-01 19.5526 -8.9768E-01 -5.0118E-01
13.2139 -1.2303E+00 -8.1441E-01 19.6478 -8.9383E-01 -4.9843E-01
13.4434 -1.2158E+00 -7.9579E-01 19.7321 -8.9045E-01 -4.9603E-01
13.6766 -1.2012E+00 -7.7778E-01 19.8050 -8.8754E-01 -4.9397E-01
13.9129 -1.1865E+00 -7.6038E-01 19.8666 -8.8509E-01 -4.9225E-01
14.1518 -1.1718E+00 -7.4361E-01 19.9167 -8.8312E-01 -4.9086E-01
14.3927 -1.1572E+00 -7.2747E-01 19.9551 -8.8160E-01 -4.8981E-01
14.6350 -1.1427E+00 -7.1196E-01 19.9817 -8.8056E-01 -4.8907E-01
14.8782 -1.1284E+00 -6.9708E-01 19.9965 -8.7998E-01 -4.8867E-01
323
Table B.23: Results for a = 50 (Kn = 0.0178)
r q2(r) q3(r) r q2(r) q3(r)
50.0035 -1.4142E+00 -1.3838E+00 55.1218 -1.3893E+00 -1.1870E+00
50.0183 -1.4142E+00 -1.3823E+00 55.3650 -1.3875E+00 -1.1795E+00
50.0449 -1.4142E+00 -1.3801E+00 55.6073 -1.3856E+00 -1.1722E+00
50.0833 -1.4141E+00 -1.3775E+00 55.8482 -1.3837E+00 -1.1649E+00
50.1334 -1.4140E+00 -1.3743E+00 56.0871 -1.3818E+00 -1.1579E+00
50.1950 -1.4139E+00 -1.3708E+00 56.3234 -1.3799E+00 -1.1510E+00
50.2679 -1.4138E+00 -1.3668E+00 56.5566 -1.3780E+00 -1.1443E+00
50.3522 -1.4136E+00 -1.3625E+00 56.7861 -1.3760E+00 -1.1379E+00
50.4474 -1.4134E+00 -1.3579E+00 57.0114 -1.3741E+00 -1.1314E+00
50.5534 -1.4132E+00 -1.3529E+00 57.2318 -1.3722E+00 -1.1253E+00
50.6700 -1.4129E+00 -1.3476E+00 57.4470 -1.3703E+00 -1.1193E+00
50.7969 -1.4125E+00 -1.3420E+00 57.6564 -1.3685E+00 -1.1136E+00
50.9337 -1.4121E+00 -1.3361E+00 57.8595 -1.3667E+00 -1.1082E+00
51.0801 -1.4117E+00 -1.3299E+00 58.0558 -1.3649E+00 -1.1029E+00
51.2359 -1.4112E+00 -1.3235E+00 58.2448 -1.3632E+00 -1.0979E+00
51.4006 -1.4106E+00 -1.3168E+00 58.4262 -1.3615E+00 -1.0932E+00
51.5738 -1.4099E+00 -1.3099E+00 58.5994 -1.3599E+00 -1.0887E+00
51.7552 -1.4092E+00 -1.3029E+00 58.7641 -1.3584E+00 -1.0845E+00
51.9442 -1.4084E+00 -1.2956E+00 58.9199 -1.3569E+00 -1.0805E+00
52.1405 -1.4075E+00 -1.2882E+00 59.0663 -1.3555E+00 -1.0768E+00
52.3436 -1.4066E+00 -1.2807E+00 59.2031 -1.3542E+00 -1.0734E+00
52.5530 -1.4055E+00 -1.2730E+00 59.3300 -1.3530E+00 -1.0702E+00
52.7682 -1.4044E+00 -1.2654E+00 59.4466 -1.3519E+00 -1.0674E+00
52.9886 -1.4032E+00 -1.2575E+00 59.5526 -1.3509E+00 -1.0648E+00
53.2139 -1.4019E+00 -1.2496E+00 59.6478 -1.3499E+00 -1.0624E+00
53.4434 -1.4006E+00 -1.2417E+00 59.7321 -1.3491E+00 -1.0603E+00
53.6766 -1.3992E+00 -1.2338E+00 59.8050 -1.3484E+00 -1.0586E+00
53.9129 -1.3977E+00 -1.2258E+00 59.8666 -1.3478E+00 -1.0571E+00
54.1518 -1.3961E+00 -1.2179E+00 59.9167 -1.3473E+00 -1.0559E+00
54.3927 -1.3945E+00 -1.2101E+00 59.9551 -1.3465E+00 -1.0549E+00
54.6350 -1.3928E+00 -1.2023E+00 59.9817 -1.3467E+00 -1.0543E+00
54.8782 -1.3911E+00 -1.1946E+00 59.9965 -1.3466E+00 -1.0540E+00
324
Table B.24: Results for a = 100 (Kn = 0.00888)
r q2(r) q3(r) r q2(r) q3(r)
100.0035 -1.4141E+00 -1.3993E+00 105.1218 -1.4074E+00 -1.2950E+00
100.0183 -1.4142E+00 -1.3986E+00 105.3650 -1.4069E+00 -1.2906E+00
100.0449 -1.4142E+00 -1.3975E+00 105.6073 -1.4063E+00 -1.2864E+00
100.0833 -1.4142E+00 -1.3962E+00 105.8482 -1.4058E+00 -1.2823E+00
100.1334 -1.4142E+00 -1.3947E+00 106.0871 -1.4052E+00 -1.2780E+00
100.1950 -1.4141E+00 -1.3930E+00 106.3234 -1.4046E+00 -1.2738E+00
100.2679 -1.4141E+00 -1.3910E+00 106.5566 -1.4040E+00 -1.2698E+00
100.3522 -1.4141E+00 -1.3889E+00 106.7861 -1.4035E+00 -1.2658E+00
100.4474 -1.4140E+00 -1.3866E+00 107.0114 -1.4029E+00 -1.2620E+00
100.5534 -1.4140E+00 -1.3841E+00 107.2318 -1.4023E+00 -1.2583E+00
100.6700 -1.4139E+00 -1.3814E+00 107.4470 -1.4017E+00 -1.2546E+00
100.7969 -1.4138E+00 -1.3786E+00 107.6564 -1.4011E+00 -1.2511E+00
100.9337 -1.4137E+00 -1.3756E+00 107.8595 -1.4005E+00 -1.2477E+00
101.0801 -1.4136E+00 -1.3724E+00 108.0558 -1.3999E+00 -1.2445E+00
101.2359 -1.4135E+00 -1.3692E+00 108.2448 -1.3994E+00 -1.2413E+00
101.4006 -1.4133E+00 -1.3657E+00 108.4262 -1.3989E+00 -1.2383E+00
101.5738 -1.4131E+00 -1.3623E+00 108.5994 -1.3984E+00 -1.2355E+00
101.7552 -1.4129E+00 -1.3585E+00 108.7641 -1.3979E+00 -1.2328E+00
101.9442 -1.4127E+00 -1.3547E+00 108.9199 -1.3974E+00 -1.2303E+00
102.1405 -1.4125E+00 -1.3508E+00 109.0663 -1.3969E+00 -1.2280E+00
102.3436 -1.4122E+00 -1.3469E+00 109.2031 -1.3965E+00 -1.2258E+00
102.5530 -1.4120E+00 -1.3427E+00 109.3300 -1.3961E+00 -1.2237E+00
102.7682 -1.4117E+00 -1.3386E+00 109.4466 -1.3957E+00 -1.2219E+00
102.9886 -1.4113E+00 -1.3343E+00 109.5526 -1.3954E+00 -1.2202E+00
103.2139 -1.4110E+00 -1.3301E+00 109.6478 -1.3951E+00 -1.2187E+00
103.4434 -1.4106E+00 -1.3257E+00 109.7321 -1.3948E+00 -1.2173E+00
103.6766 -1.4102E+00 -1.3213E+00 109.8050 -1.3946E+00 -1.2162E+00
103.9129 -1.4098E+00 -1.3170E+00 109.8666 -1.3944E+00 -1.2152E+00
104.1518 -1.4094E+00 -1.3125E+00 109.9167 -1.3942E+00 -1.2145E+00
104.3927 -1.4089E+00 -1.3082E+00 109.9551 -1.3941E+00 -1.2138E+00
104.6350 -1.4084E+00 -1.3037E+00 109.9817 -1.3940E+00 -1.2134E+00
104.8782 -1.4079E+00 -1.2993E+00 109.9965 -1.3940E+00 -1.2132E+00
325
Table B.25: Results for c1 and c2
r0 Kn c1 c2
0.01 88.8 -0.0269 -14.7212
0.025 35.5 -0.0379 -14.3333
0.05 17.8 -0.0564 -13.6869
0.075 11.8 -0.0748 -13.0404
0.1 8.88 -0.0932 -12.3940
0.25 3.55 -0.2147 -12.1216
0.5 1.78 -0.3826 -5.5702
0.75 1.18 -0.5148 -3.4101
1.0 0.888 -0.6205 -2.3438
1.25 0.710 -0.7061 -1.7167
1.5 0.592 -0.7767 -1.3043
1.75 0.5074 -0.8356 -1.0166
2.0 0.444 -0.8854 -0.8038
2.5 0.355 -0.9649 -0.5150
3.0 0.296 -1.0253 -0.3295
4.0 0.222 -1.1115 -0.1039
5.0 0.178 -1.1706 0.0268
6.0 0.148 -1.2137 0.1113
7.0 0.1269 -1.2468 0.1707
8.0 0.111 -1.2734 0.2153
9.0 0.0987 -1.2949 0.2489
10.0 0.0888 -1.3129 0.2758
50.0 0.0178 -1.4655 0.4651
100 0.00888 -1.4857 0.4858
326
Appendix C: Monte Carlo Drag and Torque Results
I have developed a Monte Carlo algorithm to compute the drag and torque on
an N -sphere aggregate in the free molecule regime. The algorithm is described in
Section 5.2.3. Here, I compare the results of my Monte Carlo algorithm to exact
results (where available) and to the drag results of Mackowski [36].
C.1 Drag on a Translating Sphere
The drag on a sphere in creeping flow in the free molecule regime F0 is given
by Epstein’s equation [32]. For purely diffuse reflection, my Monte Carlo algorithm
gives the drag as FMC = 1.001F0, and thus my MC results are in very good agree-
ment with the exact solution.
C.2 Drag on an Aggregate
Mackowski [36] developed a correlation for the drag on a fractal aggregate as
a function of the number of monomers N , the fractal dimension df , and the fractal
prefactor k0. The correlation is based on the results of his own Monte Carlo cal-
culations. Using Mackowski’s correlation [Eq. (68) of Ref. [36]], the translational
friction coefficients, normalized by Epstein’s equation, for 20- and 100-sphere aggre-
327
gates with df = 1.78 and k0 = 1.3 are 14.15 and 64.23, respectively. In comparison,
my Monte Carlo results for the 20- and 100-sphere aggregates with these fractal
dimensions are 14.51 and 64.58, respectively. Thus, my MC results are in very good
agreement with Mackowski’s correlation.
C.3 Torque on a Rotating Sphere
Epstein [32] calculates the torque on a sphere rotating about an axis through
its center. Using my Monte Carlo algorithm, I get TMC = 0.995T0, which is in very
good agreement with the exact value.
For a sphere rotating slowly around an axis located a distance R from its
center, the magnitude of the torque is given by
T = ζ 2t,0ωR + ζr,0ω (C.1)
In other words, the torque is the sum of the torque on a sphere rotating about its
center with angular velocity ω and the torque due to the linear velocity of the sphere
center ωR moving at a distance R from the origin. My Monte Carlo results for the
torque for R = a and R = 2a are 3.763ζr,0ω and 12.15ζr,0ω, respectively, where a is
the sphere radius. These results are in very good agreement with the exact results
T = 3.785ζr,0ω and T = 12.14ζr,0ω for a sphere rotating around an axis R = a and
R = 2a from its center.
328
Appendix D: Relationship between the Rotation and Coupling In-
teraction Tensors and the Flow around a Sphere
As mentioned in Section 5.2.2, the (1−δij)r−3 −2ij and (1−δij)rij terms in the ro-
tation and coupling hydrodynamic interaction tensors are related to the flow around
a sphere. We provide the derivation in this appendix. Note that this derivation is
similar to the derivation of the lower order terms in the method of reflections [59, 60].
We will start with the rotation hydrodynamic tensor Qrij. Consider a sphere
rotating in a quiescent fluid with angular velocity ωj. This angular motion is sus-
tained by applying a torque Tj = ζr,0ωj to the sphere. The velocity induced in the
fluid at a location rij from the rotating sphere can be written in spherical coordinates
as
ωa3
v(rij) = sin θê2 φ (D.1)rij
where êφ is the unit vector in the φ-direction. The vorticity in the fluid is
3
w(rij) = ∇×
ωa
v = (2 cos θêr + sin θêθ) (D.2)
r3ij
Converting to Cartesian coordinates and performing some simple manipulations, we
329
find that the vorticity can be written
( )
a3 3rijrij
w(rij) = − I ·ωj (D.3)
r3ij r
2
ij
A sphere placed in the fluid at rij would rotate with angular velocity ωi =
1w(r
2 ij
)
[49], which can be written
[ ( )]
1 a3 3rijrij
ωi = − I ·T (D.4)
2ζ r3 2
j
r,0 ij rij
The term in square brackets is the (1 − δij) term in the rotation hydrodynamic
interaction tensor [Eq. (5.24)], proving that the rotational interaction between two
spheres is related to the vorticity of the flow field around a rotating sphere.
We now turn our attention to the coupling tensor Qcij. Consider a sphere
translating through a quiescent fluid with velocity uj due to some external force
Fj = ζt,0uj. The vorticity at point rij in spherical coordinates is
3 a
wj = − uj sin θê2 φ (D.5)2 rij
We can write this more generally as
3 a 3 a
wj = − uj × rij = rij × uj (D.6)
2 r3 3ij 2 rij
We can write the cross product rij × uj as Aij ·uj, where Aij = − · rij, such that
330
the vorticity becomes ( )
− 3 awj =  · rij ·uj (D.7)
2 r3ij
Again, a sphere placed in the fluid at rij would rotate with angular velocity equal
to half the vorticity, ( )
− 3 a · · Fjωj =  rij (D.8)
4 r3ij 6πµa
Rearranging and introducing ζr,0 = 8πµa
3, we get
( )
 · r a3ij
ωj = − ·Fj (D.9)
ζ 3r,0 rij
The term in parentheses is the coupling interaction tensor Qcij given by Eq. (5.25).
This shows that the O(r−2ij ) term in the translation-rotation coupling interaction
tensor is given by the vorticity in the flow field for a translating sphere.
Alternatively, we can derive the coupling tensor by considering the velocity
field around a rotating sphere given by Eq. (D.1). Writing the velocity using the
cross product rij×ωj and converting the cross product to −( · rij) ·ωj, the velocity
becomes
a3
v(rij) = ( · r ) ·ω3 ij j (D.10)rij
Writing this equation using the torque applied on sphere j to maintain the angular
velocity ωj, we see that the fluid velocity at rij is
( )
 · r 3ij a
v(rij) = ·Tj (D.11)
ζr,0 r3ij
331
The term in parentheses is the transpose of Qcij as written in Eq. (5.25), which
shows that the O(r−2ij ) term in (Qc )†ij is given by the velocity field around a rotating
sphere.
332
Appendix E: Supplemental Material for Chapter 7
In this appendix, I provide a diagram of the body-fixed and space-fixed coor-
dinate systems used for my aggregate alignment calculations (Chapter 7), I show
probability distributions for one particle at different field strengths, I provide an ex-
ample calculation for one aggregate, and I expand upon my discussion of the effects
of Knudsen number and the number of primary spheres on fully-aligned particle
electrical mobility.
E.1 Euler Angles
The Euler angles (φ, θ, ψ) relate the body-fixed coordinates (x′, y′, z′) to the
space-fixed coordinates (x, y, z), as shown in Figure E.1. For our calculations, the
electric field is in the z-direction, and the principal axis of the polarizability tensor
is in the z′-direction. When the particle is aligned with the electric field, the z- and
z′-axes coincide.
E.2 Probability Distributions
Eq. (eqn:align:potential) gives the potential of a conducting particle in an elec-
tric field as a function of the polarizability tensor of the particle, the field strength,
333
Figure E.1: Representation of the Euler angles (φ, θ, ψ) that relate the body-fixed
coordinates (x′, y′, z′) to the space-fixed coordinates (x, y, z).
334
and the orientation of the particle relative to the field. When the electric field is
in the z-direction in space-fixed coordinates, the potential is a function of only two
of the three Euler angles shown in Figure E.1. In Figure E.2 below, I show the
probability distribution for a particle with 652 primary spheres with radius 5 nm at
three electric field strengths. The probability distribution is defined by Eq. (7.1). At
E = 1000 V/cm, the particle orientation is nearly random, while at E = 8000 V/cm,
the particle is aligned with the field. Note that the aligned particle can still rotate
around the principal axis of the polarizability tensor. Mobility results for this par-
ticle are shown in Figure 7.1.
E.3 Sample Calculation
The theory described in Chapter 7 requires one to determine the translational
friction tensor and the polarizability tensor for an aggregate in order to determine
the mobility of the particle at a given field strength. Here, I provide an example
problem for an aggregate with 10 primary spheres each having a radius a = 37.9 nm
(Kn = 1.78 for λ = 67.3 nm). The aggregate has a fractal dimension of 1.78 and
a prefactor of 1.3. The coordinates for the center of each sphere in the aggregate
are given in Table E.1. Coordinates are given relative to the center of mass of
the aggregate for an arbitrary Cartesian system. Velocity results for this Knudsen
number are available in Appendix B.
Using EKR theory (Chapters 4 and 6), one can determine the translational,
rotational, and coupling friction tensors in terms of the Cartesian coordinate system
335
Figure E.2: Probability distributions for particles with 652 primary spheres with
5 nm radii. Probability distributions are given for 3 field strengths: 1000 V/cm
(top), 3000 V/cm(middle), and 8000 V/cm (bottom). Note the different scales in
the figure: the bottom figure has a much broader scale than the top figure.
336
Table E.1: Coordinates of the center of each sphere in my sample aggregate. Coordi-
nates have been non-dimensionalized by the primary sphere radius. Since the center
of mass is at (0,0,0), simply multiply by the sphere radius to get the coordinates in
dimensional form.
Sphere x y z
1 -0.1717 -0.2663 -2.2887
2 2.0578 -0.0414 -0.2788
3 0.0587 -0.0934 -0.3096
4 0.3888 1.4840 -3.0774
5 -3.0183 -0.9897 2.3343
6 -1.7995 -1.6475 0.8915
7 -1.0953 -1.5832 -0.9794
8 -2.9090 0.6522 1.1976
9 3.1193 1.1203 2.2399
10 3.3692 1.3651 0.2707
in which the particle coordinates are given:
 
 4.490 −0.2919 0.1575 Ξt = 

− FM0.2919 4.710 0.1075 ζt,0 (E.1)
0.1575 0.1075 4.304
 
 29.66 −7.276 1.057 ΞO,r = −
2 FM
7.276 51.56 1.238 a ζt,0 (E.2)
1.057 1.238 42.77
 −0.6409 −0.3110 0.5337 Ξ FMO,c =  0.9522 0.5417 0.2755 aζt,0 (E.3)
−0.7883 0.6892 0.0884
Here, ζFMt,0 is the free molecule friction coefficient based on the primary sphere radius.
337
For consistency with my velocity results, one should assume full thermal accommo-
dation when calculating the friction coefficient. The rotational and coupling tensors
are written with respect to the center of mass of the particle.
I obtain the polarizability tensor from ZENO [62]:
  513.0 114.1 −39.24
α =  3 114.1 226.7 −31.19 0a (E.4)
−39.24 −31.19 394.1
Here, 0 is the permittivity of free space. Note that ZENO uses stochastic methods
to obtain the polarizability tensor, so the code results can change slightly depend-
ing on the number of trials used in the calculation and on the initial seed to the
random number generator. Also note that the polarizability, translational friction,
and rotational friction tensors are symmetric, as expected.
For my calculations of the mobility as a function of electric field strength, I
choose the body-fixed coordinate system based on the eigenvectors of the polariz-
ability tensor: the z′-axis is the eigenvector associated with the largest eigenvalue of
the tensor, while the x′-axis coincides with the eigenvector of the smallest eigenvalue
of the tensor. In this coordinate system, the polarizability and translational friction
tensors become
 

′ † 185.4 0 0α = V ·α ·V =  

  3 0 382.3 0  0a (E.5)
0 0 566.0
338
  4.869 0.0349 −0.1459′ † Ξt = V ·Ξt ·V =  0.0349 4.404 −
FM
0.1777 ζt,0 (E.6)
−0.1459 −0.1777 4.232
where   0.3193 0.2783 −0.9059 V = −0.9442 0.0116 −0.3293 (E.7)
−0.0811 0.9604 0.2664
is the tensor whose columns are the eigenvectors of α and V † is the transpose of that
tensor. I take the inverse of the translational friction tensor to obtain the mobility
tensor in body-fixed coordinates:
 
 0.2056 −0.0013 0.0070
M = Ξ′t = −0.0013 0.2275 0.0095
( )−1
 ζ
FM
t,0 (E.8)
0.0070 0.0095 0.2369
Note that the mobility tensor in the body-fixed coordinate system is nearly, but
not quite, diagonal. For non-skew particles like spheres or prolate spheroids, the
eigenvectors for the polarizability a nd translational friction/mobility tensors would
be the same, and thus the mobility tensor would be diagonal in the body-fixed
system corresponding to the eigenvectors of the polarizability tensor.
From Eqs. (7.6) and (7.7) and the polarizability tensor above, the potential of
339
this particle in an electric field due to the induced dipole effects is
U = −1(185.4 sin2 ψ sin2 θ + 382.3 cos2 ψ sin2 θ + 566.0 cos2 θ) 30a E2 (E.9)2
where the angles are defined in Figure E.1. From Eqs. (7.8) and (7.9) and the
mobility tensor above, the mobility of the particle as a function of field strength is
(
〈Vd,z〉 q
Z = = 0.2369〈cos2 θ〉+ 0.2275〈cos2 ψ sin2 θ〉+ 0.2056〈sin2 ψ)sin
2 θ〉
E ζFMt,0
− 0.0013〈sin 2ψ sin2 θ〉+ 0.0070〈sinψ sin 2θ〉+ 0.0095〉 cosψ sin 2θ〉 (E.10)
At very low field strengths, we get the mobility of the randomly oriented
particle, Z FMrand = 0.2233q/ζt,0 . At very high field strengths, we get the mobility
of the aligned particle, Z = 0.2369q/ζFMrand t,0 = 1.061Zrand. At a field strength of
5000 V/cm, we get Zrand = 0.2317q/ζ
FM
t,0 = 1.038Zrand.
Note that we can also use my analytic expression for the translational friction
coefficient [Eq. (4.38) of Chapter 4] to obtain Zrand. From this expression, we get
Z?rand = 0.2289q/ζ
FM
t,0 = 1.025Zrand, where Zrand is the mobility we get from the
EKR method. This shows that my analytic expression gives a good estimate of the
friction coefficient or mobility of a randomly oriented DLCA particle, without having
to deal with the complexities of the EKR method. Note that my analytic expression
does not account for different morphologies (i.e. different sphere coordinates) for the
same primary sphere diameter, same N , and same fractal dimension.
340
E.4 Effects of Knudsen Number and the Number of Primary Spheres
on Fully-Aligned Particle Mobility
To evaluate the impact of particle size and flow regime on the aligned mobility
of soot-like particles, I calculated the random (electric field strength E → 0) and
fully-aligned (E → ∞) mobilities for a wide range of N and Kn. I also calculated
the random and fully-aligned mobilities using the standard Kirkwood-Riseman ap-
proach with the Rotne-Prager-Yamakawa hydrodynamic interaction tensor [see, for
example, [30]] and using a Monte Carlo code (Chapter 5 and Appendix C) for the
continuum (Kn = 0) and free molecular (Kn =∞) limits, respectively.
Figure 7.5 in the main body of this Disseration (repeated here as Figure E.3)
shows the calculated ratio of aligned to random mobility. The choppiness of the
graphs is due to the finite sample size of the aggregates I am using for these cal-
culations. The standard deviation of the mean of each data set (i.e. the mobility
ratio of 20 cases at a given Kn and N) is on the order of 0.007, which is significant
compared to the scale of the graph and the fluctuations in the mobility ratio with
N for a given Knudsen number. Accounting for these fluctuations due to the finite
sample size, we notice a few clear trends: while the mobility ratio is approximately
constant with N in the continuum regime, there is a clear decrease in mobility ratio
with increasing N near the free molecule regime. At intermediate Knudsen numbers,
the particles exhibit more continuum-like behavior with increasing N , which is con-
sistent with my earlier results for the translational friction coefficient (Chapter 4).
341
I will consider the physical reasons for these trends in the following paragraphs.
Figure E.3: Ratio of fully-aligned to random electric mobilities for wide range of
primary sphere Knudsen numbers and the number of primaries. The Kn = 0 and
Kn = ∞ curves represent the continuum and free molecular limits, as calculated
using the standard KR theory with the RPY tensor [see e.g. Chen et al. [30]] and
using a Monte Carlo code (Chapter 6), respectively. Uncertainties of one standard
deviation of the mean (based on 20 samples with the same fractal dimension but
different morphologies) are shown for the continuum and free molecule results for
several N .
The continuum-like behavior of particles with many primary spheres at inter-
mediate Knudsen numbers has a straightforward explanation: as the particle size
increases, it has a larger effect on the velocity field of the surrounding fluid. In other
words, the particle behaves less like a collection of spheres in the transition regime
and more like an object that is large compared to the mean free path of molecules
in the fluid.
The behavior in the continuum and free molecular limits is more difficult to
explain. I will start by focusing on the continuum regime.
342
The continuum (randomly-oriented) friction coefficient can be written
ζ = 6πµRH (E.11)
where the hydrodynamic radius RH is the radius of a sphere that experiences the
same drag as the particle. Studies have shown that the hydrodynamic radius is
approximately equal to the radius of gyration for soot-like particles [4, 54, 93]. This
relationship between the hydrodynamic radius and the radius of gyration is only
valid for a randomly-oriented particle. Nevertheless, let us suppose that the drag
force on a particle in a fixed orientation moving in the z-direction is related to the
particles radius of gyration about the z-axis, which is defined as
√∑N 2
i (xi + y
2
R i
)
gz = (E.12)
N
for an aggregate of N spheres. Here, the center of the ith sphere is located at
(xi, yi, zi), while the center of mass of the particle is at the origin. Clearly, the
drag is not directly proportional to Rgz. As an example, the drag on a chain of
spheres or a rod moving parallel to its long axis increases with chain or rod length,
even though Rgz is independent of the chain or rod length. However, the drag
on a rod does increase as its radius increases (with length held constant), so a
larger Rgz corresponds to increased drag. I will look at the ratio of the radius of
gyration to the radius of gyration about the principal axis of the polarizability tensor
(i.e. the axis parallel to the particle velocity when the particle is fully-aligned with
343
the electric field) to evaluate whether trends in Rg/Rgz are correlated to trends in
the continuum mobility ratio Zalign/Z
1
rand. If this is the case, we can use trends in
Rg/Rgz to qualitatively predict the effects of alignment on mobility.
To evaluate whether Rg/Rgz and Zalign/Zrand are correlated, I calculated these
ratios for all 20 cases for eachN and plotted the results in Figure E.4. I have included
results for N = 5, 10, 50, 100, 500, and 1000; results for other N are similar. In
general, aggregates with a low radius of gyration ratio also have a low mobility ratio.
Thus, we can conclude that there is a correlation between these two ratios.
As further verification of the correlation between Rg/Rgz and Zalign/Zrand, I
have computed the average ratios for a given N , then plotted the results in Fig-
ure E.5. Once again, we see that the ratios appear to be correlated.
Given that the radius of gyration ratio and the mobility ratio appear to be
correlated, we can look at the trends in Rg/Rgz to help explain the behavior of the
continuum curve in Figure E.3. Notably, there is less spread in the mean values of
Rg/Rgz versus N than there is in Zalign/Zrand for a given N , as evidenced by the
scale of the y-axis in Figure E.4 compared to the scale in Figure E.5. For example,
the ratio of the maximum to the minimum values of the mean Rg/Rgz vs. N is 1.14,
while for a typical set of 20 particles for a given N the ratio of maximum to minimum
values of Rg/Rgz is 1.65. Likewise, the continuum mobility ratio varies from 1.076
(at N = 50) to 1.094 (at N = 236 and N = 2000), while the minimum and maximum
mobility ratios for N = 100 (which represents an average case in terms of the spread
1Since the friction coefficient is approximately proportional to the radius of gyration, the mo-
bility is inversely proportional to Rg.
344
Figure E.4: Comparison of the continuum mobility and radius of gyration ratios for
all 20 cases for N = 5, 10, 50, 100, 500, and 1000. The xs represent the mobility
ratio; the circles represent the radius of gyration ratio.
345
Figure E.5: Comparison between the average continuum mobility ratio and the
average radius of gyration ratio as a function of N . In general, a low mobility ratio
is correlated with a low radius of gyration ratio; the notable exception is for N =
5, which has a large radius of gyration ratio but a modest mobility ratio.
in mobility ratio) range from 1.05 to 1.13. Thus, the mobility ratio in the continuum
appears to be more heavily dependent on the specific configuration of spheres in the
aggregate than on the number of spheres for a given fractal dimension and prefactor.
Now I will consider the free molecular case. The friction coefficient in the
free molecular regime is roughly proportional to the orientation-averaged projected
area [41]. As I did for the continuum case, I will suppose that the projected area
of the particle in the plane normal to particle velocity when it is fully aligned is
correlated to the free molecular drag force for that particle orientation. (Again,
I use the example of a rod or chain of spheres to caution that the drag for any
particle orientation is not directly proportional to the projected area in the plane
perpendicular to the particle velocity.) I have calculated the orientation-averaged
projected area using a Monte Carlo approach, while I have calculated PAz using a
346
graphical approach (by plotting particle as a collection of spheres and counting the
number of black pixels in the projection of the particle on the xy-plane).
To evaluate whether PA/PAz and Zalign/Zrand are correlated, I have plotted
these ratios to look at how they compare for the aggregates in this study. Figure E.6
compares these ratios for the 20 cases for each of N = 5, 10, 50, 100, 500, and
1000. In general, cases with a low mobility ratio also have a low projected area
ratio. Figure E.7 shows an even better correlation between the mean values of the
projected area and mobility ratios for each N .
Figure E.6: Comparison of the free molecule mobility and projected area ratios for
all 20 cases for N = 5, 10, 50, 100, 500, and 1000. The xs represent the mobility
ratio; the circles represent the projected area ratio.
347
Figure E.7: Comparison between the average free molecule mobility ratio and the
average projected area ratio as a function of N . In general, a low mobility ratio is
correlated with a low projected area ratio. Both ratios decrease with increasing N .
Since PA/PAz and Zalign/Zrand appear to be correlated, we can use trends in
the projected area ratio to help explain the alignment behavior in Figure E.3. As the
number of primary spheres in the aggregate increases, the projected area on the plane
perpendicular to the principal axis of the polarizability tensor decreases relative to
the orientation-averaged projected area. In fact, the projected area varies little with
orientation at large N . This behavior contrasts with the continuum behavior, where
the average radius of gyration ratio is fairly constant with N .
My results demonstrate that the projected area and radius of gyration ratios
are correlated with the mobility ratio in the free molecular and continuum regimes,
respectively, and thus we can predict the trends in the mobility ratio based on the
trends of these surrogate measures. But why is this true?
In the continuum regime, the moving particle has a significant effect on the
348
velocity of the surrounding fluid. One way to think of this is as follows. The fluid
upstream of the particle must move out of the way to avoid the particle; the extent
of how far the fluid must move is related to the radius of gyration about the axis
parallel to the flow. A larger Rgz means the fluid must move further to get out of
the way of the particle, which explains the increase in drag (decrease in mobility).
In the free molecular regime, the particle has very little effect on the fluid
velocity. In fact, the particle has no effect on the velocity distribution of molecules
striking its surface (except for molecules that strike the particle more than once
between collisions with other fluid molecules), so we can use a ballistic approach for
calculating the drag [36, 54, 117, 118]. Thus, the projected area is related to the
free molecular drag: as projected area increases, gas molecules are more likely to
strike and transfer their momentum to the particle, thus increasing the drag.
In summary, the ratio of particle mobility in the limit of infinite field strength
to mobility in the limit of zero field strength decreases as the number of primary
spheres increases in the free molecular regime, while the ratio remains constant in
the continuum regime. The trends in the free molecular and continuum regimes can
be explained by trends in the projected area ratio and the radius of gyration ratio,
respectively. In the continuum regime, the ratio of particle radius of gyration to the
radius of gyration about the principal axis of the polarizability tensor (i.e. the axis
parallel to the particle velocity when the particle is fully aligned) varies little with
N . In the free molecular regime, the ratio of the orientation-averaged projected
area to the projected area in the plane perpendicular to the principal axis of the
polarizability tensor decreases with increasing N . Note that these trends are for
349
the average behavior at each N ; there are variations in the radius of gyration and
projected area ratios for aggregates with the same N , which in turn affect the
fully-aligned mobility behavior of the particles. The variations in mobility among
aggregates with the same N are large in the continuum regime compared to the
variations in the average mobility as a function of N . Finally, at intermediate
Knudsen numbers, particles exhibit more continuum-like behavior as the number
of primary spheres increases. This is because the aggregate behaves more like a
particle with characteristic dimension much larger than the mean free path than a
collection of spheres with radii smaller than the mean free path.
350
Appendix F: NGDE User Manual
This appendix serves as the User Manual for the Nodal General Dynamic
Equation (NGDE) solver described in Chapter 9. This manual refers to the MAT-
LAB version of the code; for the C version, see Prakash et al. [27]. Throughout this
documentation, text in this font represents a variable or expression in the code
or a MATLAB command.
The manual is divided into three sections: (1) instructions for running the
NGDE code and the post-processing tool; (2) detailed descriptions of input to NGDE
(ngde.m) and NGDEplot (ngde plot.m) and output from the codes; and (3) discus-
sion about the NGDE code structure. (The code for NGDEplot mostly consists of
MATLAB commands for plotting data, so discussion of NGDEplot is limited to its
input and output.) For information about the theoretical basis of NGDE and the
methods used to solve the general dynamic equation, refer to Chapter 9.
F.1 Running NGDE and NGDEplot
NGDE is distributed as a .zip archive containing the following files:
• ngde.m: the NGDE source code
• ngde plot.m: the NGDEplot source code
351
• rtpmie bohren mie.m: Mie scattering code used by NGDEplot. The Mie
scattering code is based on the Fortran source code in Appendix A of Bohren
and Huffman [89]; rtpmie bohren mie.m has been converted from the Fortran
source code provided by Professors Eugene Clothiaux and Craig Bohren from
the Pennsylvania State University.
• ngde sample problem.mat: MATLAB data file containing two MATLAB struc-
ture arrays, ngdein and plotoptions, that provide input to NGDE and
NGDEplot, respectively
To run NGDE, perform the following steps:
1. In MATLAB, navigate to the folder containing the files listed above. To do so,
click on the “Browse for folder” icon (circled in red in the screenshot below)
and open the folder containing the NGDE files. All of the files in the current
folder appear in the “Current Folder” panel.
352
2. Load the NGDE and NGDEplot input structures (ngdein and plotoptions).
This can be done by double-clicking on the ngde sample problem.mat data file
distributed with NGDE, or by double-clicking on any other MATLAB data file
that contains NGDE input. After double-clicking on the data file, you should
see the load(...) statement and the NGDE input structures appear in the
Command Window panel and in the Workspace panel, respectively. (See the
screenshot below.)
3. If desired, modify the NGDE input. To do so, double-click on the NGDE
structure array (named ngdein in the code distribution) in the Workspace
panel. This will bring up the structure array in the Variables panel. To change
any of the input parameters, select the appropriate value (e.g. the value of 300
next to the Tf field) and type in the desired value (e.g. type in 400 to end the
NGDE simulation when the temperature equals 400 K instead of 300 K).
353
4. Once all of the desired changes have been made to the input structure array,
run NGDE by entering the following in the Command Window:
[results1,results2]=ngde(ngdein);
Here, results1 and results2 are the names of the variables where the user
will store the size distribution and detailed results from NGDE, respectively,
and ngdein is the NGDE input structure array. Of course, you can choose
any name for these input and output files; these are simply the default names.
After hitting enter, the code will begin running, as shown below.
354
5. If desired, run the NGDEplot post-processing tool entering the following state-
ment in the Command Window:
[F1,F2]=ngde plot(results2,ngdein,plotoptions,saveframes);
Here, F1 and F2 are the names of the variables where the frames of the size
distribution and Mie scattering movies will be stored, respectively; results2
is the array containing the detailed output results from NGDE; ngdein is the
NGDE input structure array corresponding to the output array results2;
plotoptions is the NGDEplot input structure array; and saveframes is a
logical variable, where a non-zero value tells the code to save the frames of
the size distribution and Mie scattering movies. Note that you can show a
still of a single point in time by specifying results2(t,:), where t is the
row index of results2 corresponding to the desired time point. Likewise, you
can generate movies of a select portion of the NGDE calculation by passing
355
the appropriate rows from results2 to NGDEplot. You may also change the
NGDEplot input by double-clicking on the NGDEplot input structure array
(plotoptions by default) in the Workspace panel and changing the values of
the structure array fields.
F.2 Description of Input and Output
F.2.1 NGDE Input (ngdein)
The following parameters are included in the NGDE input structure array,
ngdein. The default values are for the “Full GDE” sample problem described in
Chapter 9.
Parameter Description
choice Calculation type. 1 = coagulation only, 2 = nucleation + co-
agulation, 3 = full GDE, 4 = surface growth only. DEFAULT
= 3
printInterval Number of timesteps between plot edits. Too small a value
will result in a very large output from the code, which may
use up excessive computer memory. DEFAULT = 1000
nodes Number of size nodes to use in the calculation. Note that the
code actually creates nodes+1 volume nodes, but the number
concentration in the final node is always zero. This is done to
prevent errors in the code. DEFAULT = 41
356
beta option Collision frequency function. 1 = free molecule regime, 2 =
Fuch’s form for FM and transition regime. DEFAULT = 2
T0 Initial temperature (K). DEFAULT = 1773
Tf Final temperature (K). Not used if choice = 1. DEFAULT
= 300
coolrate Cooldown rate for the calculation (K/s). Not used if choice
= 1. DEFAULT = 1000
P Pressure (atm). DEFAULT = 1
MW Molecular weight of the condensed species (kg/mol). DE-
FAULT = 0.027
rho Density of the condensed species (kg/m3). DEFAULT = 2700
A, B Constants for determining surface tension as a function of
temperature, given by sigma=(A-B*T)*1e-3, where sigma
has units of N/m. DEFAULT = 948, 0.202 (for aluminum)
C, D Constants for determining saturation vapor pressure
as a function of temperature and pressure, given by
Ps=exp(C-D./T).*101325, where Ps has units of Pa.
DEFAULT = 13.07, 36373 (for aluminum)
d Initial particle diameter (nm). Only used if choice = 1. DE-
FAULT = 1
N0 Initial particle number concentration (#/m3). Only used if
choice = 1 or 4. DEFAULT = 1e10
357
i0 Node in which particles are initially placed. Only used if
choice = 4. DEFAULT = 25
MWgas Molecular weight of the carrier gas (kg/mol). DEFAULT =
0.04 (for argon)
A mu, B mu Sutherland’s constants for viscosity as a function of tempera-
ture, given by mu=A mu*T^1.5/(B mu+T), where mu has units
of kg/m-s. DEFAULT = 1.9660e-6, 147.47 (for argon)
dtUser Maximum allowable time step for the calculation. DEFAULT
= 1e-4
F.2.2 NGDEplot Input (plotoptions)
The following parameters are included in the NGDEplot input structure array,
plotoptions.
Parameter Description
npar Refractive index of the particles. DEFAULT = 1 + 6.4i (for
aluminum in the visible spectrum)
nmed Refractive index of the medium. DEFAULT = 1 (for argon)
showNmovie If non-zero, NGDEplot will show a movie of how the size
distribution changes over time. DEFAULT = 1
showMie If non-zero, NGDEplot will show a movie of how the light
scattering, extinction, and absorption coefficients change over
time. DEFAULT = 1
358
showTimePlots If non-zero, NGDEplot will show plots of several calculated
parameters as a function of time. DEFAULT = 0
F.2.3 NGDE and NGDEplot Output
NGDE has two output parameters:
• results1 is a nodes-by-2 array of the particle size distribution; the first col-
umn is the volume of a particle at each node, while the second column is the
number of particles (per cubic meter) at each node.
• results2 is an array containing various results as a function of time, where
each row of the array represents the results at the time in column 1 of the
array. The number of rows will depend on the simulation time and the edit
frequency (printInterval). The columns are as follows:
359
Column Description
1 The time, t
2:nodes+2 The number concentration at nodes 1 through nodes+1, N
[#/m3]
nodes+3 The nucleation rate, Jk [#/m3s]
nodes+4 The saturation ratio, S
nodes+5 The volume mean particle diameter, dpav [m]
nodes+6 The critical particle size for nucleation, kstar [# of
monomers]; nucleated particles are placed in the node with
particle volume just larger than the volume corresponding to
kstar monomers
nodes+7 The total volume concentration of particles, Vtot [m3]; note
that this does not include the volume of monomers
nodes+8 The number concentration of particles, Ntot [#/m3]; note
that this does not include the number concentration of
monomers
For more information about how these parameters are determined, see the technical
description of NGDE in Chapter 9.
NGDEplot has two output parameters, F1 and F2, which contain the frames
for the size distribution and Mie scattering movies generated by the code. If the user
chooses not save the frames (saveframes=0) or chooses not to generate one of the
movies (showNmovie=0 or showMie=0), then F1 and/or F2 will be zero. To replay
360
a movie using the saved frames, the user should use MATLAB’s movie command.
F.3 Code Structure
The goal of this section is to give users a better understanding of how the code
works and to identify some hardwired parameters or features that one can modify
to improve the code or tailor it to a specific application. The code is divided into
three main sections: (1) reading the input structure and initializing variables, (2)
subroutines common to more than one of the possible calculation types (see choice
in Section F.2.1), and (3) the main body of the program. The initialization step is
straightforward and will not be discussed here. The following subsections discuss
the subroutines and the main body of the code. These subsections largely focus on
code structure and logic; for the technical details, refer to Chapter 9 or to the code
itself.
F.3.1 NGDE Subroutines
After reading in the input structure and initializing variables, the NGDE
code defines a number of separate subroutines or functions, which are listed be-
low. To modify the code, simply find the appropriate subroutine and make the
desired changes.
361
Subroutine Description
createNodes Establishes the volume of each node, based on the de-
sired number of nodes and a 12 order of magnitude in-
crease in volume between the smallest and largest nodes.
The smallest node is either the volume of a monomer
(choice 6= 1) or the volume a particle with diameter d0
(choice = 1). Note that the 12 order of magnitude span
in volume is hardwired in the code; to change it, mod-
ify parameter vspan. Note that this subroutine creates
an additional node larger than the largest particle size;
however, the number concentration for the extra node
remains zero throughout the calculation. This node is
created to prevent index errors in some of the for loops
in the code.
collisionFrequency Determines the frequency of collisions between particles
in each pair of nodes. The results are used to calcu-
late the rates of coagulation and surface growth for each
node. Users have two options for the collision frequency
function: the free molecule collision frequency function
based on kinetic theory (beta option = 1) or Fuchs’
form of the collision function for particles in the transi-
tion regime (beta option = 2).
362
sizeSplitting Populates the size-splitting array X that handles parti-
cles that coagulate to a size between two nodes.
dynamicTimestep Chooses a time step based on the behavior of the GDE
at each point in time. The overarching goal is to choose
the maximum time step that guarantees code stability.
The dynamic time-step algorithm selects the time step
as the minimum of five options: (1) 0.1% of the char-
acteristic coagulation time; (2) 50% of the maximum
time step that results in the number concentration in
any node decreasing to zero; (3) the time in which the
monomer concentration changes by 0.1%; (4) the time
in which the saturation ratio changes by 1%; (5) a user-
defined maximum time step, dtUser. The percentages
are hardwired parameters but can be modified by the
user if desired.
363
coagulation Calculates the coagulation rate for each node, based on
the current size distribution, the collision frequency for
each pair of nodes (see collisionFrequency above),
and the size-splitting array X.
surfaceGrowth Calculates the condensation and evaporation rates for
each node based on the current size distribution, satu-
ration monomer concentration for each node (account-
ing for the Kelvin effect), and the collision frequency
between monomers and particles.
printResults Writes results to the output array results2. See Sec-
tion F.2.3 for the structure of this array.
F.3.2 NGDE Main Program
After the above function definitions, the main body of the NGDE code begins.
This portion of the code uses if-else statements to branch into four segments, cor-
responding to the four calculation types as specified with the choice input variable.
Each branch has the same basic structure: there is an additional initalization step,
used to create the volume nodes, populate the size-splitting array X, and establish
the conditions (e.g. total particle volume Vtot, volume mean diameter dpav) at
time zero; next is a while loop, where the code marches forward in time until some
condition is met; and finally, the code stores the output in variables results1 and
results2.
364
The exit condition for each while loop depends on the calculation type. For
choice = 1, (pure coagulation), the calculation terminates at 1000 times the esti-
mated time to reach the self-preserving distribution, t SPD. The calculation starts
with an initially monodisperse aerosol. As the while loop marches the simulation
forward in time, the aerosol grows based on the conditions in the coagulation
subroutine.
For the other choices, the calculation ends when the system reaches Tf. Ini-
tially, the system is very slightly supersaturated (S = 1.001) at some user-specified
temperature (T0). Within the loop, the code calls the subroutines to determine
the time step and nucleation, coagulation, and surface growth rates, as appropriate
based on the calculation type. The code updates the size distribution and other
associated properties using the results of these subroutines.
F.4 Summary
This user manual has described the input, output, and general structure of the
NGDE code, as well as the input and output of the NGDEplot post-processing tool.
The user can control many of the parameters in the NGDE code using the ngdein
input structure; other parameters must be modified directly in the code, such as the
initial saturation ratio S, the exit conditions for the while loops (t < 1000 ∗ t SPD
for choice = 1 and T < Tf otherwise), and the span of the size distribution (12
orders of magnitude from the smallest to largest volume nodes, as specified by
vspan). To make more substantial changes to the code – such as the introduction
365
of new models for coagulation, nucleation, and growth, or changes to the constant-
cooldown conditions of the calculation – one would need to do so in the appropriate
subroutines or in the main body of the program.
366
Appendix G: MATLAB Codes Referenced in this Disseration
G.1 Code for Calculating the Velocity around a Sphere
Below I have included the source code for the MATLAB function used to calcu-
late the velocity around a sphere as a function of Knudsen number, bgk sphere par.
Note that the code takes advantage of MATLAB’s parallel computing features. For
serial execution (i.e. on one computer core), change the parfor loops to for loops.
Note that when calculating the density, velocity, and temperature perturba-
√
tions around and drag on the sphere as a function of r0 = π/[4(0.499)Kn], I have
used nodes=64 and upper bound=10. I have experimented with different values of
these parameters, but these values seem to work best based on my limited sample
size.
bgk sphere par relies on two other MATLAB functions that I have written:
gaussquad, which returns the Gaussian quadrature nodes and weights, and abram,
which returns the Tn(x) functions that appear when solving the BGK equation.
Both of these files consist of long tables of data: gaussquad(n) uses a switch, case
structure to return the n node points and weights (tj and Aj in bgk sphere par),
while abram(x,n) interpolates from tables to return Tn(x) for each value of x pro-
vided. It would not be helpful to include the file listings here because they would
367
span over 100 pages. Instead, I will provide the information necessary to recreate
these files.
The Gaussian quadrature nodes and weights are available from various sources,
including Abramowitz and Stegun [98]. For gaussquad(n), I included nodes and
weights for n=2 to n=64 nodes; each set of data is its own case in the switch, case
structure. gaussquad(n) simply returns the nodes and weights for the specified
case.
I created the tables in abram(x,n) by numerically integrating the function in
MATLAB, using the following commands:
r=[0,logspace(-2,2,99)]’;
for i=1:length(r)
for n=0:9
T(i,j)=integral(@(c) exp(-c.^2-r(i)./c).*(c.^n),0,Inf);
end
end
The file abram(x,n) consists of the values of r and T, then the following lines:
Tn=x*0;
for i=1:length(x)
Tn(i)=interp1(r,T(:,n+1),x(i),’pchip’,0);
end
The code returns T(n).
See Chapter 2 and Appendix A for technical details of the BGK equation and
its solution for a translating sphere.
368
G.1.1 Code Listing for bgk sphere par
% Solve for the density, velocity, and temperature around a sphere
% of non-dimensional radius r0, using a Gaussian quadrature with
% user-specified number of nodes between r0 and r0+upper_bound.
% This function uses the asymptotic solution of the Krook equation
% for large r from (Takata et al., 1993). Otherwise, the equations
% and overall solution strategy follow (Lea & Loyalka, 1982) and
% (Law & Loyalka, 1986). See my dissertation (Corson, 2018) for
% more details.
function [rj,eps,coefs,drag]=bgk_sphere_par(r0,nodes,...
upper_bound,outfile)
tic
if nargin < 4
outfile=[’r0=’,num2str(r0),’_results.mat’];
end
%options for integral2
Int2Opts.Method=’iterated’;
Int2Opts.AbsTol=1e-6;
Int2Opts.RelTol=1e-3;
%Coefficient in (Takata et al., 1993) trial functions
gam=1.270;
%For r between r0 (the surface of the sphere) and
%rp=r0+upper_bound, the BGK equation is solved using a Gaussian
%quadrature. The nodes and weights tj and Aj are obtained
%from a separate function.
rp=r0+upper_bound;
[tj,Aj]=gaussquad(nodes);
rj=0.5*(r0*(1-tj)+rp*(1+tj));
%Results for c1, c2, and c3 based on earlier calculations;
%these are used to calculate initial guesses for the coefficients
%in the trial functions.
guesses=[0.00888,-0.0261,-14.7502,-0.8594;
0.010,-0.0269,-14.7212,-0.8591;
0.025,-0.0379,-14.3333,-0.8545;
369
0.050,-0.0564,-13.6869,-0.8468;
0.075,-0.0748,-13.0404,-0.8392;
0.100,-0.0932,-12.3940,-0.8315;
0.250,-0.2147,-12.1216,-0.2794;
0.500,-0.3826,-5.5702,-0.2417;
0.750,-0.5148,-3.4101,-0.2094;
0.888,-0.5760,-2.7445,-0.1935;
1.000,-0.6205,-2.3438,-0.1818;
2.000,-0.8854,-0.8038,-0.1103;
3.000,-1.0253,-0.3295,-0.0740;
4.000,-1.1115,-0.1039,-0.0532;
5.000,-1.1706,0.0268,-0.0418;
6.000,-1.2137,0.1113,-0.0335;
7.000,-1.2468,0.1707,-0.0280;
8.000,-1.2734,0.2153,-0.0246;
9.000,-1.2949,0.2489,-0.0208;
10.00,-1.3129,0.2758,-0.0184;
88.80,-1.4829,0.4830,-6.418e3-4;
100.0,-1.4857,0.4858,-5.1061e-4;
1000.,-1.5, 0.5, 0];
% Initialize vectors and matrices
ar1=zeros(nodes,1); ar2=ar1; ar3=ar1;
Wg1=zeros(nodes,1); Wg2=Wg1; Wg3=Wg1; Wu1=Wg1; Wu2=Wg1; Wu3=Wg1;
S1=zeros(nodes,1); S2=S1; S3=S1;
G11=S1; G12=S1; G13=S1; G21=S1; G22=S1; G23=S1; G31=S1;
G32=S1; G33=S1;
K11=zeros(nodes); K12=K11; K13=K11; K21=K11; K22=K11;
K23=K11; K31=K11; K32=K11; K33=K11;
A=zeros(4*nodes);
% Vectors and matrices that account for temperature fluctuation
ar4=ar1; Wg4=Wg1; Wu4=Wu1; K14=K11; K24=K11; K34=K11;
K41=K11; K42=K11; K43=K11; K44=K11;
G14=G11; G24=G11; G34=G11; G41=G11; G42=G11; G43=G11;
G44=G11; S4=S1;
%Loop to calculate a(r), which in turn is used to calculate the
%constant g that appears in the source term
parfor i=1:nodes
%for i=1:nodes
ar1(i)=1/(2*r0^2*rj(i))*integral(@(t) (t.^4-2*rj(i)^2*t.^2 ...
+(rj(i)^4-r0^4)).*abram(t,2)./(t.^2),rj(i)-r0, ...
sqrt(rj(i)^2-r0^2));
ar2(i)=-1/(2*sqrt(2)*r0^2*rj(i)^2)*integral(@(t) (t.^6-t.^4 ...
*(rj(i)^2+r0^2)-t.^2*(rj(i)^2-r0^2)^2+(rj(i)^2-r0^2) ...
370
*(rj(i)^4-r0^4)).*abram(t,3)./(t.^3),rj(i)-r0, ...
sqrt(rj(i)^2-r0^2));
ar3(i)=1/(2*sqrt(2)*r0^2*rj(i)^2)*integral(@(t) (t.^6-t.^4 ...
*(3*rj(i)^2+r0^2)+t.^2*(rj(i)^2-r0^2)*(3*rj(i)^2+r0^2) ...
-(rj(i)^2-r0^2)^3).*abram(t,3)./(t.^3),rj(i)-r0, ...
sqrt(rj(i)^2-r0^2));
ar4(i)=sqrt(2/3)/(2*r0^2*rj(i))*integral(@(t) (t.^4 ...
-2*rj(i)^2*t.^2+(rj(i)^4-r0^4)).*(abram(t,4) ...
-3/2*abram(t,2))./t.^2,rj(i)-r0,sqrt(rj(i)^2-r0^2));
end
% Loop to calculate the source term W(r)
parfor i=1:nodes
%for i=1:nodes
LL=rj(i)-r0;
UL=sqrt(rj(i)^2-r0^2);
tt=0.5*(LL*(1-tj)+UL*(1+tj));
C=(UL-LL)/2;
for j=1:nodes
Wg1(i)=Wg1(i)+1/(2*sqrt(pi)*rj(i)^2*r0)*(tt(j)^4 ...
-2*rj(i)^2*tt(j)^2+(rj(i)^4-r0^4)) ...
.*abram(tt(j),2)/tt(j)^2*Aj(j)*C;
Wg2(i)=Wg2(i)+1/(2*sqrt(2*pi)*rj(i)^3*r0)*(tt(j)^6 ...
-(rj(i)^2+r0^2)*tt(j)^4-(rj(i)^2-r0^2)^2*tt(j)^2 ...
+(rj(i)^2+r0^2)*(rj(i)^2-r0^2)^2).*abram(tt(j),3) ...
./tt(j)^3*Aj(j)*C;
Wg3(i)=Wg3(i)-1/(4*sqrt(2*pi)*rj(i)^3*r0)*(tt(j)^6 ...
-(3*rj(i)^2+r0^2)*tt(j)^4+(rj(i)^2-r0^2)*(3*rj(i)^2 ...
+r0^2)*tt(j)^2-(rj(i)^2-r0^2)^3).*abram(tt(j),3) ...
./tt(j)^3*Aj(j)*C;
Wu1(i)=Wu1(i)+1/(sqrt(pi)*rj(i)^2)*(tt(j)^4 ...
-(rj(i)^2-r0^2)^2).*abram(tt(j),3)./tt(j)^3*Aj(j)*C;
Wu2(i)=Wu2(i)+1/(sqrt(2*pi)*rj(i)^3)*(tt(j)^6+tt(j)^4 ...
*(rj(i)^2-r0^2)-tt(j)^2*(rj(i)^2-r0^2)^2 ...
-(rj(i)^2-r0^2)^3).*abram(tt(j),4)./tt(j)^4*Aj(j)*C;
Wu3(i)=Wu3(i)-1/(2*sqrt(2*pi)*rj(i)^3)*(tt(j)^6-tt(j)^4 ...
*(3*rj(i)^2+r0^2)+tt(j)^2*(rj(i)^2-r0^2)*(3*rj(i)^2 ...
+r0^2)-(rj(i)^2-r0^2)^3).*abram(tt(j),4) ...
./tt(j)^4*Aj(j)*C;
Wg4(i)=Wg4(i)+sqrt(2/3)/(2*sqrt(pi)*rj(i)^2*r0) ...
*(tt(j)^4-2*rj(i)^2*tt(j)^2+(rj(i)^4-r0^4)) ...
*(abram(tt(j),4)-3/2*abram(tt(j),2))/tt(j)^2*Aj(j)*C;
Wu4(i)=Wu4(i)+sqrt(2/3)/(sqrt(pi)*rj(i)^2)*(tt(j)^4 ...
-(rj(i)^2-r0^2)^2)*(abram(tt(j),5) ...
-3/2*abram(tt(j),3))/tt(j)^3*Aj(j)*C;
371
end
end
% Loop to evaluate the kernel K(rj(i),r)
parfor i=1:nodes
%for i=1:nodes
% These are the integrands of the H functions from Lea, 1982
H11=@(r,t) -1./(rj(i)*r).*abram(t,1).*(t.^2-(rj(i)^2+r.^2))./t;
H12=@(r,t) 1./(sqrt(2)*rj(i)*r.^2).*abram(t,2) ...
.*(t.^4-2*rj(i)^2*t.^2-(r.^4-rj(i)^4))./t.^2;
H13=@(r,t) -1./(sqrt(2)*rj(i)*r.^2).*abram(t,2) ...
.*(t.^4-2*(r.^2+rj(i)^2).*t.^2+(r.^2-rj(i)^2).^2)./t.^2;
H21=@(r,t) -1./(sqrt(2)*rj(i)^2*r).*abram(t,2) ...
.*(t.^4-2*r.^2.*t.^2-(rj(i)^4-r.^4))./t.^2;
H22=@(r,t) 1./(2*rj(i)^2*r.^2).*abram(t,3).*(t.^6-t.^4 ...
.*(r.^2+rj(i)^2)-t.^2.*(rj(i)^2-r.^2).^2 ...
+(r.^2-rj(i)^2).*(r.^4-rj(i)^4))./t.^3;
H23=@(r,t) -1./(2*rj(i)^2*r.^2).*abram(t,3).*(t.^6-t.^4 ...
.*(3*r.^2+rj(i)^2)+t.^2.*(r.^2-rj(i)^2) ...
.*(3*r.^2+rj(i)^2)-(r.^2-rj(i)^2).^3)./t.^3;
H31=@(r,t) 1./(sqrt(2)*2*rj(i)^2*r).*abram(t,2) ...
.*(t.^4-2*(rj(i)^2+r.^2).*t.^2+(rj(i)^2-r.^2).^2)./t.^2;
H32=@(r,t) -1./(4*rj(i)^2*r.^2).*abram(t,3).*(t.^6-t.^4 ...
.*(3*rj(i)^2+r.^2)+t.^2.*(rj(i)^2-r.^2) ...
.*(3*rj(i)^2+r.^2)-(rj(i)^2-r.^2).^3)./t.^3;
H33=@(r,t) 1./(4*rj(i)^2*r.^2).*abram(t,3).*(t.^6-t.^4 ...
.*(3*rj(i)^2+3*r.^2)+t.^2.*(3*rj(i)^4+2*rj(i)^2*r.^2 ...
+3*r.^4)-(rj(i)^4-r.^4).*(rj(i)^2-r.^2))./t.^3;
H14=@(r,t) -sqrt(2/3)./(rj(i)*r) ...
.*(abram(t,3)-3/2*abram(t,1)).*(t.^2-(rj(i)^2+r.^2))./t;
H24=@(r,t) -1./(sqrt(3)*rj(i)^2*r).*(abram(t,4) ...
-3/2*abram(t,2)).*(t.^4-2*r.^2.*t.^2-(rj(i)^4-r.^4))./t.^2;
H34=@(r,t) 1./(2*sqrt(3)*rj(i)^2*r) ...
.*(abram(t,4)-3/2*abram(t,2)).*(t.^4-2*(rj(i)^2+r.^2) ...
.*t.^2+(rj(i)^2-r.^2).^2)./t.^2;
H41=@(r,t) -sqrt(2/3)./(rj(i)*r).*(abram(t,3) ...
-3/2*abram(t,1)).*(t.^2-(rj(i)^2+r.^2))./t;
H42=@(r,t) 1./(sqrt(3)*rj(i)*r.^2) ...
.*(abram(t,4)-3/2*abram(t,2)) ...
.*(t.^4-2*rj(i)^2*t.^2-(r.^4-rj(i)^4))./t.^2;
H43=@(r,t) -1./(sqrt(3)*rj(i)*r.^2) ...
.*(abram(t,4)-3/2*abram(t,2)).*(t.^4-2*(r.^2+rj(i)^2) ...
.*t.^2+(r.^2-rj(i)^2).^2)./t.^2;
H44=@(r,t) -2./(3*rj(i)*r).*(abram(t,5)-3*abram(t,3) ...
+9/4*abram(t,1)).*(t.^2-(rj(i)^2+r.^2))./t;
372
% The G’s represent cases when the kernel is singular,
% i.e. when r=rj(i).
G11(i)=pi^(-1/2)*(integral2(@(r,t) r/rj(i).*(H11(r,t)), ...
r0,rj(i),@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts)+integral2(@(r,t) ...
r/rj(i).*(H11(r,t)),rj(i),rp,@(r) abs(rj(i)-r), ...
@(r) sqrt(rj(i)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts));
G12(i)=pi^(-1/2)*(integral2(@(r,t) r/rj(i).*(H12(r,t)), ...
r0,rj(i),@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts)+integral2(@(r,t) ...
r/rj(i).*(H12(r,t)),rj(i),rp,@(r) abs(rj(i)-r), ...
@(r) sqrt(rj(i)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts));
G13(i)=pi^(-1/2)*(integral2(@(r,t) r/rj(i).*(H13(r,t)), ...
r0,rj(i),@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts)+integral2(@(r,t) ...
r/rj(i).*(H13(r,t)),rj(i),rp,@(r) abs(rj(i)-r), ...
@(r) sqrt(rj(i)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts));
G21(i)=pi^(-1/2)*(integral2(@(r,t) r/rj(i).*(H21(r,t)), ...
r0,rj(i),@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts)+integral2(@(r,t) ...
r/rj(i).*(H21(r,t)),rj(i),rp,@(r) abs(rj(i)-r), ...
@(r) sqrt(rj(i)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts));
G22(i)=pi^(-1/2)*(integral2(@(r,t) r/rj(i).*(H22(r,t)), ...
r0,rj(i),@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts)+integral2(@(r,t) ...
r/rj(i).*(H22(r,t)),rj(i),rp,@(r) abs(rj(i)-r), ...
@(r) sqrt(rj(i)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts));
G23(i)=pi^(-1/2)*(integral2(@(r,t) r/rj(i).*(H23(r,t)), ...
r0,rj(i),@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts)+integral2(@(r,t) ...
r/rj(i).*(H23(r,t)),rj(i),rp,@(r) abs(rj(i)-r), ...
@(r) sqrt(rj(i)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts));
G31(i)=pi^(-1/2)*(integral2(@(r,t) r/rj(i).*(H31(r,t)), ...
r0,rj(i),@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts)+integral2(@(r,t) ...
r/rj(i).*(H31(r,t)),rj(i),rp,@(r) abs(rj(i)-r),...
@(r) sqrt(rj(i)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts));
G32(i)=pi^(-1/2)*(integral2(@(r,t) r/rj(i).*(H32(r,t)), ...
r0,rj(i),@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts)+integral2(@(r,t) ...
r/rj(i).*(H32(r,t)),rj(i),rp,@(r) abs(rj(i)-r), ...
@(r) sqrt(rj(i)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts));
G33(i)=pi^(-1/2)*(integral2(@(r,t) r/rj(i).*(H33(r,t)), ...
r0,rj(i),@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts)+integral2(@(r,t) ...
373
r/rj(i).*(H33(r,t)),rj(i),rp,@(r) abs(rj(i)-r), ...
@(r) sqrt(rj(i)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts));
G14(i)=pi^(-1/2)*(integral2(@(r,t) r/rj(i).*(H14(r,t)), ...
r0,rj(i),@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts)+integral2(@(r,t) ...
r/rj(i).*(H14(r,t)),rj(i),rp,@(r) abs(rj(i)-r), ...
@(r) sqrt(rj(i)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts));
G24(i)=pi^(-1/2)*(integral2(@(r,t) r/rj(i).*(H24(r,t)), ...
r0,rj(i),@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts)+integral2(@(r,t) ...
r/rj(i).*(H24(r,t)),rj(i),rp,@(r) abs(rj(i)-r), ...
@(r) sqrt(rj(i)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts));
G34(i)=pi^(-1/2)*(integral2(@(r,t) r/rj(i).*(H34(r,t)), ...
r0,rj(i),@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts)+integral2(@(r,t) ...
r/rj(i).*(H34(r,t)),rj(i),rp,@(r) abs(rj(i)-r), ...
@(r) sqrt(rj(i)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts));
G41(i)=pi^(-1/2)*(integral2(@(r,t) r/rj(i).*(H41(r,t)), ...
r0,rj(i),@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts)+integral2(@(r,t) ...
r/rj(i).*(H41(r,t)),rj(i),rp,@(r) abs(rj(i)-r), ...
@(r) sqrt(rj(i)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts));
G42(i)=pi^(-1/2)*(integral2(@(r,t) r/rj(i).*(H42(r,t)), ...
r0,rj(i),@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts)+integral2(@(r,t) ...
r/rj(i).*(H42(r,t)),rj(i),rp,@(r) abs(rj(i)-r), ...
@(r) sqrt(rj(i)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts));
G43(i)=pi^(-1/2)*(integral2(@(r,t) r/rj(i).*(H43(r,t)), ...
r0,rj(i),@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts)+integral2(@(r,t) ...
r/rj(i).*(H43(r,t)),rj(i),rp,@(r) abs(rj(i)-r), ...
@(r) sqrt(rj(i)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts));
G44(i)=pi^(-1/2)*(integral2(@(r,t) r/rj(i).*(H44(r,t)), ...
r0,rj(i),@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts)+integral2(@(r,t) ...
r/rj(i).*(H44(r,t)),rj(i),rp,@(r) abs(rj(i)-r), ...
@(r) sqrt(rj(i)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts));
for j=1:nodes
LL=abs(rj(i)-rj(j));
UL=sqrt(rj(i)^2-r0^2)+sqrt(rj(j)^2-r0^2);
tt=0.5*(LL*(1-tj)+UL*(1+tj));
C=(UL-LL)/2;
K11(i,j)=pi^(-1/2)*sum(rj(j)/rj(i)*H11(rj(j),tt).*Aj*C);
K12(i,j)=pi^(-1/2)*sum(rj(j)/rj(i)*H12(rj(j),tt).*Aj*C);
K13(i,j)=pi^(-1/2)*sum(rj(j)/rj(i)*H13(rj(j),tt).*Aj*C);
374
K21(i,j)=pi^(-1/2)*sum(rj(j)/rj(i)*H21(rj(j),tt).*Aj*C);
K22(i,j)=pi^(-1/2)*sum(rj(j)/rj(i)*H22(rj(j),tt).*Aj*C);
K23(i,j)=pi^(-1/2)*sum(rj(j)/rj(i)*H23(rj(j),tt).*Aj*C);
K31(i,j)=pi^(-1/2)*sum(rj(j)/rj(i)*H31(rj(j),tt).*Aj*C);
K32(i,j)=pi^(-1/2)*sum(rj(j)/rj(i)*H32(rj(j),tt).*Aj*C);
K33(i,j)=pi^(-1/2)*sum(rj(j)/rj(i)*H33(rj(j),tt).*Aj*C);
K14(i,j)=pi^(-1/2)*sum(rj(j)/rj(i)*H14(rj(j),tt).*Aj*C);
K24(i,j)=pi^(-1/2)*sum(rj(j)/rj(i)*H24(rj(j),tt).*Aj*C);
K34(i,j)=pi^(-1/2)*sum(rj(j)/rj(i)*H34(rj(j),tt).*Aj*C);
K41(i,j)=pi^(-1/2)*sum(rj(j)/rj(i)*H41(rj(j),tt).*Aj*C);
K42(i,j)=pi^(-1/2)*sum(rj(j)/rj(i)*H42(rj(j),tt).*Aj*C);
K43(i,j)=pi^(-1/2)*sum(rj(j)/rj(i)*H43(rj(j),tt).*Aj*C);
K44(i,j)=pi^(-1/2)*sum(rj(j)/rj(i)*H44(rj(j),tt).*Aj*C);
end
end
K={K11,K12,K13,K14;
K21,K22,K23,K24;
K31,K32,K33,K34;
K41,K42,K43,K44};
G={G11,G12,G13,G14;
G21,G22,G23,G24;
G31,G32,G33,G34;
G41,G42,G43,G44};
C=(rp-r0)/2;
% Set up coefficient matrix A(i,j)
for p=1:4
for q=1:4
for m=1:nodes
for n=1:nodes
i=(p-1)*nodes+m;
j=(q-1)*nodes+n;
if m==n
if p==q
A(i,j)=1;
end
A(i,j)=A(i,j)-G{p,q}(m);
for l=1:m-1
A(i,j)=A(i,j)+C*Aj(l)*K{p,q}(m,l);
end
for l=m+1:nodes
A(i,j)=A(i,j)+C*Aj(l)*K{p,q}(m,l);
end
375
else
A(i,j)=-C*Aj(n)*K{p,q}(m,n);
end
end
end
end
end
% Initial guesses for c1, c2, and c3
guess=interp1(guesses(:,1),guesses(:,2:4),r0,’linear’,’extrap’)’;
%Solve for the perturbations to the density, velocity, and
%temperature around the sphere, based on the values of the
%coefficients to the trial functions ("guess") for the solution
%far from the sphere. The function returns the difference
%between the calculated values of the perturbations from the
%gaussian quadrature solution and the values from the trial
%functions at the boundary between the inner and outer domains
%(r = r0 + upper_bound).
J=0;
function difference=findQ(guess)
J=J+1;
J %Print out the iteration number and the calculation time
toc
s1=zeros(nodes,1);s2=s1;s3=s1;s4=s1;
%Trial functions based on Takata (1993)
q1t=@(r) (gam*guess(1)/r0-guess(3))./(r/r0).^2;
q2t=@(r) sqrt(2)*guess(1)./(r/r0) ...
+sqrt(2)*guess(2)./(r/r0).^3;
q3t=@(r) 1/sqrt(2)*guess(1)./(r/r0) ...
-1/sqrt(2)*guess(2)./(r/r0).^3;
q4t=@(r) sqrt(3/2)*guess(3)./(r/r0).^2;
%Integrals involving q from r0+upper_bound to infinity
far1=integral2(@(r,t) r.*q1t(r)./(2*r0^2.*r) ...
.*(t.^4-2*r.^2.*t.^2+(r.^4-r0^4)).*abram(t,2) ...
./(t.^2),rp,Inf,@(r) r-r0,@(r) sqrt(r.^2-r0^2), ...
Int2Opts);
far2=-integral2(@(r,t) r.*q2t(r)./(2*sqrt(2)*r0^2*r.^2) ...
.*(t.^6-t.^4.*(r.^2+r0^2)-t.^2.*(r.^2-r0^2).^2 ...
+(r.^2-r0^2).*(r.^4-r0^4)).*abram(t,3)./(t.^3), ...
rp,Inf,@(r) r-r0,@(r) sqrt(r.^2-r0^2),Int2Opts);
far3=integral2(@(r,t) r.*q3t(r)./(2*sqrt(2)*r0^2*r.^2) ...
.*(t.^6-t.^4.*(3*r.^2+r0^2)+t.^2.*(r.^2-r0^2) ...
.*(3*r.^2+r0^2)-(r.^2-r0^2).^3).*abram(t,3)./(t.^3),...
rp,Inf,@(r) r-r0,@(r) sqrt(r.^2-r0^2),Int2Opts);
376
far4=integral2(@(r,t) r.*q4t(r)*sqrt(2/3)./(2*r0^2.*r) ...
.*(t.^4-2*r.^2.*t.^2+(r.^4-r0^4)).*(abram(t,4) ...
-1.5*abram(t,2))./(t.^2),rp,Inf,@(r) r-r0, ...
@(r) sqrt(r.^2-r0^2),Int2Opts);
src=-sqrt(pi)+2/r0*(sum(rj.*(q1t(rj).*ar1+q2t(rj).*ar2 ...
+q3t(rj).*ar3+q4t(rj).*ar4).*Aj*(rp-r0)/2) ...
+far1+far2+far3+far4);
W1=Wu1+src*Wg1;
W2=Wu2+src*Wg2;
W3=Wu3+src*Wg3;
W4=Wu4+src*Wg4;
parfor I=1:nodes
%for I=1:nodes
H11=@(r,t) -1./(rj(I)*r).*abram(t,1) ...
.*(t.^2-(rj(I)^2+r.^2))./t;
H12=@(r,t) 1./(sqrt(2)*rj(I)*r.^2).*abram(t,2) ...
.*(t.^4-2*rj(I)^2*t.^2-(r.^4-rj(I)^4))./t.^2;
H13=@(r,t) -1./(sqrt(2)*rj(I)*r.^2).*abram(t,2) ...
.*(t.^4-2*(r.^2+rj(I)^2).*t.^2 ...
+(r.^2-rj(I)^2).^2)./t.^2;
H21=@(r,t) -1./(sqrt(2)*rj(I)^2*r).*abram(t,2) ...
.*(t.^4-2*r.^2.*t.^2-(rj(I)^4-r.^4))./t.^2;
H22=@(r,t) 1./(2*rj(I)^2*r.^2).*abram(t,3).*(t.^6 ...
-t.^4.*(r.^2+rj(I)^2)-t.^2.*(rj(I)^2-r.^2).^2 ...
+(r.^2-rj(I)^2).*(r.^4-rj(I)^4))./t.^3;
H23=@(r,t) -1./(2*rj(I)^2*r.^2).*abram(t,3).*(t.^6 ...
-t.^4.*(3*r.^2+rj(I)^2)+t.^2.*(r.^2-rj(I)^2) ...
.*(3*r.^2+rj(I)^2)-(r.^2-rj(I)^2).^3)./t.^3;
H31=@(r,t) 1./(sqrt(2)*2*rj(I)^2*r).*abram(t,2) ...
.*(t.^4-2*(rj(I)^2+r.^2).*t.^2 ...
+(rj(I)^2-r.^2).^2)./t.^2;
H32=@(r,t) -1./(4*rj(I)^2*r.^2).*abram(t,3).*(t.^6 ...
-t.^4.*(3*rj(I)^2+r.^2)+t.^2.*(rj(I)^2-r.^2) ...
.*(3*rj(I)^2+r.^2)-(rj(I)^2-r.^2).^3)./t.^3;
H33=@(r,t) 1./(4*rj(I)^2*r.^2).*abram(t,3).*(t.^6 ...
-t.^4.*(3*rj(I)^2+3*r.^2)+t.^2.*(3*rj(I)^4 ...
+2*rj(I)^2*r.^2+3*r.^4)-(rj(I)^4-r.^4) ...
.*(rj(I)^2-r.^2))./t.^3;
H14=@(r,t) -sqrt(2/3)./(rj(I)*r).*(abram(t,3) ...
-3/2*abram(t,1)).*(t.^2-(rj(I)^2+r.^2))./t;
H24=@(r,t) -1./(sqrt(3)*rj(I)^2*r).*(abram(t,4) ...
-3/2*abram(t,2)).*(t.^4-2*r.^2.*t.^2 ...
-(rj(I)^4-r.^4))./t.^2;
H34=@(r,t) 1./(2*sqrt(3)*rj(I)^2*r).*(abram(t,4) ...
-3/2*abram(t,2)).*(t.^4-2*(rj(I)^2+r.^2) ...
377
.*t.^2+(rj(I)^2-r.^2).^2)./t.^2;
H41=@(r,t) -sqrt(2/3)./(rj(I)*r).*(abram(t,3) ...
-3/2*abram(t,1)).*(t.^2-(rj(I)^2+r.^2))./t;
H42=@(r,t) 1./(sqrt(3)*rj(I)*r.^2).*(abram(t,4) ...
-3/2*abram(t,2)).*(t.^4-2*rj(I)^2*t.^2 ...
-(r.^4-rj(I)^4))./t.^2;
H43=@(r,t) -1./(sqrt(3)*rj(I)*r.^2).*(abram(t,4) ...
-3/2*abram(t,2)).*(t.^4-2*(r.^2+rj(I)^2).*t.^2 ...
+(r.^2-rj(I)^2).^2)./t.^2;
H44=@(r,t) -2./(3*rj(I)*r).*(abram(t,5)-3*abram(t,3)...
+9/4*abram(t,1)).*(t.^2-(rj(I)^2+r.^2))./t;
s1(I)=pi^(-1/2)*integral2(@(r,t) r/rj(I).*(q1t(r) ...
.*H11(r,t)+q2t(r).*H12(r,t)+q3t(r).*H13(r,t) ...
+q4t(r).*H14(r,t)),rp,Inf,@(r) abs(rj(I)-r), ...
@(r) sqrt(rj(I)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts);
s2(I)=pi^(-1/2)*integral2(@(r,t) r/rj(I).*(q1t(r) ...
.*H21(r,t)+q2t(r).*H22(r,t)+q3t(r).*H23(r,t) ...
+q4t(r).*H24(r,t)),rp,Inf,@(r) abs(rj(I)-r), ...
@(r) sqrt(rj(I)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts);
s3(I)=pi^(-1/2)*integral2(@(r,t) r/rj(I).*(q1t(r) ...
.*H31(r,t)+q2t(r).*H32(r,t)+q3t(r).*H33(r,t) ...
+q4t(r).*H34(r,t)),rp,Inf,@(r) abs(rj(I)-r), ...
@(r) sqrt(rj(I)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts);
s4(I)=pi^(-1/2)*integral2(@(r,t) r/rj(I).*(q1t(r) ...
.*H41(r,t)+q2t(r).*H42(r,t)+q3t(r).*H43(r,t) ...
+q4t(r).*H44(r,t)),rp,Inf,@(r) abs(rj(I)-r), ...
@(r) sqrt(rj(I)^2-r0^2)+sqrt(r.^2-r0^2),Int2Opts);
end
S=[W1+s1;W2+s2;W3+s3;W4+s4];
Q=A\S;
oldsrc=0;
while abs((oldsrc-src)/src)>1e-6
oldsrc=src;
src=-sqrt(pi)+2/r0*(sum(rj.*(Q(1:nodes).*ar1 ...
+Q(nodes+1:2*nodes).*ar2+Q(2*nodes+1:3*nodes) ...
.*ar3+Q(3*nodes+1:4*nodes).*ar4).*Aj*(rp-r0)/2) ...
+far1+far2+far3+far4);
S=[Wu1+src*Wg1+s1;
Wu2+src*Wg2+s2;
Wu3+src*Wg3+s3;
Wu4+src*Wg4+s4];
Q=A\S;
end
difference=[Q(nodes)-q1t(rj(end));
Q(2*nodes)-q2t(rj(end));
378
Q(3*nodes)-q3t(rj(end));
Q(4*nodes)-q4t(rj(end))];
end
%Find coefficients in the trial functions for the perturbations
%in the density, velocity, and temperature far from the sphere
coefs=fsolve(@(x) findQ(x),guess);
%Based on the coefficients calculated above, find the drag on the
%sphere.
q1t=@(r) (gam*coefs(1)/r0-coefs(3))./(r/r0).^2;
q2t=@(r) sqrt(2)*coefs(1)./(r/r0)+sqrt(2)*coefs(2)./(r/r0).^3;
q3t=@(r) 1/sqrt(2)*coefs(1)./(r/r0)-1/sqrt(2)*coefs(2)./(r/r0).^3;
q4t=@(r) sqrt(3/2)*coefs(3)./(r/r0).^2;
far1=integral2(@(r,t) r.*q1t(r)./(2*r0^2.*r).*(t.^4-2*r.^2 ...
.*t.^2+(r.^4-r0^4)).*abram(t,2)./(t.^2),rp,Inf, ...
@(r) r-r0,@(r) sqrt(r.^2-r0^2),Int2Opts);
far2=-integral2(@(r,t) r.*q2t(r)./(2*sqrt(2)*r0^2*r.^2) ...
.*(t.^6-t.^4.*(r.^2+r0^2)-t.^2.*(r.^2-r0^2).^2+(r.^2-r0^2) ...
.*(r.^4-r0^4)).*abram(t,3)./(t.^3),rp,Inf,@(r) r-r0, ...
@(r) sqrt(r.^2-r0^2),Int2Opts);
far3=integral2(@(r,t) r.*q3t(r)./(2*sqrt(2)*r0^2*r.^2) ...
.*(t.^6-t.^4.*(3*r.^2+r0^2)+t.^2.*(r.^2-r0^2) ...
.*(3*r.^2+r0^2)-(r.^2-r0^2).^3).*abram(t,3)./(t.^3), ...
rp,Inf,@(r) r-r0,@(r) sqrt(r.^2-r0^2),Int2Opts);
far4=integral2(@(r,t) r.*q4t(r)*sqrt(2/3)./(2*r0^2.*r) ...
.*(t.^4-2*r.^2.*t.^2+(r.^4-r0^4)).*(abram(t,4) ...
-1.5*abram(t,2))./(t.^2),rp,Inf,@(r) r-r0, ...
@(r) sqrt(r.^2-r0^2),Int2Opts);
g=-sqrt(pi)+2/r0*(sum(rj.*(q1t(rj).*ar1+q2t(rj).*ar2 ...
+q3t(rj).*ar3+q4t(rj).*ar4).*Aj*(rp-r0)/2)...
+far1+far2+far3+far4);
W1=Wu1+g*Wg1;
W2=Wu2+g*Wg2;
W3=Wu3+g*Wg3;
W4=Wu4+g*Wg4;
parfor i=1:nodes
%for i=1:nodes
H11=@(r,t) -1./(rj(i)*r).*abram(t,1) ...
.*(t.^2-(rj(i)^2+r.^2))./t;
H12=@(r,t) 1./(sqrt(2)*rj(i)*r.^2).*abram(t,2) ...
.*(t.^4-2*rj(i)^2*t.^2-(r.^4-rj(i)^4))./t.^2;
H13=@(r,t) -1./(sqrt(2)*rj(i)*r.^2).*abram(t,2) ...
.*(t.^4-2*(r.^2+rj(i)^2).*t.^2+(r.^2-rj(i)^2).^2)./t.^2;
H21=@(r,t) -1./(sqrt(2)*rj(i)^2*r).*abram(t,2) ...
379
.*(t.^4-2*r.^2.*t.^2-(rj(i)^4-r.^4))./t.^2;
H22=@(r,t) 1./(2*rj(i)^2*r.^2).*abram(t,3) ...
.*(t.^6-t.^4.*(r.^2+rj(i)^2)-t.^2.*(rj(i)^2-r.^2).^2 ...
+(r.^2-rj(i)^2).*(r.^4-rj(i)^4))./t.^3;
H23=@(r,t) -1./(2*rj(i)^2*r.^2).*abram(t,3) ...
.*(t.^6-t.^4.*(3*r.^2+rj(i)^2)+t.^2.*(r.^2-rj(i)^2) ...
.*(3*r.^2+rj(i)^2)-(r.^2-rj(i)^2).^3)./t.^3;
H31=@(r,t) 1./(sqrt(2)*2*rj(i)^2*r).*abram(t,2) ...
.*(t.^4-2*(rj(i)^2+r.^2).*t.^2+(rj(i)^2-r.^2).^2)./t.^2;
H32=@(r,t) -1./(4*rj(i)^2*r.^2).*abram(t,3) ...
.*(t.^6-t.^4.*(3*rj(i)^2+r.^2)+t.^2.*(rj(i)^2-r.^2) ...
.*(3*rj(i)^2+r.^2)-(rj(i)^2-r.^2).^3)./t.^3;
H33=@(r,t) 1./(4*rj(i)^2*r.^2).*abram(t,3).*(t.^6-t.^4 ...
.*(3*rj(i)^2+3*r.^2)+t.^2.*(3*rj(i)^4+2*rj(i)^2*r.^2 ...
+3*r.^4)-(rj(i)^4-r.^4).*(rj(i)^2-r.^2))./t.^3;
H14=@(r,t) -sqrt(2/3)./(rj(i)*r).*(abram(t,3) ...
-3/2*abram(t,1)).*(t.^2-(rj(i)^2+r.^2))./t;
H24=@(r,t) -1./(sqrt(3)*rj(i)^2*r).*(abram(t,4) ...
-3/2*abram(t,2)).*(t.^4-2*r.^2.*t.^2-(rj(i)^4-r.^4))./t.^2;
H34=@(r,t) 1./(2*sqrt(3)*rj(i)^2*r).*(abram(t,4) ...
-3/2*abram(t,2)).*(t.^4-2*(rj(i)^2+r.^2).*t.^2 ...
+(rj(i)^2-r.^2).^2)./t.^2;
H41=@(r,t) -sqrt(2/3)./(rj(i)*r).*(abram(t,3) ...
-3/2*abram(t,1)).*(t.^2-(rj(i)^2+r.^2))./t;
H42=@(r,t) 1./(sqrt(3)*rj(i)*r.^2).*(abram(t,4) ...
-3/2*abram(t,2)).*(t.^4-2*rj(i)^2*t.^2 ...
-(r.^4-rj(i)^4))./t.^2;
H43=@(r,t) -1./(sqrt(3)*rj(i)*r.^2).*(abram(t,4) ...
-3/2*abram(t,2)).*(t.^4-2*(r.^2+rj(i)^2).*t.^2 ...
+(r.^2-rj(i)^2).^2)./t.^2;
H44=@(r,t) -2./(3*rj(i)*r).*(abram(t,5)-3*abram(t,3) ...
+9/4*abram(t,1)).*(t.^2-(rj(i)^2+r.^2))./t;
S1(i)=pi^(-1/2)*integral2(@(r,t) r/rj(i).*(q1t(r).*H11(r,t) ...
+q2t(r).*H12(r,t)+q3t(r).*H13(r,t)+q4t(r).*H14(r,t)), ...
rp,Inf,@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts);
S2(i)=pi^(-1/2)*integral2(@(r,t) r/rj(i).*(q1t(r).*H21(r,t) ...
+q2t(r).*H22(r,t)+q3t(r).*H23(r,t)+q4t(r).*H24(r,t)), ...
rp,Inf,@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts);
S3(i)=pi^(-1/2)*integral2(@(r,t) r/rj(i).*(q1t(r).*H31(r,t) ...
+q2t(r).*H32(r,t)+q3t(r).*H33(r,t)+q4t(r).*H34(r,t)), ...
rp,Inf,@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts);
S4(i)=pi^(-1/2)*integral2(@(r,t) r/rj(i).*(q1t(r).*H41(r,t) ...
380
+q2t(r).*H42(r,t)+q3t(r).*H43(r,t)+q4t(r).*H44(r,t)), ...
rp,Inf,@(r) abs(rj(i)-r),@(r) sqrt(rj(i)^2-r0^2) ...
+sqrt(r.^2-r0^2),Int2Opts);
end
oldg=0;
S=[Wu1+g*Wg1+S1;Wu2+g*Wg2+S2;Wu3+g*Wg3+S3;Wu4+g*Wg4+S4];
eps=A\S;
while (abs(oldg-g)/abs(g))>1e-6
oldg=g;
g=-sqrt(pi)+2/r0*(sum(rj.*(eps(1:nodes).*ar1 ...
+eps(nodes+1:2*nodes).*ar2+eps(2*nodes+1:3*nodes).*ar3 ...
+eps(3*nodes+1:4*nodes).*ar4).*Aj*(rp-r0)/2) ...
+far1+far2+far3+far4);
S=[Wu1+g*Wg1+S1;Wu2+g*Wg2+S2;Wu3+g*Wg3+S3;Wu4+g*Wg4+S4];
eps=A\S;
end
C=(rp-r0)/2;
fardrag1=2/r0^2*integral2(@(r,t) q1t(r).*(t.^4-(r.^2-r0^2).^2) ...
.*abram(t,3)./t.^3,rp,Inf,@(r) r-r0,@(r) sqrt(r.^2-r0^2), ...
Int2Opts);
fardrag2=sqrt(2)/r0^2*integral2(@(r,t) q2t(r)./r.*(t.^6+t.^4 ...
.*(r.^2-r0^2)-t.^2.*(r.^2-r0^2).^2-(r.^2-r0^2).^3) ...
.*abram(t,4)./t.^4,rp,Inf,@(r) r-r0,@(r) sqrt(r.^2-r0^2), ...
Int2Opts);
fardrag3=sqrt(2)/r0^2*integral2(@(r,t) -q3t(r)./r.*(t.^6-t.^4 ...
.*(3*r.^2+r0^2)+t.^2.*(r.^2-r0^2).*(3*r.^2+r0^2) ...
-(r.^2-r0^2).^3).*abram(t,4)./t.^4,rp,Inf,@(r) r-r0, ...
@(r) sqrt(r.^2-r0^2),Int2Opts);
fardrag4=2/r0^2*integral2(@(r,t) sqrt(2/3)*q4t(r).*(t.^4 ...
-(r.^2-r0^2).^2).*(abram(t,5)-3/2*abram(t,3))./t.^3,rp, ...
Inf,@(r) r-r0,@(r) sqrt(r.^2-r0^2),Int2Opts);
drag=(fardrag1-fardrag2-fardrag3+fardrag4+2*sqrt(pi)/r0^2*C*Aj’ ...
*(rj.^2.*(eps(1:nodes).*Wu1-eps(nodes+1:2*nodes).*Wu2 ...
-2*eps(2*nodes+1:3*nodes).*Wu3+eps(3*nodes+1:4*nodes).*Wu4))...
+8-g*sqrt(pi))/(8+pi);
%Save the results of the calculation
save(outfile,’eps’,’coefs’,’drag’,’rj’)
end
381
G.2 Codes for Calculating the Friction and Diffusion Tensors
Below I have included the source code for the MATLAB functions used to cal-
culate the friction and diffusion tensors for an aggregate ofN spheres of unit radius in
the continuum and transition regimes. continuum tensors uses Kirkwood-Riseman
theory with the Rotne-Prager-Yamakawa tensor for translational interactions but ig-
nores rotational and coupling interactions. One can also use the Stokes’ velocity ten-
sor in place of the RPY tensor; simply comment in the appropriate lines, as indicated
in the source code. continuum tensors 3rd uses Kirkwood-Riseman theory with
translational, rotational, and coupling interaction tensors accurate to O(r−3ij ); this
is referred to as the 3RD approach. bgk tensors uses extended Kirkwood-Riseman
theory to solve for the friction and diffusion tensors in the transition regime.
See Chapters 3-6 for technical details of these calculations.
G.2.1 Code Listing for continuum tensors
%Solve for the translational, rotational, and coupling friction
%and diffusion tensors of an aggregate in the continuum flow
%regime using the Kirkwood-Riseman method with the
%Rotne-Prager-Yamakawa tensor. The results are given in
%non-dimensional form. For dimensional results, multiply the
%translational, coupling, and rotational friction tensors by
%6*pi*mu*a, 6*pi*mu*a^2, and 6*pi*mu*a^3, %respectively, where a
%is the radius of the primary spheres and mu is the gas viscosity.
%
%Input is as follows:
% bodfile: Text file containing the x,y,z coordinates of the
% spheres in the aggregate, or an N-by-3 matrix, where N is
% the number of spheres in the aggregate. See a sample file
% for the required format if a text file is to be provided.
382
%For details of the KR method, see the following sources:
% Carrasco, B. & Garcia de la Torre, J.,
% Journal of Chemical Physics 111 (1999): 4817.
% Corson, J. et al., Physical Review E 95 (2017): 013103.
% Corson, J., PhD. Dissertation, University of Maryland (2017).
%
function [Xit,Xic,Xir,DOt,DOc,Dr,rOD,DDt] ...
= continuum_tensors(bodfile)
%Read in the x,y,z coordinates of the spheres in the aggregate
if ischar(bodfile)
f=fopen(bodfile);
coords=textscan(f,’%6c %10.4f %10.4f %10.4f %5d’);
coords=[coords{2},coords{3},coords{4}];
fclose(f);
else
coords=bodfile;
end
M=size(coords,1); %Number of spheres
a=1; %Set sphere radius to unity
%Populate lower triangular portion of interaction matrix; because
%T is symmetric, we get complete matrix from T=T+T’. The factor
%of 0.5 used to create the identity matrix ensures that the values
%on the diagonal of T are all 1.
T=eye(3*M)*0.5;
for i=1:M
for j=1:i-1
rij=coords(j,:)-coords(i,:);
Rij=rij’*rij;
rij=norm(rij);
%Use the Rotne-Prager-Yamakawa tensor
T(3*i-2:3*i,3*j-2:3*j)=3/4*a/rij*(eye(3)+Rij/rij^2)...
+a^3/(2*rij^3)*(eye(3)-3*Rij/rij^2);
%Use velocity field for Stokes flow around a sphere
%T(3*i-2:3*i,3*j-2:3*j)=3/4*a/rij*(eye(3)+Rij/rij^2)...
% +a^3/(4*rij^3)*(eye(3)-3*Rij/rij^2);
end
end
%Form the symmetrical matrix T using the lower-triangular matrix
%populated above, invert it, and determine the friction tensors
%from the inverted matrix.
T=T+T’;
383
S=inv(T);
clear T
Xit=zeros(3); Xir=zeros(3); Xic=zeros(3);
for i=1:M
for j=1:M
Sij=S(3*i-2:3*i,3*j-2:3*j);
Ai=[0 -coords(i,3) coords(i,2);
coords(i,3) 0 -coords(i,1);
-coords(i,2) coords(i,1) 0];
Aj=[0 -coords(j,3) coords(j,2);
coords(j,3) 0 -coords(j,1);
-coords(j,2) coords(j,1) 0];
Xit=Xit+Sij; % Friction tensor;
Xic=Xic+Ai*Sij; % Coupling tensor at the origin
Xir=Xir-Ai*Sij*Aj; % Rotational friction tensor at the origin
end
end
Xir=Xir+eye(3)*M*4/3;
% Calculate diffusion tensors and vector from the origin
% to the center of diffusion
DOt=inv(Xit-Xic’*(Xir\Xic));
DOc=-inv(Xir)*Xic*inv(Xit-Xic’*(Xir\Xic));
Dr=inv(Xir-Xic*(Xit\Xic’));
rOD=[Dr(2,2)+Dr(3,3), -Dr(1,2), -Dr(1,3);
-Dr(1,2), Dr(1,1)+Dr(3,3), -Dr(2,3);
-Dr(1,3), -Dr(2,3), Dr(1,1)+Dr(2,2)]...
\[DOc(2,3)-DOc(3,2);
DOc(3,1)-DOc(1,3);
DOc(1,2)-DOc(2,1)];
A=[0 -rOD(3) rOD(2);rOD(3) 0 -rOD(1);-rOD(2) rOD(1) 0];
DDt=DOt-A*Dr*A+DOc’*A-A*DOc;
end
G.2.2 Code Listing for continuum tensors 3rd
%Solve for the translational, rotational, and coupling friction
%and diffusion tensors of an aggregate in the continuum flow
%regime using the Kirkwood-Riseman method with the
%terms up to order r_{ij}^{-3} in the translational, rotational,
%and coupling hydrodynamic interaction tensors (i.e. the 3RD method
384
%mentioned by Carrasco & Garcia de la Torre, 1999). Results are
%given in non-dimensional form. For dimensional results, multiply
%the translational, coupling, and rotational friction tensors by
%6*pi*mu*a, 6*pi*mu*a^2, and 6*pi*mu*a^3, %respectively, where a
%is the radius of the primary spheres and mu is the gas viscosity.
%
%Input is as follows:
% bodfile: Text file containing the x,y,z coordinates of the
% spheres in the aggregate, or an N-by-3 matrix, where N is
% the number of spheres in the aggregate. See a sample file
% for the required format if a text file is to be provided.
%For details of the KR method, see the following sources:
% Carrasco, B. & Garcia de la Torre, J.,
% Journal of Chemical Physics 111 (1999): 4817.
% Corson, J. et al., Physical Review E 95 (2017): 013103.
% Corson, J., PhD. Dissertation, University of Maryland (2017).
%
function [Xit,Xic,Xir,DOt,DOc,Dr,rOD,DDt] ...
= continuum_tensors_3rd(bodfile)
%Read in the x,y,z coordinates of the spheres in the aggregate
if ischar(bodfile)
f=fopen(bodfile);
coords=textscan(f,’%6c %10.4f %10.4f %10.4f %5d’);
coords=[coords{2},coords{3},coords{4}];
fclose(f);
else
coords=bodfile;
end
M=size(coords,1); %Number of spheres
a=1; %Set sphere radius to unity
%Populate the hydrodynamic interaction matrix
T=zeros(6*M);
for i=1:M
% 3x3 matrices on the diagonal of T
T(3*i-2:3*i,3*i-2:3*i)=eye(3)/(6*a);
T(3*i-2+3*M:3*i+3*M,3*i-2+3*M:3*i+3*M)=eye(3)/(8*a^3);
for j=1:M
if i==j
continue
end
rij=coords(j,:)-coords(i,:);
385
Rij=rij’*rij;
epsrij=[0 rij(3) -rij(2);
-rij(3) 0 rij(1);
rij(2) -rij(1) 0]/norm(rij);
rij=norm(rij);
%Use the RPY tensor for translational interactions
T(3*i-2:3*i,3*j-2:3*j)=(3/4*a/rij*(eye(3)+Rij/rij^2)...
+a^3/(2*rij^3)*(eye(3)-3*Rij/rij^2))/(6*a);
%Rotation portion of the T, mu_rr
T(3*M+3*i-2:3*M+3*i,3*M+3*j-2:3*M+3*j)=(3*Rij/rij^2 ...
-eye(3))/(16*rij^3);
%Rotation-translation coupling, mu_rt
T(3*M+3*i-2:3*M+3*i,3*j-2:3*j)=-epsrij/(8*rij^2);
T(3*j-2:3*j,3*M+3*i-2:3*M+3*i)=epsrij/(8*rij^2);
end
end
%Invert the hydrodynamic interaction matrix T and determine
%the friction tensors from the inverted matrix.
S=inv(T);
clear T
Xit=zeros(3); Xir=zeros(3); Xic=zeros(3);
for i=1:M
for j=1:M
lam_tt=S(3*i-2:3*i,3*j-2:3*j);
lam_rt=S(3*i-2+3*M:3*i+3*M,3*j-2:3*j);
lam_tr=S(3*i-2:3*i,3*j-2+3*M:3*j+3*M);
lam_rr=S(3*i-2+3*M:3*i+3*M,3*j-2+3*M:3*j+3*M);
Ai=[0 -coords(i,3) coords(i,2);
coords(i,3) 0 -coords(i,1);
-coords(i,2) coords(i,1) 0];
Aj=[0 -coords(j,3) coords(j,2);
coords(j,3) 0 -coords(j,1);
-coords(j,2) coords(j,1) 0];
Xit=Xit+lam_tt; %Friction tensor
Xic=Xic+lam_rt+Ai*lam_tt; %Coupling tensor at the origin
Xir=Xir+lam_rr-lam_rt*Aj+Ai*lam_tr...
-Ai*lam_tt*Aj; %Rotational friction tensor at the origin
end
end
Xit=Xit/(6*a); Xic=Xic/(6*a); Xir=Xir/(6*a);
% Calculate diffusion tensors and vector from the origin
% to the center of diffusion
DOt=inv(Xit-Xic’*(Xir\Xic));
DOc=-inv(Xir)*Xic*inv(Xit-Xic’*(Xir\Xic));
Dr=inv(Xir-Xic*(Xit\Xic’));
386
rOD=[Dr(2,2)+Dr(3,3), -Dr(1,2), -Dr(1,3);
-Dr(1,2), Dr(1,1)+Dr(3,3), -Dr(2,3);
-Dr(1,3), -Dr(2,3), Dr(1,1)+Dr(2,2)]...
\[DOc(2,3)-DOc(3,2);DOc(3,1)-DOc(1,3);DOc(1,2)-DOc(2,1)];
A=[0 -rOD(3) rOD(2);rOD(3) 0 -rOD(1);-rOD(2) rOD(1) 0];
DDt=DOt-A*Dr*A+DOc’*A-A*DOc;
end
G.2.3 Code Listing for bgk tensors
%Solve for the translational, rotational, and coupling friction
%and diffusion tensors of an aggregate in the transition flow
%regime using the EKR method. The results are given in
%non-dimensional form. For dimensional results, multiply the
%translational, coupling, and rotational friction tensors by the
%following factors:
% Xit = Xit*zeta_{0,epstein}
% Xic = Xic*zeta_{0,epstein}*a
% Xir = Xir*zeta_{0,epstein}*a^2
%where
% zeta_{0,epstein}=pi*(8+pi)/2.994*mu/lambda*a^2
% a=r0*lambda*1.996/sqrt(pi) is the sphere radius
% lambda is the gas mean free path
% mu is the gas viscosity
%
%Input is as follows:
% bodfile: Text file containing the x,y,z coordinates of the
% spheres in the aggregate, or an N-by-3 matrix, where N is
% the number of spheres in the aggregate. See a sample file
% for the required format if a text file is to be provided.
% datafile: Matlab data file containing BGK results. This file
% should have four variables: rj, eps, coefs, drag. rj
% is a vector containing the radial nodes at which the BGK
% equation is solved. eps is a vector containing the BGK
% results for the density, radial velocity, tangential
% velocity, and temperature perturbations around the sphere.
% The first m elements of eps correspond to the density at
% nodes rj, the next m correspond to the radial velocity, etc.
% coefs is a vector of coefficients c_1, c_2, and c_3 that
% appear in the asymptotic solution of the BGK equation far
% from the sphere. drag is the drag on the sphere, normalized
387
% by zeta_{0,epstein}, as defined above.
% r0: Non-dimensional sphere radius, related to the sphere Knudsen
% number by r0=sqrt(pi)/(4*0.499*Kn). Note that the BGK
% results in datafile should correspond to r0.
%For details of the EKR method, see the following sources:
% Corson, J. et al., Physical Review E 95 (2017): 013103.
% Corson, J., PhD. Dissertation, University of Maryland (2017).
%
function [Xit,Xic,Xir,DOt,DOc,Dr,rOD,DDt] = bgk_tensors(bodfile,...
datafile,r0)
%Read in the x,y,z coordinates of the spheres in the aggregate
if ischar(bodfile)
f=fopen(bodfile);
coords=textscan(f,’%6c %10.4f %10.4f %10.4f %5d’);
coords=[coords{2},coords{3},coords{4}];
fclose(f);
else
coords=bodfile;
end
%Read BGK results from datafile
data=load(datafile);
radii=data.rj;
eps=data.eps;
coefs=data.coefs;
drag=data.drag;
%Results for the torque ratio T/T_fm from Loyalka (1992); results
%for R=0 to R=10 are from Table IV; results for R=25 to R=100 are
%from the slip formula, Eq. (42), with the slip coefficient equal
%to 0.9875.
Loyalka=[0,1;
0.1, 0.9901;
0.25,0.9803;
0.5, 0.9601;
0.75,0.9427;
1.0, 0.9206;
2.0, 0.8362;
3.0, 0.7514;
5.0, 0.6080;
7.0, 0.5044;
10.0,0.3996;
25.0,0.1902;
50.0,0.1004;
388
75.0,0.0682;
100.,0.0516];
M=size(coords,1); %Number of spheres
%Number of points at which the velocities are given
nodes=length(radii);
%Resize radii relative to the sphere radius
radii=radii/r0;
%Extract q2(r) and q3(r) from eps
q2=eps(nodes+1:2*nodes);
q3=eps(2*nodes+1:3*nodes);
%Define functions for the asymptotic (large r) solution to
%the BGK equation.
q2t=@(r) sqrt(2)*coefs(1)./(r)+sqrt(2)*coefs(2)./(r).^3;
q3t=@(r) 1/sqrt(2)*coefs(1)./(r)-1/sqrt(2)*coefs(2)./(r).^3;
%Populate lower triangular portion of interaction matrix; because
%T is symmetric, we get complete matrix from T=T+T’. The factor
%of 0.5 used to create the identity matrix ensures that the values
%on the diagonal of T are all 1.
T=eye(3*M)*0.5;
for i=1:M
for j=1:i-1
rij=coords(j,:)-coords(i,:);
Rij=rij’*rij;
rij=norm(rij);
%Interpolate from q2 and q3 rij < radii(end);
%otherwise, use the functions for the asymptotic
%solution for large r.
if rij < radii(end)
T(3*i-2:3*i,3*j-2:3*j)=-interp1(radii,q2,rij,...
’pchip’,’extrap’)/sqrt(2)*(Rij/rij^2)...
-interp1(radii,q3,rij,’pchip’,’extrap’)...
*(eye(3)-Rij/rij^2)/sqrt(2);
else
T(3*i-2:3*i,3*j-2:3*j)=-q2t(rij)/sqrt(2)...
*(Rij/rij^2)-q3t(rij)*(eye(3)-Rij/rij^2)...
/sqrt(2);
end
end
end
%Form the symmetrical matrix T using the lower-triangular matrix
%populated above, invert it, and determine the friction tensors
%from the inverted matrix.
T=T+T’;
389
S=drag*inv(T);
clear T
Xit=zeros(3); Xir=zeros(3); Xic=zeros(3);
for i=1:M
for j=1:M
Sij=S(3*i-2:3*i,3*j-2:3*j);
Ai=[0 -coords(i,3) coords(i,2);
coords(i,3) 0 -coords(i,1);
-coords(i,2) coords(i,1) 0];
Aj=[0 -coords(j,3) coords(j,2);
coords(j,3) 0 -coords(j,1);
-coords(j,2) coords(j,1) 0];
Xit=Xit+Sij; %Friction tensor;
Xic=Xic+Ai*Sij; %Coupling tensor at the origin
Xir=Xir-Ai*Sij*Aj; %Rotational friction tensor at the origin
end
end
Xir=Xir+eye(3)*M*4/(8+pi)...
*interp1(Loyalka(:,1),Loyalka(:,2),r0,’pchip’,’extrap’);
%Calculate diffusion tensors and vector from the origin to
%the center of diffusion
DOt=inv(Xit-Xic’*(Xir\Xic));
DOc=-inv(Xir)*Xic*inv(Xit-Xic’*(Xir\Xic));
Dr=inv(Xir-Xic*(Xit\Xic’));
rOD=[Dr(2,2)+Dr(3,3), -Dr(1,2), -Dr(1,3);
-Dr(1,2), Dr(1,1)+Dr(3,3), -Dr(2,3);
-Dr(1,3), -Dr(2,3), Dr(1,1)+Dr(2,2)]...
\[DOc(2,3)-DOc(3,2);DOc(3,1)-DOc(1,3);DOc(1,2)-DOc(2,1)];
A=[0 -rOD(3) rOD(2);rOD(3) 0 -rOD(1);-rOD(2) rOD(1) 0];
DDt=DOt-A*Dr*A+DOc’*A-A*DOc;
end
390
G.3 Codes for Calculating the Average Friction Coefficient of a Par-
ticle in an Electric Field
Below I have included the source code for the MATLAB functions used to
calculate the orientation-averaged translational friction coefficient for an aggre-
gate of N spheres of unit radius in an external electric field. The first function
(avg bgk velocity) uses the average-drift-velocity approach of Li et al. [105]; the
second function (avg bgk drag) uses Li et al.’s averaged-drag-force approach. The
user must provide the electric field strength; the polarizability tensor, or a Zeno out-
put file containing the tensor; and either the translational friction tensor normalized
by the free molecule friction coefficient of a primary sphere in the aggregate, or the
necessary information to perform the EKR calculation for the friction tensor. (See
the listing for bgk tensors above.)
See Chapter 7 for technical details of the orientation-averaged friction coeffi-
cient calculations.
G.3.1 Code Listing for avg bgk velocity
%This function calculates the scalar friction coefficient of an
%aggregate using the averaged-drift velocity approach of
%Li et al, 2014.
%
%Input parameters are as follows:
% ee: Electric field strength [W/cm**2].
% r0: Non-dimensional sphere radius, related to the sphere Knudsen
% number by r0=sqrt(pi)/(4*0.499*Kn). Note that the BGK
% results in datafile should correspond to r0.
391
% zenofile: File containing Zeno results (for the polarizability
% tensor), or a 3-by-3 matrix containing the polarizability
% tensor. If the polarizability tensor is provided, it
% must be provided in non-dimensional form, such that the
% dimensional polarizability tensor alpha is related to
% the non-dimensional zenofile by
% alpha=zenofile*a^3*epsilon_0, where epsilon_0 is the
% permittivity of free space and a is the primary sphere
% radius.
% bodfile: Text file containing the x,y,z coordinates of the
% spheres in the aggregate, or a 3-by-3 matrix containing
% the translational friction tensor. See a sample file
% for the required format if a text file is to be provided.
% If the translational friction tensor is provided, it must
% be in non-dimensional form, Xit=Xit/zeta_FM, where
% zeta_FM is the free molecule monomer friction coefficient
% from Epstein’s equation.
% datafile: Matlab data file containing BGK results. This file
% should have four variables: rj, eps, coefs, drag. rj
% is a vector containing the radial nodes at which the BGK
% equation is solved. eps is a vector containing the BGK
% results for the density, radial velocity, tangential
% velocity, and temperature perturbations around the sphere.
% The first m elements of eps correspond to the density at
% nodes rj, the next m correspond to the radial velocity, etc.
% coefs is a vector of coefficients c_1, c_2, and c_3 that
% appear in the asymptotic solution of the BGK equation far
% from the sphere. drag is the drag on the sphere, normalized
% by zeta_{0,epstein}, as defined above. If the
% translational friction tensor is provided (see entry for
% bodfile above), datafile is not used.
%
function [favg,F] = avg_bgk_velocity(ee,r0,zenofile,...
bodfile,datafile)
%Check to see whether or not the polarizability tensor
%has been provided
if size(zenofile) == [3,3]
alpha=zenofile;
else %If not, get polarizability tensor from specified file
f=fopen(zenofile);
C=textscan(f,’%s’);
if C{1,1}{61}(1) == ’P’
alpha(:,1)=[str2double(C{1,1}{63}(2:end-1));
str2double(C{1,1}{64}(1:end-1));
392
str2double(C{1,1}{65}(1:end-1))];
alpha(:,2)=[str2double(C{1,1}{66}(1:end-1));
str2double(C{1,1}{67}(1:end-1));
str2double(C{1,1}{68}(1:end-1))];
alpha(:,3)=[str2double(C{1,1}{69}(1:end-1));
str2double(C{1,1}{70}(1:end-1));
str2double(C{1,1}{71}(1:end-1))];
elseif C{1,1}{66}(1) == ’P’
alpha(:,1)=[str2double(C{1,1}{68}(2:end-1));
str2double(C{1,1}{69}(1:end-1));
str2double(C{1,1}{70}(1:end-1))];
alpha(:,2)=[str2double(C{1,1}{71}(1:end-1));
str2double(C{1,1}{72}(1:end-1));
str2double(C{1,1}{73}(1:end-1))];
alpha(:,3)=[str2double(C{1,1}{74}(1:end-1));
str2double(C{1,1}{75}(1:end-1));
str2double(C{1,1}{76}(1:end-1))];
else
alpha(:,1)=[str2double(C{1,1}{80}(2:end-1));
str2double(C{1,1}{81}(1:end-1));
str2double(C{1,1}{82}(1:end-1))];
alpha(:,2)=[str2double(C{1,1}{83}(1:end-1));
str2double(C{1,1}{84}(1:end-1));
str2double(C{1,1}{85}(1:end-1))];
alpha(:,3)=[str2double(C{1,1}{86}(1:end-1));
str2double(C{1,1}{87}(1:end-1));
str2double(C{1,1}{88}(1:end-1))];
end
fclose(f);
end
%Diagonalize the polarizability tensor
[V,alpha]=eig(alpha);
if alpha(3,3)==max(alpha*[1;1;1])
P=eye(3);
elseif alpha(2,2)==max(alpha*[1;1;1])
P=[1,0,0;0,0,-1;0,1,0];
else
P=[0,0,-1;0,1,0;1,0,0];
end
alpha=P’*alpha*P;
%Convert to appropriate units
lambda=67.3e-9; %MFP [nm]
mu=1.85e-5; %viscosity [kg/m-s]
a=4*0.499*r0*lambda/sqrt(pi); %Primary sphere radius [m]
alpha=alpha*a^3*8.854e-12; %Polarizability tensor [C-m^2/V]
393
E=[0;0;1]; %Electric field is in the positive z-direction
ee=ee*100; %Convert electric field strength to W/m
kT=298*1.38e-23; %Brownian energy [J]; assume T = 298 K
pmax=0.5*ee^2*alpha(3,3)/kT; %Max potential energy between
%the aggregate and the field; normalized by the
%Brownian energy
zetafm=pi*(8+pi)/(6*0.499)*mu/lambda*a^2; %Monomer friction
%coefficient in the FM regime; used to dimensionalize the
%results of this Matlab function [kg/s]
%Check to see whether or not the friction tensor has been provided
if size(bodfile) == [3,3]
F=P’*V’*bodfile*V*P;
%If not, perform BGK calculation and rotate to the body-fixed axes
%defined by the polarizability tensor
else
F=bgk_tensors(bodfile,datafile,r0);
F=P’*V’*F*V*P;
end
K=inv(F); %K is the mobility matrix
%(i.e. the inverse of the friction matrix)
%Define functions for the probability and the z-component of
%the drift velocity as a function of orientation
p=@(ps,th) exp(0.5*(sin(ps).^2.*sin(th).^2*alpha(1,1)...
+cos(ps).^2.*sin(th).^2*alpha(2,2)...
+(cos(th).^2-1)*alpha(3,3))*ee^2/kT);
vz=@(ps,th) K(3,3)*cos(th).^2+K(2,2)*cos(ps).^2.*sin(th).^2 ...
+K(1,1)*sin(th).^2.*sin(ps).^2+K(1,2)*sin(th).^2.*sin(2*ps)...
+K(1,3)*sin(2*th).*sin(ps)+K(2,3)*cos(ps).*sin(2*th);
%Calculate the average friction coefficient
Q=integral2(@(ps,th) p(ps,th).*sin(th),0,2*pi,0,pi); %
%Q == partition function, used to normalize probability p
vd=integral2(@(ps,th) p(ps,th).*vz(ps,th).*sin(th),0,2*pi,0,pi)/Q;
favg=1/vd*zetafm; %Orientation-averaged friction coef [kg/s]
end
G.3.2 Code Listing for avg bgk drag
%This function calculates the scalar friction coefficient of an
%aggregate using the averaged-drag force approach of
394
%Li et al, 2014.
%
%Input parameters are as follows:
% ee: Electric field strength [W/cm**2].
% r0: Non-dimensional sphere radius, related to the sphere Knudsen
% number by r0=sqrt(pi)/(4*0.499*Kn). Note that the BGK
% results in datafile should correspond to r0.
% zenofile: File containing Zeno results (for the polarizability
% tensor), or a 3-by-3 matrix containing the polarizability
% tensor. If the polarizability tensor is provided, it
% must be provided in non-dimensional form, such that the
% dimensional polarizability tensor alpha is related to
% the non-dimensional zenofile by
% alpha=zenofile*a^3*epsilon_0, where epsilon_0 is the
% permittivity of free space and a is the primary sphere
% radius.
% bodfile: Text file containing the x,y,z coordinates of the
% spheres in the aggregate, or a 3-by-3 matrix containing
% the translational friction tensor. See a sample file
% for the required format if a text file is to be provided.
% If the translational friction tensor is provided, it must
% be in non-dimensional form, Xit=Xit/zeta_FM, where
% zeta_FM is the free molecule monomer friction coefficient
% from Epstein’s equation.
% datafile: Matlab data file containing BGK results. This file
% should have four variables: rj, eps, coefs, drag. rj
% is a vector containing the radial nodes at which the BGK
% equation is solved. eps is a vector containing the BGK
% results for the density, radial velocity, tangential
% velocity, and temperature perturbations around the sphere.
% The first m elements of eps correspond to the density at
% nodes rj, the next m correspond to the radial velocity, etc.
% coefs is a vector of coefficients c_1, c_2, and c_3 that
% appear in the asymptotic solution of the BGK equation far
% from the sphere. drag is the drag on the sphere, normalized
% by zeta_{0,epstein}, as defined above. If the
% translational friction tensor is provided (see entry for
% bodfile above), datafile is not used.
%
function [favg,F] = avg_bgk_drag(ee,r0,zenofile,bodfile,datafile)
%Check to see whether or not the polarizability tensor
%has been provided
if size(zenofile) == [3,3]
alpha=zenofile;
395
else %If not, get polarizability tensor from specified file
f=fopen(zenofile);
C=textscan(f,’%s’);
if C{1,1}{61}(1) == ’P’
alpha(:,1)=[str2double(C{1,1}{63}(2:end-1));
str2double(C{1,1}{64}(1:end-1));
str2double(C{1,1}{65}(1:end-1))];
alpha(:,2)=[str2double(C{1,1}{66}(1:end-1));
str2double(C{1,1}{67}(1:end-1));
str2double(C{1,1}{68}(1:end-1))];
alpha(:,3)=[str2double(C{1,1}{69}(1:end-1));
str2double(C{1,1}{70}(1:end-1));
str2double(C{1,1}{71}(1:end-1))];
elseif C{1,1}{66}(1) == ’P’
alpha(:,1)=[str2double(C{1,1}{68}(2:end-1));
str2double(C{1,1}{69}(1:end-1));
str2double(C{1,1}{70}(1:end-1))];
alpha(:,2)=[str2double(C{1,1}{71}(1:end-1));
str2double(C{1,1}{72}(1:end-1));
str2double(C{1,1}{73}(1:end-1))];
alpha(:,3)=[str2double(C{1,1}{74}(1:end-1));
str2double(C{1,1}{75}(1:end-1));
str2double(C{1,1}{76}(1:end-1))];
else
alpha(:,1)=[str2double(C{1,1}{80}(2:end-1));
str2double(C{1,1}{81}(1:end-1));
str2double(C{1,1}{82}(1:end-1))];
alpha(:,2)=[str2double(C{1,1}{83}(1:end-1));
str2double(C{1,1}{84}(1:end-1));
str2double(C{1,1}{85}(1:end-1))];
alpha(:,3)=[str2double(C{1,1}{86}(1:end-1));
str2double(C{1,1}{87}(1:end-1));
str2double(C{1,1}{88}(1:end-1))];
end
fclose(f);
end
%Diagonalize the polarizability tensor
[V,alpha]=eig(alpha);
if alpha(3,3)==max(alpha*[1;1;1])
P=eye(3);
elseif alpha(2,2)==max(alpha*[1;1;1])
P=[1,0,0;0,0,-1;0,1,0];
else
P=[0,0,-1;0,1,0;1,0,0];
end
396
alpha=P’*alpha*P;
%Convert to appropriate units
lambda=67.3e-9; %MFP [nm]
mu=1.85e-5; %viscosity [kg/m-s]
a=4*0.499*r0*lambda/sqrt(pi); %Primary sphere radius [m]
alpha=alpha*a^3*8.854e-12; %Polarizability tensor [C-m^2/V]
E=[0;0;1]; %Electric field is in the positive z-direction
ee=ee*100; %Convert electric field strength to W/m
kT=298*1.38e-23; %Brownian energy [J]; assume T = 298 K
pmax=0.5*ee^2*alpha(3,3)/kT; %Max potential energy between
%the aggregate and the field; normalized by the
%Brownian energy
zetafm=pi*(8+pi)/(6*0.499)*mu/lambda*a^2; %Monomer friction
%coefficient in the FM regime; used to dimensionalize the
%results of this Matlab function [kg/s]
%Check to see whether or not the friction tensor has been provided
if size(bodfile) == [3,3]
F=P’*V’*bodfile*V*P;
%If not, perform BGK calculation and rotate to the body-fixed axes
%defined by the polarizability tensor
else
F=bgk_tensors(bodfile,datafile,r0);
F=P’*V’*F*V*P;
end
%Define functions for the probability and the z-component of
%the drag force as a function of orientation
p=@(ps,th) exp(0.5*(sin(ps).^2.*sin(th).^2*alpha(1,1)...
+cos(ps).^2.*sin(th).^2*alpha(2,2)...
+(cos(th).^2-1)*alpha(3,3))*ee^2/kT);
Fz=@(ps,th) F(3,3)*cos(th).^2+F(2,2)*cos(ps).^2.*sin(th).^2 ...
+F(1,1)*sin(th).^2.*sin(ps).^2+F(1,2)*sin(th).^2.*sin(2*ps)...
+F(1,3)*sin(2*th).*sin(ps)+F(2,3)*cos(ps).*sin(2*th);
%Calculate the average friction coefficient
Q=integral2(@(ps,th) p(ps,th).*sin(th),0,2*pi,0,pi); %
%Q == partition function, used to normalize probability p
Favg=integral2(@(ps,th) p(ps,th).*Fz(ps,th).*sin(th),0,2*pi,0,pi)/Q;
favg=Favg*zetafm; %Orientation-averaged friction coef [kg/s]
end
397
G.4 Codes for Hydrodynamic Interactions between Particles
Below I have included the source code for the MATLAB functions used to
determine the hydrodynamic interactions between particles (Chapter 8). The first
function (bgk two particles) determines the drag on each particle in a two particle
system for the specified coordinates of the spheres in each particle, the primary
sphere Knudsen number, the distance between the center of mass of each particle,
and the particle velocities. The second function (bgk cloud) calculates the velocity
of each particle in a spherical cloud based on the radius of the cloud and on the
particle volume fraction, friction coefficient, and Knudsen number.
See Chapter 8 for technical details of the hydrodynamic interaction calcula-
tions.
G.4.1 Code Listing for bgk two particles
%This function calculates the hydrodynamic force on each particle
%in a two-particle system. The interactions between primary spheres
%in the particles are determined using the extended
%Kirkwood-Riseman method (Corson et al., Phys. Rev. E. 95(1), 2017).
%Results are also given for the drag each particle would experience
%if it was isolated in an infinite fluid (i.e. when the separation
%distance between particles goes to infinity). The results are
%given in non-dimensional form. For dimensional
%results, multiply the forces following factor:
% F = F*zeta_{0,epstein}*U
%where
% zeta_{0,epstein}=pi*(8+pi)/2.994*mu/lambda*a^2
% a=r0*lambda*1.996/sqrt(pi) is the sphere radius
% lambda is the gas mean free path
% mu is the gas viscosity
% U is the unit used to specify particle velocities U1 and U2
398
%
%Input parameters are as follows:
% bodfile1: Text file containing the x,y,z coordinates of the
% spheres in aggregate 1, or an N-by-3 matrix, where N is
% the number of spheres in the aggregate. See a sample file
% for the required format if a text file is to be provided.
% bodfile2: Text file containing the x,y,z coordinates of the
% spheres in aggregate 2, or an N-by-3 matrix, where N is
% the number of spheres in the aggregate. See a sample file
% for the required format if a text file is to be provided.
% datafile: Matlab data file containing BGK results. This file
% should have four variables: rj, eps, coefs, drag. rj
% is a vector containing the radial nodes at which the BGK
% equation is solved. eps is a vector containing the BGK
% results for the density, radial velocity, tangential
% velocity, and temperature perturbations around the sphere.
% The first m elements of eps correspond to the density at
% nodes rj, the next m correspond to the radial velocity, etc.
% coefs is a vector of coefficients c_1, c_2, and c_3 that
% appear in the asymptotic solution of the BGK equation far
% from the sphere. drag is the drag on the sphere, normalized
% by zeta_{0,epstein}, as defined above.
% r0: Non-dimensional sphere radius, related to the sphere Knudsen
% number by r0=sqrt(pi)/(4*0.499*Kn). Note that the BGK
% results in datafile should correspond to r0.
% separation: Vector connecting the center of mass of particle 1
% to the center of mass of particle 2. The distance
% is in units of primary sphere radius.
% U1: Velocity of particle 1 in arbitrary units
% U2: Velocity of particle 2 in arbitrary units
%
function [F1,F2,F01,F02]=bgk_two_particles(bodfile1,bodfile2,...
datafile,r0,separation,U1,U2)
%Make sure separation is a row vector to prevent errors
separation=reshape(separation,[1 3]);
%Get coordinates for particles 1 and 2
if ischar(bodfile1)
f=fopen(bodfile1);
coords1=textscan(f,’%6c %10.4f %10.4f %10.4f %5d’);
coords1=[coords1{2},coords1{3},coords1{4}];
fclose(f);
else
coords1=bodfile1;
399
end
if ischar(bodfile2)
f=fopen(bodfile2);
coords2=textscan(f,’%6c %10.4f %10.4f %10.4f %5d’);
coords2=[coords2{2},coords2{3},coords2{4}];
fclose(f);
else
coords2=bodfile2;
end
%Add and subtract separation vectors from the coordinates
coords1=coords1+repmat(separation,length(coords1(:,1)),1)/2;
coords2=coords2-repmat(separation,length(coords2(:,1)),1)/2;
%Combine coordinates from particles 1 and 2 into one data structure
coords=[coords1;coords2];
data=load(datafile);
radii=data.rj;
eps=data.eps;
coefs=data.coefs;
drag=data.drag;
M=size(coords,1); %Total number of spheres in both particles
M1=size(coords1,1); %Number of spheres in particle 1
M2=size(coords2,1); %Number of spheres in particle 2
%Number of points at which the velocities are given
nodes=length(radii);
%Resize radii relative to the sphere radius
radii=radii/r0;
%Extract q2(r) and q3(r) from eps
q2=eps(nodes+1:2*nodes);
q3=eps(2*nodes+1:3*nodes);
q2t=@(r) sqrt(2)*coefs(1)./(r)+sqrt(2)*coefs(2)./(r).^3;
q3t=@(r) 1/sqrt(2)*coefs(1)./(r)-1/sqrt(2)*coefs(2)./(r).^3;
%Make matrix to store velocity of each sphere
U=zeros(3,M);
U(1,1:M1)=drag*U1(1);
U(2,1:M1)=drag*U1(2);
U(3,1:M1)=drag*U1(3);
U(1,M1+1:M)=drag*U2(1);
U(2,M1+1:M)=drag*U2(2);
U(3,M1+1:M)=drag*U2(3);
%Convert matrix to vectors to facilitate linear algebra solution
U=reshape(U,[3*M,1]);
400
%Populate lower triangular portion of interaction matrix;
%because T is symmetric, we get complete matrix from T=T+T’.
%The factor of 0.5 used to create the identity matrix ensures
%that the values on the diagonal of T are all 1.
T=eye(3*M)*0.5;
for i=1:M
for j=1:i-1
rij=coords(j,:)-coords(i,:);
Rij=rij’*rij;
rij=norm(rij);
%Use data from Lea and Loyalka, 1982, along with derived
%formula
if rij < radii(end)
T(3*i-2:3*i,3*j-2:3*j)=-interp1(radii,q2,rij,...
’pchip’,’extrap’)/sqrt(2)*(Rij/rij^2)...
-interp1(radii,q3,rij,’pchip’,’extrap’)...
*(eye(3)-Rij/rij^2)/sqrt(2);
else
T(3*i-2:3*i,3*j-2:3*j)=-q2t(rij)/sqrt(2)...
*(Rij/rij^2)-q3t(rij)*(eye(3)-Rij/rij^2)...
/sqrt(2);
end
end
end
%Form the symmetrical matrix T using the lower-triangular matrix
%we populated above
T=T+T’;
F=T\U;
F=reshape(F,[3,M])’;
%Determine the force on each particle
F1=sum(F(1:M1,:),1);
F2=sum(F(M1+1:M,:),1);
%Determine the force on each particle if it is isolated
S1=inv(T(1:3*M1,1:3*M1))*drag;
F01=zeros(3);
for i=1:M1
for j=1:M1
Sij=S1(3*i-2:3*i,3*j-2:3*j);
F01=F01+Sij; % Friction tensor for isolated particle 1
end
end
S2=inv(T(3*M1+1:3*M,3*M1+1:3*M))*drag;
F02=zeros(3);
for i=1:M2
for j=1:M2
401
Sij=S2(3*i-2:3*i,3*j-2:3*j);
F02=F02+Sij; % Friction tensor for isolated particle 2
end
end
end
G.4.2 Code Listing for bgk cloud
%This function calculates the velocity of each particle in
%a spherical cloud. The particles are arranged in a regular
%rectangular grid. Each particle is the same size and experiences
%the same external force (e.g. the force of gravity). The code
%uses a point force method for non-continuum particles.
%The results are given in non-dimensional form by dividing the
%velocity by the velocity each particle would have if it was
%alone in an infinite fluid experiencing the same external force.
%
%Input parameters are as follows:
% phi: Particle volume fraction in the cloud
% for the required format if a text file is to be provided.
% zeta: Friction coefficient for each particle, normalized by
% Epstein’s equation,
% zeta_{0,epstein}=pi*(8+pi)/2.994*mu/lambda*a^2
% N: Number of spheres in each particle; if N>1, each particle
% is an aggregate
% Kn: Primary sphere Knudsen number
% cloudrad: Radius of the circular cloud of particles
%
function [U,coords] = bgk_cloud(phi,zeta,N,Kn,cloudrad)
%Node spacing for aggregates with N spheres
L=(4*pi*N/(3*phi))^(1/3);
%Node points
nodes=0:L:L+cloudrad;
nodes=[-nodes(end:-1:2),nodes];
%Number of nodes. The cloud consists of a 3D, n-by-n-by-n grid of
%equally-spaced particles.
n=length(nodes);
%Set the grid coordinates
402
%coords=zeros(M,3);
M=0;
for i=1:n
for j=1:n
for k=1:n
if norm([nodes(i),nodes(j),nodes(k)])<=cloudrad
M=M+1;
coords(M,:)=[nodes(i),nodes(j),nodes(k)];
end
end
end
end
%Determine the effective sphere size and the coefficient c1
%in the hydrodynamic interaction term
Cc=@(Kn) 1+Kn.*(1.257+0.4*exp(-1.1./Kn));
aeff=fsolve(@(a) 1-(8+pi)*Cc(Kn./a)*zeta./(36*.499*Kn*a),1);
Kneff=Kn/aeff;
c1=get_c1(Kneff);
U=0*coords;
%Determine the velocities; the velocity of sphere i is
% u_i=[0;0;1]+sum_{i\neq j}T_{ij}*[0;0;1]
%where [0;0;1] is the normalized velocity the sphere
%would have if it was alone in an infinite fluid and T_{ij}
%is the velocity tensor around sphere j
for i=1:M
U(i,:)=[0,0,1];
for j=1:i-1
rij=coords(j,:)-coords(i,:);
Rij=rij’*rij;
rij=norm(rij);
T=-c1/2*aeff/rij*(eye(3)+Rij/rij^2)*[0;0;1];
U(i,:)=U(i,:)+T’;
end
for j=i+1:M
rij=coords(j,:)-coords(i,:);
Rij=rij’*rij;
rij=norm(rij);
T=-c1/2*aeff/rij*(eye(3)+Rij/rij^2)*[0;0;1];
U(i,:)=U(i,:)+T’;
end
end
403
end
function c1=get_c1(Kneff)
results=[0.00100000330103084,-1.50000000000000;
0.00888002931315389,-1.48570344760713;
0.0100000330103084,-1.48289594184936;
0.0177600586263078,-1.46549558211423;
0.0307267450282141,-1.43445528240899;
0.0319425514861651,-1.43196708006806;
0.0321740192505576,-1.43143453851274;
0.0439605411542272,-1.40531002377626;
0.0457733469750200,-1.40107279504826;
0.0462501526726765,-1.40004970203708;
0.0498878051300780,-1.39260388823354;
0.0710402345052311,-1.34737220217562;
0.0733886720095363,-1.34271424782340;
0.0746220950685201,-1.34015845200810;
0.0752544857046940,-1.33883481743332;
0.0888002931315389,-1.31287863666467;
0.0914524131117805,-1.30796980781124;
0.0934739927700409,-1.30416195054598;
0.0986669923683765,-1.29489226387913;
0.100000330103084,-1.29259732469631;
0.103256154804115,-1.28681575294805;
0.106347656444957,-1.28147952131604;
0.108293040404316,-1.27800956474164;
0.111000366414424,-1.27340641424302;
0.112833917575018,-1.27028453245384;
0.148000488552565,-1.21371512310943;
0.153368381919756,-1.20564257152255;
0.158571952020605,-1.19790999542810;
0.177600586263078,-1.17058108398838;
0.222000732828847,-1.11152705469468;
0.224810868687440,-1.10796421368653;
0.248740316895067,-1.07877454770160;
0.259649979916780,-1.06602375573901;
0.296000977105130,-1.02526711131977;
0.309408686869473,-1.01090376654656;
0.312677088491334,-1.00749077881340;
0.319425514861651,-1.00054637096953;
0.330112613871892,-0.989656069389966;
0.337643700119920,-0.982110085768983;
0.355201172526155,-0.964899381185821;
404
0.360976801347719,-0.959351241563024;
0.370001221381412,-0.950650149617769;
0.444001465657694,-0.885439122815643;
0.450762909297152,-0.879881745753469;
0.462501526726765,-0.870274716693120;
0.480001584494805,-0.856574960658491;
0.507430246465936,-0.835629618679835;
0.522354665479641,-0.824727150242144;
0.541465202021579,-0.811001567718485;
0.592001954210259,-0.776725108676454;
0.608221185832458,-0.766318500586142;
0.672729493420749,-0.727041807705684;
0.710402345052311,-0.706087358715459;
0.853848972418643,-0.635349895187173;
0.862138768267368,-0.631686617740032;
0.870591109132734,-0.627973793024235;
0.888002931315389,-0.620465383723801;
1.00000330103084,-0.575967854009263;
1.04348170542349,-0.560310261228538;
1.18400390842052,-0.514765064439723;
1.34545898684150,-0.470715703993602;
1.43226279244418,-0.449900766293788;
1.64141022424286,-0.406436008989048;
1.77600586263078,-0.382577551475560;
1.89339644203708,-0.363913509990032;
2.00000660206169,-0.348442640233600;
2.22000732828847,-0.320287048255757;
2.29458121786922,-0.311735817784993;
2.68278831213108,-0.273641427999317;
2.80127107670470,-0.263783862704728;
3.00000990309253,-0.248735495332713;
3.22910156841960,-0.231501041361636;
3.26471665924775,-0.231161177027969;
3.36364746710375,-0.224408494612492;
3.55201172526156,-0.214680751106139;
3.84416853383285,-0.200153427272282;
4.24881785318368,-0.182982441031353;
4.62501526726765,-0.169458890693224;
4.80001584494805,-0.163823640031794;
4.82610288758364,-0.163015430672425;
4.98878051300780,-0.158149023776157;
5.16280774020575,-0.153254881401797;
5.95975121688180,-0.134209619067310;
6.72729493420749,-0.117429350671894;
6.99214906547550,-0.115605358650687;
405
8.88002931315389,-0.0932081818613417;
10.0000330103084,-0.0849534017314433;
10.1139286026810,-0.0842245513328470;
11.2833917575018,-0.0775245651729286;
11.8400390842052,-0.0747839505227221;
13.4545898684150,-0.0681752177326948;
13.5988197751208,-0.0676599360292581;
16.8501504993432,-0.0583854353709664;
17.7600586263078,-0.0563648468965600;
20.0000660206169,-0.0522394912334894;
26.9909705566987,-0.0437872519905655;
34.0231008166816,-0.0387726176692985;
35.5201172526156,-0.0379489334389416;
88.8002931315389,-0.0268922634420460;
100.000330103084,-0.0260747426475103];
c1=interp1(results(:,1),results(:,2),Kneff);
end
406
Bibliography
[1] Fuchs, N. A. The Mechanics of Aerosols. Pergamon Press, Oxford, England,
1964.
[2] Friedlander, S. K. Smoke, Dust, and Haze: Fundamentals of Aerosol Dynam-
ics. Topics in chemical engineering. Oxford University Press, New York, 2
edition, 2000.
[3] Williams, M. M. R. and Loyalka, S. K. Aerosol Science: Theory and Practice:
With Special Applications to the Nuclear Industry. Pergamon Press, Oxford,
1991.
[4] Sorensen, C. M. The mobility of fractal aggregates: A review. Aerosol Science
and Technology, 45(7):765–779, 2011.
[5] Li, M., Mulholland, G. W., and Zachariah, M. R. The effect of orientation on
the mobility and dynamic shape factor of charged axially symmetric particles
in an electric field. Aerosol Science and Technology, 46(9):1035–1044, 2012.
[6] Li, M., Mulholland, G. W., and Zachariah, M. R. The effect of alignment on
the electric mobility of soot. Aerosol Science and Technology, 50(10):1003–
1016, 2016.
[7] Pratsinis, S. E. Flame aerosol synthesis of ceramic powders. Progress in
Energy and Combustion Science, 24(3):197–219, 1998.
[8] Mädler, L., Kammler, H. K., Mueller, R., and Pratsinis, S. E. Controlled
synthesis of nanostructured particles by flame spray pyrolysis. Journal of
Aerosol Science, 33(2):369–389, 2002.
[9] Johannessen, T., Jensen, J. R., Mosleh, M., Johansen, J., Quaade, U., and
Livbjerg, H. Flame synthesis of nanoparticles: Applications in catalysis and
product/process engineering. Chemical Engineering Research and Design, 82
(11):1444–1452, 2004.
[10] Strobel, R., Baiker, A., and Pratsinis, S. E. Aerosol flame synthesis of cata-
lysts. Advanced Powder Technology, 17(5):457–480, 2006.
407
[11] Bluth, G. J. S., Doiron, S. D., Schnetzler, C. C., Krueger, A. J., and Walter,
L. S. Global tracking of the SO2 clouds from the June, 1991 Mount Pinatubo
eruptions. Geophysical Research Letters, 19(2):151–154, 1992.
[12] Minnis, P., Harrison, E. F., Stowe, L. L., Gibson, G. G., Denn, F. M., Doelling,
D. R., and Smith, W. L. Radiative climate forcing by the Mount Pinatubo
eruption. Science, 259(5100):1411–1415, 1993.
[13] Jacobson, M. Z. Strong radiative heating due to the mixing state of black
carbon in atmospheric aerosols. Nature, 409(6821):695–697, 2001.
[14] Menon, S., Hansen, J., Nazarenko, L., and Luo, Y. Climate effects of black
carbon aerosols in China and India. Science, 297(5590):2250–2253, 2002.
[15] Novakov, T., Ramanathan, V., Hansen, J. E., Kirchstetter, T. W., Sato, M.,
Sinton, J. E., and Sathaye, J. A. Large historical changes of fossil-fuel black
carbon aerosols. Geophysical Research Letters, 30(6), 2003.
[16] IPCC. Climate Change 2013: the physical science basis: Working Group I
contribution to the Fifth assessment report of the Intergovernmental Panel on
Climate Change. Cambridge University Press, Cambridge, United Kingdom
and New York, NY, USA, 2013. [Stocker, T.F., D. Qin, G.-K. Plattner, M.
Tignor, S.K. Allen, J. Boschung, A. Nauels, Y. Xia, V. Bex and P.M. Midgley
(eds.)].
[17] Brunner, T. J., Wick, P., Manser, P., Spohn, P., Grass, R. N., Limbach, L. K.,
Bruinink, A., and Stark, W. J. In vitro cytotoxicity of oxide nanoparticles:
Comparison to asbestos, silica, and the effect of particle solubility. Environ-
mental science & technology, 40(14):4374–4381, 2006.
[18] Rojas-Martinez, R., Perez-Padilla, R., Olaiz-Fernandez, G., Mendoza-
Alvarado, L., Moreno-Macias, H., Fortoul, T., McDonnell, W., Loomis, D.,
and Romieu, I. Lung function growth in children with long-term exposure to
air pollutants in Mexico City. American Journal of Respiratory and Critical
Care Medicine, 176(4):377–384, 2007.
[19] Branche, C. M., Schulte, P., and Geraci, C. Approaches to safe nanotech-
nology – managing the health and safety concerns associated with engineered
nanomaterials, 2009.
[20] Albanese, A., Tang, P. S., and Chan, W. C. W. The effect of nanoparticle
size, shape, and surface chemistry on biological systems. Annual review of
biomedical engineering, 14:1–16, 2012.
[21] Loyalka, S. K. Mechanics of aerosols in nuclear reactor safety: A review.
Progress in Nuclear Energy, 12(1):1–56, 1983.
408
[22] Devell, L., Tovedal, H., Bergström, U., Appelgren, A., Chyssler, J., and Ander-
sson, L. Initial observations of fallout from the reactor accident at Chernobyl.
Nature, 321(6067):192–193, 1986.
[23] Dubrova, Y. E., Nesterov, V. N., Krouchinsky, N. G., Ostapenko, V. A., et al.
Human minisatellite mutation rate after the Chernobyl accident. Nature, 380
(6576):683, 1996.
[24] Chino, M., Nakayama, H., Nagai, H., Terada, H., Katata, G., and Yamazawa,
H. Preliminary estimation of release amounts of 131I and 137Cs accidentally
discharged from the Fukushima Daiichi nuclear power plant into the atmo-
sphere. Journal of nuclear science and technology, 48(7):1129–1134, 2011.
[25] Morino, Y., Ohara, T., and Nishizawa, M. Atmospheric behavior, deposition,
and budget of radioactive materials from the Fukushima Daiichi nuclear power
plant in March 2011. Geophysical Research Letters, 38(7), 2011.
[26] Gelbard, F. MAEROS User Manual. Technical Report NUREG/CR-1391,
Sandia National Laboratories, December 1982.
[27] Prakash, A., Bapat, A. P., and Zachariah, M. R. A simple numerical algo-
rithm and software for solution of nucleation, surface growth, and coagulation
problems. Aerosol Science and Technology, 37(11):892–898, 2003.
[28] Kirkwood, J. G. and Riseman, J. The intrinsic viscosities and diffusion con-
stants of flexible macromolecules in solution. The Journal of Chemical Physics,
16(6):565–573, 1948.
[29] Brenner, H. Coupling between the translational and rotational Brownian mo-
tions of rigid particles of arbitrary shape: II. General theory. Journal of colloid
and interface science, 23(3):407–436, 1967.
[30] Chen, Z.-Y., Deutch, J. M., and Meakin, P. Translational friction coefficient
of diffusion limited aggregates. The Journal of chemical physics, 80(6):2982–
2983, 1984.
[31] Hubbard, J. B. and Douglas, J. F. Hydrodynamic friction of arbitrarily shaped
Brownian particles. Physical Review E, 47(5):R2983, 1993.
[32] Epstein, P. S. On the resistance experienced by spheres in their motion through
gases. Physical Review, 23(6):710, 1924.
[33] Dahneke, B. E. Slip correction factors for nonspherical bodies – II Free
molecule flow. Journal of Aerosol Science, 4(2):147–161, 1973.
[34] Chan, P. and Dahneke, B. Free-molecule drag on straight chains of uniform
spheres. Journal of Applied Physics, 52(5):3106–3110, 1981.
409
[35] Meakin, P., Donn, B., and Mulholland, G. W. Collisions between point masses
and fractal aggregates. Langmuir, 5(2):510–518, 1989.
[36] Mackowski, D. W. Monte Carlo simulation of hydrodynamic drag and ther-
mophoresis of fractal aggregates of spheres in the free-molecule flow regime.
Journal of aerosol science, 37(3):242–259, 2006.
[37] Rogak, S. N., Flagan, R. C., and Nguyen, H. V. The mobility and structure
of aerosol agglomerates. Aerosol Science and Technology, 18(1):25–47, 1993.
[38] Lall, A. A. and Friedlander, S. K. On-line measurement of ultrafine aggre-
gate surface area and volume distributions by electrical mobility analysis: I.
Theoretical analysis. Journal of Aerosol Science, 37(3):260–271, 2006.
[39] Eggersdorfer, M. L., Gröhn, A. J., Sorensen, C. M., McMurry, P. H., and
Pratsinis, S. E. Mass-mobility characterization of flame-made ZrO2 aerosols:
Primary particle diameter and extent of aggregation. Journal of colloid and
interface science, 387(1):12–23, 2012.
[40] Bird, G. A. Molecular gas dynamics, Clarendon. Oxford, 1976.
[41] Zhang, C., Thajudeen, T., Larriba, C., Schwartzentruber, T. E., and Hogan Jr,
C. J. Determination of the scalar friction factor for nonspherical particles and
aggregates across the entire Knudsen number range by direct simulation monte
carlo (DSMC). Aerosol Science and Technology, 46(10):1065–1078, 2012.
[42] Köylü, Ü. Ö. and Faeth, G. M. Structure of overfire soot in buoyant turbulent
diffusion flames at long residence times. Combustion and Flame, 89(2):140–
156, 1992.
[43] Shin, W. G., Mulholland, G. W., Kim, S., Wang, J., Emery, M. S., and Pui,
D. Y. H. Friction coefficient and mass of silver agglomerates in the transition
regime. Journal of Aerosol Science, 40(7):573–587, 2009.
[44] Shin, W. G., Mulholland, G. W., and Pui, D. Y. H. Determination of volume,
scaling exponents, and particle alignment of nanoparticle agglomerates using
tandem differential mobility analyzers. Journal of Aerosol Science, 41(7):665–
681, 2010.
[45] Seigneur, C., Hudischewskyj, A. B., Seinfeld, J. H., Whitby, K. T., Whitby,
E. R., Brock, J. R., and Barnes, H. M. Simulation of aerosol dynamics: A
comparative review of mathematical models. Aerosol Science and Technology,
5(2):205–222, 1986.
[46] Vincenti, W. G. and Kruger, C. H. Introduction to Physical Gas Dynamics,
volume 1. Wiley, 1965.
[47] Chapman, S. and Cowling, T. The Mathematical Theory of Non-Uniform
Gases. Cambridge University Press, 2 edition, 1952.
410
[48] Deen, W. M. Analysis of Transport Phenomena. Topics in chemical engi-
neering. Oxford University Press, 2012. ISBN 9780199740284. URL https:
//books.google.com/books?id=60YsAwEACAAJ.
[49] Happel, J. and Brenner, H. Low Reynolds Number Hydrodynamics: With
Special Applications to Particulate Media. Prentice-Hall international series
in the physical and chemical engineering sciences. Prentice-Hall, Eaglewood
Cliffs, NJ, 1965.
[50] Stokes, G. G. On the effect of the internal friction of fluids on the motion of
pendulums. Transactions of the Cambridge Philosophical Society, 9, 1851.
[51] Bernal, J. M. G., La Torre, D., and Garćıa, J. Transport properties and hydro-
dynamic centers of rigid macromolecules with arbitrary shapes. Biopolymers,
19(4):751–766, 1980.
[52] Garćıa de La Torre, J., Jimenez, A., and Freire, J. J. Monte Carlo calcu-
lation of hydrodynamic properties of freely jointed, freely rotating, and real
polymethylene chains. Macromolecules, 15(1):148–154, 1982.
[53] Carrasco, B. and Garcıa de la Torre, J. Improved hydrodynamic interaction
in macromolecular bead models. The Journal of chemical physics, 111(10):
4817–4826, 1999.
[54] Meakin, P., Chen, Z.-Y., and Deutch, J. M. The translational friction coeffi-
cient and time dependent cluster size distribution of three dimensional cluster–
cluster aggregation. The Journal of Chemical Physics, 82(8):3786–3789, 1985.
[55] Meakin, P. and Deutch, J. M. Properties of the fractal measure describing the
hydrodynamic force distributions for fractal aggregates moving in a quiescent
fluid. The Journal of chemical physics, 86(8):4648–4656, 1987.
[56] Lattuada, M., Wu, H., and Morbidelli, M. Hydrodynamic radius of fractal
clusters. Journal of colloid and interface science, 268(1):96–105, 2003.
[57] Rotne, J. and Prager, S. Variational treatment of hydrodynamic interaction
in polymers. The Journal of Chemical Physics, 50(11):4831–4837, 1969.
[58] Yamakawa, H. Transport properties of polymer chains in dilute solution:
Hydrodynamic interaction. The Journal of Chemical Physics, 53(1):436–443,
1970.
[59] Reuland, P., Felderhof, B. U., and Jones, R. B. Hydrodynamic interaction of
two spherically symmetric polymers. Physica A: Statistical Mechanics and its
Applications, 93(3-4):465–475, 1978.
[60] Mazur, P. and Van Saarloos, W. Many-sphere hydrodynamic interactions and
mobilities in a suspension. Physica A: Statistical Mechanics and its Applica-
tions, 115(1-2):21–57, 1982.
411
[61] Goldstein, R. F. Macromolecular diffusion constants: a calculational strategy.
The Journal of chemical physics, 83(5):2390–2397, 1985.
[62] Mansfield, M. L., Douglas, J. F., and Garboczi, E. J. Intrinsic viscosity and
the electrical polarizability of arbitrarily shaped objects. Phys. Rev. E, 64:
061401, Nov 2001. doi: 10.1103/PhysRevE.64.061401. URL http://link.
aps.org/doi/10.1103/PhysRevE.64.061401.
[63] Douglas, J. F., Zhou, H.-X., and Hubbard, J. B. Hydrodynamic friction and
the capacitance of arbitrarily shaped objects. Phys. Rev. E, 49:5319–5331,
1994.
[64] Millikan, R. A. The general law of fall of a small spherical body through a gas,
and its bearing upon the nature of molecular reflection from surfaces. Physical
Review, 22(1):1, 1923.
[65] Davies, C. N. Definitive equations for the fluid resistance of spheres. Proceed-
ings of the Physical Society, 57(4):259, 1945.
[66] Allen, M. D. and Raabe, O. G. Slip correction measurements of spherical
solid aerosol particles in an improved Millikan apparatus. Aerosol Science and
Technology, 4(3):269–286, 1985.
[67] Dahneke, B. E. Slip correction factors for nonspherical bodies – III The form
of the general law. Journal of Aerosol Science, 4(2):163–170, 1973.
[68] Thajudeen, T., Jeon, S., and Hogan, C. J. The mobilities of flame synthesized
aggregates/agglomerates in the transition regime. Journal of Aerosol Science,
80:45–57, 2015.
[69] Gopalakrishnan, R., McMurry, P. H., and Hogan, C. J. The electrical mobili-
ties and scalar friction factors of modest-to-high aspect ratio particles in the
transition regime. Journal of Aerosol Science, 82:24–39, 2015.
[70] Cercignani, C. and Pagani, C. D. Flow of a rarefied gas past an axisymmetric
body. I. General remarks. Physics of Fluids (1958-1988), 11(7):1395–1399,
1968.
[71] Bhatnagar, P. L., Gross, E. P., and Krook, M. A model for collision processes
in gases. I. Small amplitude processes in charged and neutral one-component
systems. Physical Review, 94(3):511–525, 1954.
[72] Kogan, M. N. On the equations of motion of a rarefied gas. Journal of Applied
Mathematics and Mechanics, 22(4):597–607, 1958.
[73] Cercignani, C., Pagani, C. D., and Bassanini, P. Flow of a rarefied gas past
an axisymmetric body. II. Case of a sphere. Physics of Fluids (1958-1988), 11
(7):1399–1403, 1968.
412
[74] Lea, K.-C. Mass, Heat and Momentum Transfer to a Sphere in a Rarefied
Gas. PhD thesis, University of Missouri-Columbia, August 1980.
[75] Lea, K. C. and Loyalka, S. K. Motion of a sphere in a rarefied gas. Physics of
Fluids (1958-1988), 25(9):1550–1557, 1982.
[76] Law, W. S. and Loyalka, S. K. Motion of a sphere in a rarefied gas. II. Role
of temperature variation in the knudsen layer. Physics of Fluids (1958-1988),
29(11):3886–3888, 1986.
[77] Loyalka, S. K. Motion of a sphere in a gas: Numerical solution of the linearized
Boltzmann equation. Physics of Fluids A: Fluid Dynamics, 4(5):1049–1056,
1992.
[78] Takata, S., Sone, Y., and Aoki, K. Numerical analysis of a uniform flow of a
rarefied gas past a sphere on the basis of the Boltzmann equation for hard-
sphere molecules. Physics of Fluids A: Fluid Dynamics (1989-1993), 5(3):
716–737, 1993.
[79] Melas, A. D., Isella, L., Konstandopoulos, A. G., and Drossinos, Y. Fric-
tion coefficient and mobility radius of fractal-like aggregates in the transition
regime. Aerosol Science and Technology, 48(12):1320–1331, 2014.
[80] Melas, A. D., Isella, L., Konstandopoulos, A. G., and Drossinos, Y. A method-
ology to calculate the friction coefficient in the transition regime: Application
to straight chains. Journal of Aerosol Science, 82:40–50, 2015.
[81] Tandon, P. and Rosner, D. E. Translation Brownian diffusion coefficient of
large (multiparticle) suspended aggregates. Industrial & engineering chemistry
research, 34(10):3265–3277, 1995.
[82] Rosner, D. E. and Tandon, P. Aggregation-and rarefaction-effects on parti-
cle mass deposition rates by convective-diffusion, thermophoresis or inertial
impaction: Consequences of multi-spherule ’momentum shielding’. Aerosol
Science and Technology, pages 1–17, 2017.
[83] Flagan, R. C. History of electrical aerosol measurements. Aerosol Science and
Technology, 28(4):301–380, 1998.
[84] Cheng, Y.-S., Allen, M. D., Gallegos, D. P., Yeh, H.-C., and Peterson, K. Drag
force and slip correction of aggregate aerosols. Aerosol Science and Technology,
8(3):199–214, 1988.
[85] Ehara, K., Hagwood, C., and Coakley, K. J. Novel method to classify aerosol
particles according to their mass-to-charge ratioaerosol particle mass analyser.
Journal of Aerosol Science, 27(2):217–234, 1996.
413
[86] Radney, J. G., Ma, X., Gillis, K. A., Zachariah, M. R., Hodges, J. T., and
Zangmeister, C. D. Direct measurements of mass-specific optical cross sections
of single-component aerosol mixtures. Analytical chemistry, 85(17):8319–8325,
2013.
[87] Zelenyuk, A. and Imre, D. On the effect of particle alignment in the DMA.
Aerosol science and technology, 41(2):112–124, 2007.
[88] Li, M., You, R., Mulholland, G. W., and Zachariah, M. R. Development of
a pulsed-field differential mobility analyzer: A method for measuring shape
parameters for nonspherical particles. Aerosol Science and Technology, 48(1):
22–30, 2014.
[89] Bohren, C. F. and Huffman, D. R. Absorption and Scattering of Light by
Small Particles. John Wiley & Sons, 1983.
[90] Weiss, R. E., Kapustin, V. N., and Hobbs, P. V. Chain-aggregate aerosols
in smoke from the Kuwait oil fires. Journal of Geophysical Research: Atmo-
spheres, 97(D13):14527–14531, 1992.
[91] Colbeck, I., Atkinson, B., and Johar, Y. The morphology and optical prop-
erties of soot produced by different fuels. Journal of aerosol science, 28(5):
715–723, 1997.
[92] Corson, J., Mulholland, G. W., and Zachariah, M. R. Friction factor for
aerosol fractal aggregates over the entire Knudsen range. Physical Review E,
95(1):013103, 2017.
[93] Corson, J., Mulholland, G. W., and Zachariah, M. R. Analytical expression
for the friction coefficient of DLCA aggregates based on extended Kirkwood-
Riseman theory. Aerosol Science and Technology, 51(6):766–777, 2017.
[94] Corson, J., Mulholland, G. W., and Zachariah, M. R. Calculating the rota-
tional friction coefficient of fractal aerosol particles in the transition regime
using extended Kirkwood-Riseman theory. Physical Review E, 96(1):013110,
2017.
[95] Corson, J., Mulholland, G. W., and Zachariah, M. R. Analytical expression for
the rotational friction coefficient of DLCA aggregates over the entire knudsen
regime. Aerosol Science and Technology, 52(2):209–221, 2018. doi: 10.1080/
02786826.2017.1390544.
[96] Corson, J., Mulholland, G. W., and Zachariah, M. R. The effect of electric
field induced alignment on the electrical mobility of fractal aggregates. Aerosol
Science and Technology, page in press, 2018.
[97] Felderhof, B. U. Hydrodynamic interaction between two spheres. Physica A:
Statistical Mechanics and its Applications, 89(2):373–384, 1977.
414
[98] Abramowitz, M. and Stegun, I. A. Handbook of Mathematical Functions: With
Formulas, Graphs, and Mathematical Tables. Applied mathematics series.
Dover Publications, 1964. ISBN 9780486612720.
[99] Loyalka, S. K. On the motion of aerosols in nonuniform gases. I. The Journal
of Chemical Physics, 55(1):1–4, 1971.
[100] Kreyszig, E. Advanced Engineering Mathematics. John Wiley & Sons, 2010.
ISBN 9780470458365.
[101] Zhou, H.-X., Szabo, A., Douglas, J. F., and Hubbard, J. B. A Brownian
dynamics algorithm for calculating the hydrodynamic friction and the electro-
static capacitance of an arbitrarily shaped object. The Journal of Chemical
Physics, 100(5):3821–3826, 1994.
[102] Shvartsburg, A. A. and Jarrold, M. F. An exact hard-spheres scattering model
for the mobilities of polyatomic ions. Chemical Physics Letters, 261(1):86–91,
1996.
[103] Cho, K., Hogan Jr, C. J., and Biswas, P. Study of the mobility, surface area,
and sintering behavior of agglomerates in the transition regime by tandem
differential mobility analysis. Journal of Nanoparticle Research, 9(6):1003–
1012, 2007.
[104] Sone, Y. and Aoki, K. Forces on a spherical particle in a slightly rarefied gas.
Rarefied Gas Dynamics, 51(pt 1):417–433, 1977.
[105] Li, M., Mulholland, G. W., and Zachariah, M. R. Understanding the mobility
of nonspherical particles in the free molecular regime. Physical Review E, 89
(2):022112, 2014.
[106] Swanson, E., Teller, D. C., and de Haën, C. Creeping flow translational
resistance of rigid assemblies of spheres. The Journal of Chemical Physics, 72
(3):1623–1628, 1980.
[107] Garćıa de la Torre, J. and Rodes, V. Effects from bead size and hydrodynamic
interactions on the translational and rotational coefficients of macromolecular
bead models. The Journal of chemical physics, 79(5):2454–2460, 1983.
[108] Sorensen, C. M. and Wang, G. M. Note on the correction for diffusion and
drag in the slip regime. Aerosol Science and Technology, 33(4):353–356, 2000.
[109] Cheng, M. T., Xie, G. W., Yang, M., and Shaw, D. T. Experimental character-
ization of chain-aggregate aerosol by electrooptic scattering. Aerosol Science
and Technology, 14(1):74–81, 1991.
[110] Chen, S.-C., Wang, J., Fissan, H., and Pui, D. Y. H. Exposure assessment
of nanosized engineered agglomerates and aggregates using Nuclepore filter.
Journal of nanoparticle research, 15(10):1955, 2013.
415
[111] Lattuada, M. Predictive model for diffusion-limited aggregation kinetics of
nanocolloids under high concentration. The Journal of Physical Chemistry B,
116(1):120–129, 2011.
[112] Ortega, A. and Garcıa de la Torre, J. Hydrodynamic properties of rodlike and
disklike particles in dilute solution. The Journal of Chemical Physics, 119(18):
9914–9919, 2003.
[113] Halbritter, J. Torque on a rotating ellipsoid in a rarefied gas. Zeitschrift für
Naturforschung A, 29(12):1717–1722, 1974.
[114] Li, M., Mulholland, G. W., and Zachariah, M. R. Rotational diffusion co-
efficient (or rotational mobility) of a nanorod in the free-molecular regime.
Aerosol Science and Technology, 48(2):139–141, 2014.
[115] Garćıa de la Torre, J., del Rio Echenique, G., and Ortega, A. Improved
calculation of rotational diffusion and intrinsic viscosity of bead models for
macromolecules and nanoparticles. The Journal of Physical Chemistry B, 111
(5):955–961, 2007.
[116] Mulholland, G. W., Hagwood, C. R., Li, M., and Zachariah, M. R. Effect
of particle rotation on the drift velocity for nonspherical aerosol particles.
Journal of Aerosol Science, 101:65–76, 2016.
[117] Larriba, C. and Hogan Jr, C. J. Ion mobilities in diatomic gases: Measurement
versus prediction with non-specular scattering models. The Journal of Physical
Chemistry A, 117(19):3887–3901, 2013.
[118] Shrivastav, V., Nahin, M., Hogan, C. J., and Larriba-Andaluz, C. Benchmark
comparison for a multi-processing ion mobility calculator in the free molecular
regime. Journal of The American Society for Mass Spectrometry, 28:1540–
1551, 2017.
[119] Tekasakul, P., Bentz, J. A., Tompson, R. V., and Loyalka, S. K. The spinning
rotor gauge: Measurements of viscosity, velocity slip coefficients, and tangen-
tial momentum accommodation coefficients. Journal of Vacuum Science &
Technology A: Vacuum, Surfaces, and Films, 14(5):2946–2952, 1996.
[120] Bentz, J. A., Tompson, R. V., and Loyalka, S. K. Measurements of viscosity,
velocity slip coefficients, and tangential momentum accommodation coeffi-
cients using a modified spinning rotor gauge. Journal of Vacuum Science &
Technology A: Vacuum, Surfaces, and Films, 19(1):317–324, 2001.
[121] Mountain, R. D., Mulholland, G. W., and Baum, H. Simulation of aerosol
agglomeration in the free molecular and continuum flow regimes. Journal of
Colloid and Interface Science, 114(1):67–81, 1986.
416
[122] Heinson, W. R., Pierce, F., Sorensen, C. M., and Chakrabarti, A. Crossover
from ballistic to Epstein diffusion in the free-molecular regime. Aerosol Science
and Technology, 48(7):738–746, 2014.
[123] Kousaka, Y., Endo, Y., Ichitsubo, H., and Alonso, M. Orientation-specific
dynamic shape factors for doublets and triplets of spheres in the transition
regime. Aerosol science and technology, 24(1):36–44, 1996.
[124] Bottcher, C. J. F. and Belle, O. C. V. Theory of Electric Polarization, Vol. 1:
Dielectrics in Static Fields. Elsevier Scientific Publishing Company, Amster-
dam, 1973.
[125] Dobbins, R. A. Soot inception temperature and the carbonization rate of
precursor particles. Combustion and Flame, 130(3):204–214, 2002.
[126] Park, K., Kittelson, D. B., Zachariah, M. R., and McMurry, P. H. Measure-
ment of inherent material density of nanoparticle agglomerates. Journal of
Nanoparticle Research, 6(2):267–272, 2004.
[127] Fernandes, M. X. and Garćıa de la Torre, J. Brownian dynamics simulation
of rigid particles of arbitrary shape in external fields. Biophysical journal, 83
(6):3039–3048, 2002.
[128] Batchelor, G. K. Sedimentation in a dilute dispersion of spheres. Journal of
fluid mechanics, 52(2):245–268, 1972.
[129] Sorensen, C. M., Hageman, W. B., Rush, T. J., Huang, H., and Oh, C. Aero-
gelation in a flame soot aerosol. Physical Review Letters, 80(8):1782–1785,
1998.
[130] Stimson, M. and Jeffery, G. B. The motion of two spheres in a viscous fluid.
Proceedings of the Royal Society of London. Series A, Containing Papers of a
Mathematical and Physical Character, 111(757):110–116, 1926.
[131] Burgers, J. M. Hydrodynamics. – On the influence of the concentration of
a suspension upon the sedimentation velocity (in particular for a suspension
of spherical particles). In Selected Papers of JM Burgers, pages 452–477.
Springer, 1995.
[132] Brenner, H. The slow motion of a sphere through a viscous fluid towards a
plane surface. Chemical engineering science, 16(3-4):242–251, 1961.
[133] Goldman, A., Cox, R., and Brenner, H. The slow motion of two identical
arbitrarily oriented spheres through a viscous fluid. Chemical Engineering
Science, 21(12):1151–1170, 1966.
[134] Batchelor, G. K. Brownian diffusion of particles with hydrodynamic interac-
tion. Journal of Fluid Mechanics, 74(1):1–29, 1976.
417
[135] Ermak, D. L. and McCammon, J. A. Brownian dynamics with hydrodynamic
interactions. The Journal of chemical physics, 69(4):1352–1360, 1978.
[136] Brady, J. F. and Bossis, G. Stokesian dynamics. Annual review of fluid
mechanics, 20(1):111–157, 1988.
[137] Smoluchowski, M. On the practical applicability of Stokes’ law of resistance,
and the modifications of it required in certain cases. University Press, 1912.
[138] Mackaplow, M. B. and Shaqfeh, E. S. G. A numerical study of the sedimen-
tation of fibre suspensions. Journal of Fluid Mechanics, 376:149–182, 1998.
[139] Loyalka, S. K. Slip and jump coefficients for rarefied gas flows: Variational
results for Lennard-Jones and n (r)-6 potentials. Physica A: Statistical Me-
chanics and its Applications, 163(3):813–821, 1990.
[140] Peterson, T. W., Gelbard, F., and Seinfeld, J. H. Dynamics of source-
reinforced, coagulating, and condensing aerosols. Journal of colloid and inter-
face science, 63(3):426–445, 1978.
[141] Gelbard, F. and Seinfeld, J. H. Exact solution of the general dynamic equation
for aerosol growth by condensation. Journal of Colloid and Interface Science,
68(1):173–183, 1979.
[142] Gelbard, F., Tambour, Y., and Seinfeld, J. H. Sectional representations for
simulating aerosol dynamics. Journal of Colloid and Interface Science, 76(2):
541–556, 1980.
[143] Gelbard, F. and Seinfeld, J. H. Simulation of multicomponent aerosol dynam-
ics. Journal of colloid and Interface Science, 78(2):485–501, 1980.
[144] Suck, S. and Brock, J. Evolution of atmospheric aerosol particle size distri-
butions via Brownian coagulation numerical simulation. Journal of Aerosol
Science, 10(6):581–590, 1979.
[145] Whitby, K. T. Determination of aerosol growth rates in the atmosphere using
lumped mode aerosol dynamics. Journal of Aerosol Science, 12(3):173–178,
1981.
[146] Seinfeld, J. H. and Pandis, S. N. Atmospheric Chemistry and Physics: From
Air Pollution to Climate Change. John Wiley & Sons, New York, NY, 1998.
[147] Girshick, S. L. and Chiu, C.-P. Kinetic nucleation theory: A new expression
for the rate of homogeneous nucleation from an ideal supersaturated vapor.
The journal of chemical physics, 93(2):1273–1277, 1990.
[148] Friedlander, S. K. and Wang, C. S. The self-preserving particle size distri-
bution for coagulation by Brownian motion. Journal of Colloid and Interface
Science, 22(2):126–132, 1966.
418
[149] Vemury, S., Kusters, K. A., and Pratsinis, S. E. Time-lag for attainment of
the self-preserving particle size distribution by coagulation. Journal of Colloid
and Interface Science, 165(1):53–59, 1994.
[150] Brewster, M. Q. Thermal Radiative Transfer and Properties. John Wiley &
Sons, 1992.
[151] Spyrogianni, A., Karadima, K. S., Goudeli, E., Mavrantzas, V. G., and Pratsi-
nis, S. E. Mobility and settling rate of agglomerates of polydisperse nanopar-
ticles. The Journal of chemical physics, 148(6):064703, 2018.
[152] Kelesidis, G. A., Goudeli, E., and Pratsinis, S. E. Morphology and mobility
diameter of carbonaceous aerosols during agglomeration and surface growth.
Carbon, 121:527–535, 2017.
[153] Feng, Y. and Kleinstreuer, C. Analysis of non-spherical particle transport in
complex internal shear flows. Physics of Fluids, 25(9):091904, 2013.
[154] Kolb, M., Botet, R., and Jullien, R. Scaling of kinetically growing clusters.
Physical Review Letters, 51(13):1123, 1983.
[155] Meakin, P. The effects of rotational diffusion on the fractal dimensionality
of structures formed by cluster–cluster aggregation. The Journal of Chemical
Physics, 81(10):4637–4639, 1984.
[156] Tian, L., Ahmadi, G., and Tu, J. Brownian diffusion of fibers. Aerosol Science
and Technology, 50(5):474–486, 2016.
419
Publications and Presentations
Accepted publications
1. Corson, J., Mulholland, G. W., and Zachariah, M. R. Friction factor for
aerosol fractal aggregates over the entire Knudsen range. Physical Review E,
95(1):013103, 2017. DOI: 10.1103/PhysRevE.95.013103.
2. Corson, J., Mulholland, G. W., and Zachariah, M. R. Analytical expression
for the friction coefficient of DLCA aggregates based on extended Kirkwood-
Riseman theory. Aerosol Science and Technology, 51(6):766–777, 2017. DOI:
10.1080/02786826.2017.1300635.
3. Corson, J., Mulholland, G. W., and Zachariah, M. R. Calculating the rota-
tional friction coefficient of fractal aerosol particles in the transition regime
using extended Kirkwood-Riseman theory. Physical Review E, 96(1):013110,
2017. DOI: 10.1103/PhysRevE.96.013110.
4. Corson, J., Mulholland, G. W., and Zachariah, M. R. Analytical expres-
sion for the rotational friction coefficient of DLCA aggregates over the entire
knudsen regime. Aerosol Science and Technology, 52(2):209–221, 2018. DOI:
10.1080/02786826.2017.1390544.
5. Corson, J., Mulholland, G. W., and Zachariah, M. R. The effect of electric
field induced alignment on the electrical mobility of fractal aggregates. Aerosol
420
Science and Technology, 2018. DOI: 10.1080/02786826.2018.1427210.
Submitted publications
1. Corson, J., Mulholland, G. W., and Zachariah, M. R. Hydrodynamic Inter-
actions between Aerosol Particles in the Transition Regime. Journal of Fluid
Mechanics.
Delivered presentations (presenting author underlined)
1. Corson, J., Zachariah, M. R., Mulholland, G. W., and Baum, H. Extension
of Kirkwood-Riseman Theory across the Entire Range of Knudsen Numbers.
Annual Meeting of the American Physical Society Division of Fluid Dynamics.
Portland, OR. November 2016. (Platform presentation)
2. Corson, J., Mulholland, G. W., and Zachariah, M. R. Calculating the Trans-
lational Friction Coefficient of DLCA Aggregates in the Transition Regime.
American Association for Aerosol Research Annual Meeting. Raleigh, NC.
October 2017. (Platform presentation)
3. Corson, J., Mulholland, G. W., and Zachariah, M. R. Drag and Torque
on Fractal Aggregates in the Transition Regime. American Association for
Aerosol Research Annual Meeting. Raleigh, NC. October 2017. (Poster)
4. Corson, J., Mulholland, G. W., and Zachariah, M. R. Solving the GDE for
Nucleation, Surface Growth, and Coagulation using the Nodal Method. Amer-
ican Association for Aerosol Research Annual Meeting. Raleigh, NC. October
2017. (Poster)
421
5. Corson, J., Mulholland, G. W., and Zachariah, M. R. Drag and Torque on
Fractal Aerosol Aggregates Across the Entire Knudsen Regime. Fourteenth
Annual Symposium of the Burgers Program for Fluid Dynamics. College Park,
MD. November 2017. (Poster)
Planned presentations (abstract submitted)
1. Corson, J., Mulholland, G. W., and Zachariah, M. R. NGDE: A Simple,
MATLAB-based Code for Solving the General Dynamic Equation. 10th In-
ternational Aerosol Conference. St. Louis, MO. September 2018. (Platform
presentation)
2. Corson, J., Mulholland, G. W., and Zachariah, M. R. The Effect of Elec-
tric Field Induced Alignment on the Electrical Mobility of Fractal Aggregates.
10th International Aerosol Conference. St. Louis, MO. September 2018. (Plat-
form presentation)
422