ABSTRACT
Title of Dissertation: FLUID DYNAMICS OF EXTENSIONAL
DEFORMATION AND CAPILLARY-DRIVEN
BREAKUP OF DROPS AT
LOW REYNOLDS NUMBER
Aditya Narendrababu Sangli
Doctor of Philosophy, 2021
Dissertation Directed by: Professor David I. Bigio
Department of Mechanical Engineering
In this dissertation, extensional deformation and capillary-driven breakup of
drops at low Reynolds number is investigated using a combination of theoretical,
experimental, and numerical techniques. The dissertation introduces a new non-
dimensional measure for drop deformation, rationalizes previously unseen drop
breakup behavior, and extends our overall understanding of the fluid dynamics
behind drop deformation and breakup.
First, non-stagnant extensional deformation of Silicone oil drops in Castor oil
is experimentally studied over a wide range of capillary numbers by injecting the
drops along the centerline of a flow through a hyperbolic converging channel. The
unique design of the channel is capable of imposing a constant extensional rate and is
validated using lubrication theory. Based on a careful analysis of drop deformation
at both small and large capillary numbers compared to the critical capillary number,
a new measure called the semi-minor capillary number is introduced to characterize
drop behavior. Critical semi-minor capillary number is presented for a wide range
of viscosity ratios and the significance of the new measure over the conventional
capillary number measure is discussed.
During the course of the experiments, it was observed that drops undergoing
non-stagnant extension exhibited a lag in velocity compared to the background
flow velocity at the same point. This lag in velocity is attributed to flow induced
deformation of the drops and the phenomenon is rationalized for a wide range of
capillary numbers.
When drops are injected offset of the centerline of the channel, an anomalously
large degree of deformation is observed even at low flow rates. A careful investigation
of the phenomenon revealed that the strain rates along offset lines were at least an
order of magnitude larger than the extensional rates along the centerline. A model
is developed based on lubrication theory to predict the large deformation of drops
and is successfully validated with experimental measurements.
Finally, in the absence of a forcing external flow, when slender drops are
allowed to develop under the effect of interfacial tension they either retract into
a sphere or breakup into multiple drops. This phenomenon is investigated using
direct numerical simulations in a previously unexplored part of the parametric space
where both inertial and viscous effects in the outside fluid are considered. A stability
diagram is presented that shows a transition of drop states from asymptotically
unstable to asymptotically stable states at different viscosity ratios. The drop
behavior in different regimes is discussed and the significance of the balance between
inertial and viscous forces is thoroughly described.
FLUID DYNAMICS OF EXTENSIONAL DEFORMATION
AND CAPILLARY-DRIVEN BREAKUP OF
DROPS AT LOW REYNOLDS NUMBER
by
Aditya Narendrababu Sangli
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
2021
Advisory Committee:
Professor David I. Bigio, Chair & Advisor
Professor Amir Riaz
Professor James H. Duncan
Professor Kenneth Kiger
Professor Richard V. Calabrese, Dean?s Representative
? Copyright by
Aditya Narendrababu Sangli
2021
To my parents
ii
Acknowledgments
I would like to thank several people who have supported me through the years
as I completed this dissertation. First of all, I would like to thank my advisor,
Professor David Bigio, for his mentorship throughout my graduate education. He
afforded me tremendous freedom to pursue research topics that interested me and
I have benefited greatly from his counsel in every area. I will forever cherish our
wide-ranging conversations on science and philosophy and thank him deeply for his
role in my personal development that had led me to where I am today. I would
also like to thank Professor Amir Riaz who played an important role in advancing
my computational fluid dynamics education during the final years of my PhD. He
was patient with me as I learnt the techniques of the field and introduced me to
interesting problems.
Members of the Advanced Manufacturing and Polymer Processing Lab have
been of great support to me. In particular, I would like to thank Marcelo Arispe-
Guzman and Connor Armstrong who helped me build a nice experiment that lasted
six years. Together we developed as scientists and learned to do good experiments.
I also thank Artiom Kostiouk and Margaret Lo who spent a summer generating
lots of useful data for my research. Jiazhen Qiao and Bernard Chang from the
Riaz group were always around to discuss interesting ideas and helped me setup
numerical experiments.
iii
I would also like to acknowledge the support from UMD?s broader fluid
dynamics community. In particular, the Calabrese, Duncan, and Kiger research
groups have played a supporting role in my development as a member of the
community. Derrick Ko was of great help early in my career and was patient with me
as he shared his experimental expertise. I am indebted to Professor Howard Stone at
Princeton who played a big role in my fluid dynamics education by introducing me
to several new ideas, and inspired me to work on many problems in this dissertation.
I learnt from him how to think about problems in fluid dynamics and I am grateful
for his support. I also acknowledge the support from the University of Maryland
supercomputing resources which I extensively used in my research.
The generous support from the Mechanical Engineering department allowed
me to be financially stable through the years. Having been a teaching assistant
throughout my tenure, I would like to thank Professor Gary Pertmer for his
mentoring role in my development as a teacher.
My housemates and friends over the years have always been there for me
when I needed them the most. I am particularly grateful to Alok Bharadwaj for his
friendship.
Finally to my family, I say thank you. I am truly appreciative for your belief
in me and the constant love and support through the years.
iv
Table of Contents
Dedication ii
Acknowledgements iii
Table of Contents v
List of Figures vii
Chapter 1: Introduction to the dissertation 1
Chapter 2: A new measure for drop deformation in extensional flows at low
Reynolds number 4
2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.1 Drop deformation at Cai < Cacrit . . . . . . . . . . . . . . . . 13
2.4.2 Drop deformation at Cai > Cacrit . . . . . . . . . . . . . . . . 18
2.4.3 Significance of semi-minor length b in extensional deformation
of drops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.4 Critical Casm at different viscosity ratios . . . . . . . . . . . . 24
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Chapter 3: Velocity of suspended fluid particles in a low Reynolds number
converging flow 27
3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Lubrication analysis for a planar converging flow . . . . . . . . . . . . 32
3.3.1 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . 36
3.3.2 Experimental measurement of velocity . . . . . . . . . . . . . 37
3.4 Effect of flow on suspended immiscible droplets . . . . . . . . . . . . 38
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Chapter 4: Modelling shear-influenced deformation of drops in a converging
channel using lubrication theory 48
4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
v
4.3 Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.5.1 Spatial-temporal phase space . . . . . . . . . . . . . . . . . . 59
4.5.2 Comparison with experiments . . . . . . . . . . . . . . . . . . 60
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Chapter 5: Effect of inertia on extensional deformation and capillary-driven
breakup of drops 64
5.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . 72
5.3.3 Numerical scheme and boundary conditions . . . . . . . . . . 74
5.3.4 Parametric space for numerical experiments . . . . . . . . . . 77
5.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4.1 Inertial effects on drop deformation in stagnant extensional flow 79
5.4.2 Inertial effects on drop deformation in non-stagnant extensional
flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4.3 Inertial effects on capillary-driven breakup . . . . . . . . . . . 82
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Appendix A:Supplementary material to chapter 3 96
A.1 Numerical simulation in ANSYS FLUENT . . . . . . . . . . . . . . . 96
A.1.1 Specifications of numerical solver . . . . . . . . . . . . . . . . 97
A.1.2 Results and verification . . . . . . . . . . . . . . . . . . . . . . 97
A.2 Raw experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Appendix B: Supplementary material to chapter 5 104
B.1 Grid independence study . . . . . . . . . . . . . . . . . . . . . . . . . 104
B.2 Inertial effect on breakup of slender drop in [1] . . . . . . . . . . . . . 104
B.3 Influence of the end-shape on capillary-driven flow . . . . . . . . . . . 109
B.4 Non-trivial breakup modes under large inertia . . . . . . . . . . . . . 110
B.4.1 Entrapment of surrounding fluid . . . . . . . . . . . . . . . . . 111
B.4.2 Torus formation . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Bibliography 114
vi
List of Figures
2.1 (a) Front view, (b) side view, and (c) isometric view of a planar
channel with flow streamwise along x. All dimensions are in mm.
The origin of the coordinate system is equidistant from the two planar
bounding walls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 (a) The experimental setup showing the planar converging channel
and the flow control mechanism. Flow rate of Castor oil is controlled
through the motion of the reciprocating piston into the channel.
(b) shows the schematic of an initially spherical drop undergoing
deformation due to the external flow. L is the semi-major length
of the deformed drop and b is the semi-minor length (also referred to
as the width). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Drop deformation at Cai = 0.12. (a) is the initial state where the
flow is initiated. (b) to (f) shows subsequent deformation. The drop
deforms and quickly attains a steady prolate spheroidal shape. . . . . 14
2.4 Drop deformation at various Cai versus ??t where ?? is the extensional
rate and t is time. Note the attainment of steady states for low Cai
and the continuous deformation for Cai > Cacrit. Sample error bar
is shown for Cai = 0.14. . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Tracking the L/b of deforming drops as a function of Casm. Each
curve represents a single experiment carried out at a fixed Cai. Drops
start from an initial state of L/b = 1 where Cai = Casm. Subsequent
deformation is tracked till drops reached a steady state for Cai <
Cacrit, or till drops exit from the flow for Cai > Cacrit. The vertical
dashed line is the lower bound (and a unique value for a given viscosity
ratio) of Casm for all Cai. Sample error bar is shown for Cai = 0.14. 16
2.6 Data extracted from [2]. The plot shows Casm at steady states
calculated for various Cai. The three curves represent different viscosity
ratios (?). Note that as Cai is increased, Casm becomes independent
of Cai and is referred to as the critical Casm. The largest Cai for
each viscosity ratio is the Cacrit. . . . . . . . . . . . . . . . . . . . . . 17
2.7 Drop deformation at Cai = 0.60. (a) is the initial state where the
flow is initiated. (b) to (f) shows subsequent deformation. The drop
deforms continuously in the flow and no steady state is observed. . . 18
2.8 Self-similar behavior of drops at Cai > Cacrit. . . . . . . . . . . . . . 19
2.9 Kinematics of drop deformation from initial state (L/b = 1) to deformed
state. b? = b/a where a is the initial radius of the drop. . . . . . . . . 20
vii
2.10 Instantaneous cross sections in the xy plane of a deforming drop at
Cai = 0.44. b? = b/a and L? = L/a where a is the initial radius of the
drop. (a) is the initial state and (b) to (j) shows the subsequent
deformation in time. Note that b? at the center of mass location
changes by a small amount with increasing deformation. . . . . . . . 22
2.11 Cacrit and critical Casm at different viscosity ratios (?) using data
extracted from [2]. The red diamond represents critical Casm obtained
experimentally in this study for ? = 0.28. . . . . . . . . . . . . . . . . 24
3.1 (a) Front view and (b) isometric view of a planar channel with flow
streamwise along x. The spanwise width reduces inversely as the
streamwise coordinate. All dimensions are in mm. The origin of
the coordinate system is equidistant from the two planar bounding
walls. Half-width of the channel at the outlet (where x = 160 mm) is
3.53 mm. We use an axial length scale ? = 160 mm, and a transverse
length scale h = 10 mm. . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Non dimensional velocity (u? = u/(Q/h2)) vs. Non dimensional streamwise
coordinate (x? = x/?) along the centerline of the flow obtained using
lubrication approximation (solid line) and numerical simulation (dashed
line). Error bars on experimental data represent standard deviation
from the mean value. Inset graph plots the slope of the lubrication
approximation and is equivalent to the variation of extensional rate. . 35
3.3 (a) The experimental apparatus to measure droplet deformation in
the flow channel. Flow rate of Castor oil is controlled through the
motion of the linear actuator into the channel along the downward
direction. The motion of a Silicone oil droplet (highlighted black
circle) in the region of interest (black dotted line) is captured using
high speed photography. Viscosity ratio is around 0.28. Fig. b1
and c1 shows the initial state for Capillary number equal to 0.15 and
0.60 respectively. Fig. b2 to b6 and c2 to c6 shows the subsequent
deformation induced by the motion. . . . . . . . . . . . . . . . . . . . 40
3.4 L/b vs x? for different initial Capillary numbers (Ca). The x? refers
to the instantaneous center of mass location of the droplet. Note
the attainment of a steady elongated shape for low initial Capillary
numbers (hollow symbols) and continuous elongation for high initial
Capillary numbers (solid symbols). . . . . . . . . . . . . . . . . . . . 41
3.5 Comparison of droplet translational velocity and the background flow
velocity. Ca refers to initial Capillary number and x? refers to the
instantaneous center of mass location of the droplet. Note the large
initial slope for low initial Capillary numbers (hollow symbols) compared
with the high initial Capillary numbers (solid symbols). . . . . . . . . 42
viii
3.6 Motion induced deformation of droplet at initial Capillary number of
0.17. Crosses represent variation of L/b and circles represent variation
of u?. Solid line is the lubrication approximation of velocity of the
single phase background flow. The droplet attains a steady shape at
x? ? 0.3. The x? refers to instantaneous center of mass location of the
droplet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.7 Motion induced deformation of droplet at initial Capillary number
of 0.67. Squares represent variation of L/b and circles represent
variation of u?. Solid line is the lubrication approximation of velocity
of the single phase background flow. The droplet continuously deforms
in the domain with asymptotic reduction in b after reaching bcrit. The
x? refers to instantaneous center of mass location of the droplet. . . . 45
4.1 (a) Front view, (b) side view, and (c) isometric view of a planar
channel with flow streamwise along x. All dimensions are in mm.
The origin of the coordinate system is equidistant from the two planar
bounding walls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Side view of channel (as shown in Figure 4.1(b)) shown with a drop
initially suspended at (x? = x?0, y? = 0, z? = (z?1 + z?2)/2). When the flow
is actuated, the drop undergoes shear-influenced deformation but is
restricted between z?1 and z?1. . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Shear-influenced deformation of a Silicone oil drop in Castor oil flow.
(a) shows the initial configuration at rest where the diameter of the
drop is approximately 4.5 mm and the drop is located at z? = 0.5. (b)
to (g) show the subsequent motion induced deformation. See Figure
4.2 for a kinematic description of the drop behavior from the side
view. In (g) the drop has attained a length of approximately 44 mm.
Experiment was repeated for different z? initial drop locations. . . . . 58
4.4 Space-time phase space of trajectories of points in flow described by
equation 4.4 and following notation shown in Figure 4.1. At t? = 0,
x? = 0. As time progresses, particle paths follow the background
velocity field. Points at the center of the channel (z? = 0) travel the
furthest while points at the wall do not move. . . . . . . . . . . . . . 59
4.5 Trajectories for z?1 = 0.4 and z?1 = 0.6 extracted from Figure 4.4. The
two points separate from each other along the plotted trajectories.
Difference in x? at any time instant provides a measure of their separation. 61
4.6 Comparison of model prediction of drop lengths |x?1?x?2| vs experimental
measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1 Domain for the stagnant extensional flow problem. The drop is shown
suspended at the center of a square 8 X 8 domain. All dimensions are
non-dimensionalized using a length scale equal to the initial radius of
the drop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
ix
5.2 Domain for the non-stagnant extensional flow problem. The drop is
shown suspended in a converging channel. All dimensions are non-
dimensionalized using a length scale equal to the inlet width of the
channel (120 mm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.3 Quarter domain for the capillary-driven flow problem. A long slender
drop is shown suspended in another quiescent fluid. The drop has
an aspect ratio = 30. Such slender drops are unstable at moderate
Reynolds numbers when the effect of the surrounding fluid is ignored. 72
5.4 Discretization scheme for the numerical method. In the shown staggered
cell with grid spacing ?x = ?y, the vector variables are defined on
the cell faces and scalar variables are defined on the cell centers. . . . 75
5.5 The parametric space for investigation of capillary-driven breakup of
slender drops. The space comprises viscosity ratio (?) and Ohnesorge
number (Oh). In this study, both inertial and viscous effects of the
outside fluid are considered. . . . . . . . . . . . . . . . . . . . . . . . 78
5.6 (a) to (d) shows the steady states of drops at Wes = 0.05, Wes =
0.07, Wes = 0.09, and Wes = 0.11 respectively. Reynolds number
(Res) for all cases is equal to 1. . . . . . . . . . . . . . . . . . . . . . 79
5.7 Semi-major length L and semi-minor length b of a deformed drop. . . 80
5.8 Effect of slight inertia at Res = 1 on drops undergoing stagnant
extension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.9 (a) to (f) represent transient deformation of drops in a non-stagnant
extensional flow at Rens = 2.53 and Wens = 7.99. The left panel
in every image is the experimental observation and the right panel is
the computational result. . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.10 Comparison of simulation and experimental results for non-stagnant
extension of a drop at Rens = 2.53 and Wens = 7.99. . . . . . . . . . 82
5.11 Comparison of (a) experimental results [3] and (b) simulations of
capillary-driven breakup at Oh = 0.0057. . . . . . . . . . . . . . . . . 84
5.12 Capillary driven flow at Rec = 25 and ? = 1.00. The drop is
conditionally unstable ? breaking up into two daughter drops ? but
asymptotically stable. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.13 Capillary driven flow atRe = 25 and ? = 0.10. The drop is asymptotically
unstable and results in two daughter drops with a small satellite drop
suspended between them. . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.14 Capillary driven flow at Rec = 25 and ? = 0.01. This drop is
asymptotically unstable ? resulting in three daughter drops with
satellite drops between them. . . . . . . . . . . . . . . . . . . . . . . 87
5.15 Asymptotic fates of the slender drop for different combinations of
viscosity ratio (?) and Reynolds number (Rec). . . . . . . . . . . . . 88
5.16 Vorticity distribution in the system at Rec = 100 and ? = 1.00 before
attainment of minimum neck radius. Regions of opposite signed
vortices can be seen. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
x
5.17 Vorticity distribution in the system at Rec = 100 and ? = 1.00 at
minimum neck radius. Onset of the separation of the jet flow from
the neck to the bulb can be observed. . . . . . . . . . . . . . . . . . . 91
5.18 Vorticity distribution in the system at Rec = 100 and ? = 1.00 after
reopening of the neck. A prominent vortex ring can be seen persisting
in the bulb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.19 Vorticity distribution in the system just before monotonic pinch-off of
the neck. Note that vorticity is diffused and no concentrated vortex
rings are observed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
A.1 (a) Front view and (b) isometric view of a planar channel with flow
streamwise along x. The flow domain was discretized into 179,080
hexahedral elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
A.2 Non dimensional velocity of the flow u? along the centerline x? at
various non dimensional timesteps. The time is non dimensionalized
using the velocity and length scale. Flow rapidly develops to the
steady state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
A.3 Streamwise velocity along the centerline at two levels of flow domain
discretization. Solid line represents 23,544 elements and dashed line
represents 179,080 elements. . . . . . . . . . . . . . . . . . . . . . . . 99
A.4 Measurement of speed of tracer particle in a flow with inlet speed of
2.4 mm s?1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
A.5 Non-normalized data for speed of tracer particle in a flow with inlet
speed of 3.2 mm s?1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A.6 Non-normalized data for speed of tracer particle in a flow with inlet
speed of 4.8 mm s?1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A.7 Speed of droplet with initial Capillary number of 0.12 compared with
the lubrication theory prediction. . . . . . . . . . . . . . . . . . . . . 102
A.8 Speed of droplet with initial Capillary number of 0.15 compared with
the lubrication theory prediction. . . . . . . . . . . . . . . . . . . . . 102
A.9 Speed of droplet with initial Capillary number of 0.60 compared with
the lubrication theory prediction. . . . . . . . . . . . . . . . . . . . . 103
A.10 Speed of droplet with initial Capillary number of 0.77 compared with
the lubrication theory prediction. . . . . . . . . . . . . . . . . . . . . 103
B.1 RMS velocity in the domain at Rec = 50 and ? = 0.10 for two levels
of domain discretization. Red is at 1176 X 84 resolution and green is
at 980 X 70. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
B.2 Initial configuration for the capillary-driven flow problem in [1] . . . . 106
B.3 Capillary-driven breakup in a slender drop at Rc ? 0 and ? = 0.05.
Figure is reproduced from Stone et al. [1] with permission. . . . . . . 107
B.4 (a) to (k) show the capillary-driven breakup in a slender drop at
Rec = 100 and ? = 0.05. Comparing with figure at Rec ? 0, only
five daughter drops were obtained at steady state (k). . . . . . . . . . 108
xi
B.5 Steady state after capillary-driven breakup at Rec = 250 and ? = 0.05
producing four daughter drops. . . . . . . . . . . . . . . . . . . . . . 109
B.6 Steady state after capillary-driven breakup at Rec = 500 and ? = 0.05
producing three daughter drops. . . . . . . . . . . . . . . . . . . . . . 109
B.7 Transient flow in a slender drop with a large radius of curvature near
the tips. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
B.8 Inertial capillary flow in a slender drop with aspect ratio = 10 resulting
in entrapment of the surrounding phase. . . . . . . . . . . . . . . . . 112
B.9 Formation of a torus structure in the transient flow. In the bottom
frame, the torus is surrounded by two daughter drops. The initial
slender drop had an aspect ratio = 30. For the inertial-capillary flow,
Rec = 100 and ? = 100. . . . . . . . . . . . . . . . . . . . . . . . . . 113
xii
Chapter 1: Introduction to the dissertation
This dissertation is a detailed study of extensional drop deformation
and capillary-driven breakup at low Reynolds number. Many industrially
relevant phenomena such as preparation of emulsions, dispersive and distributive
mixing of liquid additives, and incorporating immiscible liquids into additive
manufacturing applications involve deformation and breakup of drops. Extending
our understanding of the dynamics behind such processes will be a useful scientific
endeavor and will benefit many industries. In this regard, this dissertation revisits
the classical problem to better understand it.
Extensional deformation of drops at low Reynolds number was formally studied
first by G. I. Taylor [4] in a device called the four-roll mill. It is a testament to
the usefulness and the simplicity of the device that many classical papers [1, 2, 5]
over the decades have used the device to study drop deformation in stagnant
extensional flows. Among them is a paper by Grace [5] which first published the
so-called Grace curve ? a widely used measure in mixing industries ? to characterize
the critical capillary number at different viscosity ratios. While extensional flows
with a stagnation point find better use in labs, non-stagnant extensional flows are
more common in industries than stagnation point extensional flows. Further, the
1
applicability of the grace curve has been questioned [6] under certain circumstances
and in this dissertation, the problem is revisited using a new approach. Drop
behavior in non-stagnant extensional flows are studied over a wide range of capillary
numbers both above and below the critical capillary number. A unique channel
design that can imposing a constant extension rate is introduced. Thus, the
experiments in this dissertation are similar to the classical studies [1, 2, 5] but
is easily translatable to an industrial setting.
Lubrication theory [7, 8] is used to understand the non-stagnant extensional
flow in the channel. While single phase flows are typically studied using lubrication
theory, experiments in this dissertation involve a drop deforming in the flow. Thus,
it is important to fully understand the effect of the drop present in the flow and the
implications of applying the theory. An objective of this dissertation is to apply the
theory to predict some anomalous behavior of drops in different parts of the flow
and compare drop velocity to the lubrication theory prediction.
Drop deformation at low Reynolds number is primarily caused by application
of external viscous force. When the force is removed, the tendency of drops is to
retract into a sphere or breakup into many drops. While the study of such transient
behavior is a classical problem [1, 9] by itself, it fits into a general parametric space
comprising of viscosity ratio and Reynolds number. Different parts of this space has
received attention from researchers over the years [10, 11, 12, 13, 14, 15] who have
all considered effects of viscosity ratio and inertia separately. In line with keeping
this study relevant to industrial applications, another objective of this study is to
explore behavior of slender drops in the new part of the parametric space.
2
The dissertation is organized into four independent studies. In chapter 2, the
behavior of drops in a non-stagnant extensional flow is analyzed and a new non-
dimensional measure is introduced. Chapter 3 is a verbatim reproduction of an
article published in Physics of Fluids [16] and rationalizes a previously unobserved
behavior of velocity lag in drops undergoing non-stagnant extension. In chapter
4, a model is developed based on lubrication theory and validated to predict drop
deformation under shear-influenced deformation. Chapter 5 focuses on capillary-
driven breakup of slender drops in a previously unexplored part of the parametric
space. Finally, the appendix provides supplementary data from experiments and
computations, and presents some non-trivial drop behavior in different domains.
3
Chapter 2: A new measure for drop deformation in extensional flows
at low Reynolds number
2.1 Abstract
The critical capillary number of immiscible drops, which represents the state
where the drop cannot be stable in a surrounding flow, is usually determined in low
Reynolds number extensional flows by suspending them at the stagnation point of a
four-roll mill device and subjecting them to different steady states. In this study, we
subject drops to non-stagnant extensional flow in a converging channel and present
a new measure ? called the semi-minor capillary number ? to describe the drop
deformation process. The measure uses the instantaneous semi-minor dimension of
the deforming drop as the length scale in calculating the capillary number. Our
experiments at small initial capillary numbers, compared to the critical capillary
number, yielded steady drops with a constant value of semi-minor capillary number.
For large initial capillary numbers, the same constant represented an asymptotic
limit of the self-similar deformation. The new measure of semi-minor capillary
number thus helped rationalize drop behavior at both small and large initial capillary
numbers. More importantly, it provided significance to drop behavior at large initial
4
capillary numbers which is a largely unstudied parametric space in the context of
determining the critical capillary number. Finally, we present the critical semi-minor
capillary number at different viscosity ratios and discuss its significance.
2.2 Introduction
Understanding and improving mixing of two immiscible liquids is of interest to
a range of industries that manufacture food products [17, 18], cosmetics [19], paints
[20], and even vaccines [21]. Studying the fundamental fluid dynamics problem
behind immiscible fluid mixing ? dispersion of a single drop of fluid into another
? allows for process improvement at various scales that are of interest to these
industries. In this chapter, we experimentally study the fundamental problem
and unveil a new non-dimensional measure that predicts drop deformation in such
systems.
Assuming both fluids are Newtonian and incompressible, the problem involves
a drop of viscosity ?? and density ?? that is immiscible in a second fluid of viscosity
? and density ?. The initial radius of the drop is assumed as a, and the interfacial
tension between the fluids is assumed as ?. Making a low Reynolds number
approximation for both the fluids, the motion is governed by the incompressible
Stokes equation and continuity equation both inside and outside the drop. At the
boundary of the drop, continuity of velocity and tangential stress between the two
fluids is assumed. Due to the interfacial tension, the normal stress undergoes a
jump across the interface and the shape of the drop can be described by the non-
5
dimensional boundary condition [8]
?p?? 1p+ [(? ? ??? ) ? n] ? n = (? ? n). (2.1)
Ca
Here, ? and ?? are the dimensionless stress tensors, and p and p? are the dimensionless
fluid pressures with capped variables corresponding to the fluid inside the drop.
The ratio of viscosity of the two phases is ? (= ??/?) and n is the normal vector
to the interface directed outwards from the drop. Thus, ? ? n is the curvature
of the interface at the base of the normal vector. Ca is the capillary number ? a
dimensionless quantity ? defined as the ratio of external viscous force on the drop to
the interfacial force holding the shape together. The balance between the two forces
dictates the shape of the drop in the flow. Capillary number is given by equation
2.2
?V
Ca = , (2.2)
?
where V is a velocity scale of the external flow that causes motion of the drop.
While the description of the problem is complete, the drop behavior in this system
is non-trivial and many approaches have been used to study the problem [22, 23,
24]. Small deformation analyses [25, 26, 27, 28], slender body theories [29, 30, 31],
and numerical methods [9, 32] all represent different attempts to uncover the fluid
dynamics of deforming drops. Theoretical methods typically use known solutions of
low Reynolds number flows to seek details about the flow. When detailed solutions
provided by such theoretical methods are not needed, experimental methods [1,
6
2, 4, 5, 33] have provided useful insights. Visualizing flow fields within deforming
drops [34, 35] helps understand the effect of external flow on drop motion. While
such experiments are useful visualization exercises, a particular class of experiments
[1, 2, 5, 33] aimed to classify drop behavior into stable and unstable regimes and
were based on direct probing of equation 2.1.
From equation 2.1, it can be seen that only a small change in ? ? n is needed
when the capillary number is small. This means that the deviation in shape of the
drop from a sphere is small for small capillary numbers and is consistent with the
fact that the interfacial tension force that resists deformation is high. The deviation
of drop shape from sphericity also increases with increase in capillary number as per
the same equation. This inspires an experiment where the capillary number can be
systematically varied and the critical capillary number (Cacrit) ? where no stable
shape can exist ? can be discerned. Many experiments [1, 2, 5, 33] accomplished
this is by subjecting the drop to increasing amounts of shear rate ??, starting from a
low value, in a linear flow and found the condition where the drop became unstable.
The velocity scale in equation 2.2 was written as a?? making the capillary number
?a??
Ca = . (2.3)
?
The experimental approach did not necessitate subjecting the drops to shear rate
beyond the critical value since it was generally accepted that large capillary numbers
always made the drop unstable. This unstable behavior of drops has been studied
with a focus on breakup mechanisms after flow stoppage [36, 37], but without a
7
focus on the drop behavior during deformation. Since the regime of large capillary
numbers was unexplored from the perspective of understanding transient drop
deformation, we aimed to fill this gap in literature and sought a generalized mode
of drop behavior over all capillary numbers ? both above and below the critical
capillary number. To accomplish this, we devised a new measure called the semi-
minor capillary number (Casm) defined as
?b??
Casm = , (2.4)
?
where b is the instantaneous semi-minor dimension at the center of mass location
of a deforming drop. Thus, Casm represents the ratio of external viscous force
and the interfacial force on a drop of radius b. Since drop deformation at any
capillary number is always characterized by a reduction in semi-minor dimension,
Casm can potentially capture drop behavior over all capillary numbers. To study
the variation in Casm at various capillary numbers, we subjected drops to steady
extensional rates for a finite time in a non-stagnant extensional flow. These flows
have been investigated before [38, 39, 40] but in channels that did not impose a
steady extensional rate. Achieving a steady extensional rate was important to
maintain consistency with the methods in literature [1, 2, 5, 33]. We thus designed a
flow converging channel whose spanwise width reduced as inverse of the streamwise
coordinate [41, 42, 43, 44] which rendered the extensional rate along the centerline
of the channel near constant [7, 16]. A wide range of extensional rates were obtained
by simply controlling the flow rate in the channel. We studied the variation of Casm
8
of deforming drops in the flow and discovered the existence of a unique Casm that
fully captured drop behavior over all capillary numbers.
The chapter is organized as follows. In section 2.3 we describe the experimental
setup, fluids used in the study, our experimental methods, and the data analysis
technique. Next, in section 2.4 we analyse drop behavior at various capillary
numbers and discuss them in three parts. First, we describe drop behavior in the
channel at small capillary numbers which revealed the existence of a critical Casm.
This is followed by discussion of drop behavior at large initial capillary numbers
where the critical Casm represented an asymptotic limit. Finally, we use data from
literature to plot and discuss the critical Casm over different viscosity ratios. In the
conclusions section, we assert that Casm provides kinematic significance to the drop
deformation phenomenon in the parametric space of large capillary numbers and
discuss the significance of the metric in industrial applications.
2.3 Experiments
A 3D model of the flow channel used in this study is shown in figure 2.1. The
channel had two bounding walls parallel to each other and the other two bounding
walls slowly varying as f(x) starting from the indicated origin. The fluid flow was
streamwise along positive x direction.
9
120 20
?????
70
? ?
? ?
600
160 ? = ? + 10
(?) (?) (?) ??????
Figure 2.1: (a) Front view, (b) side view, and (c) isometric view of a planar
channel with flow streamwise along x. All dimensions are in mm. The origin of
the coordinate system is equidistant from the two planar bounding walls.
The channel was machined from a transparent block of acrylic and was
supported using an aluminium frame. A reciprocating piston was used to push
the bulk fluid through the channel, and a rack and pinion mechanism was used to
control its motion. An Arduino was used to control the input voltage to the motor
coupled with the pinion gear. This regulated the linear speed of the piston and
allowed fluid flow rate control in the channel. From the outlet of the channel, a pipe
channeled the liquid into a long column where static pressure of the liquid head
ensured a steady state of the fluid in the flow channel. The experimental setup is
shown in figure 2.2(a). The channel had an injection port, as seen in figure 2.2(a),
that allowed for the injection of a single drop of an immiscible fluid using a syringe
into the bulk flow. The injection port consisted of a resealable membrane which
10
ensured no fluid leaked out of the channel during the drop injection process.
?????????
?????
???? ????
?
?????????????
?????? ?
???? ?????????
????
???? ???????
???? ?????? ???
?
?????? ?????
?? ???? ?
?
?????? ????
(?)
50 ??
(?)
Figure 2.2: (a) The experimental setup showing the planar converging channel and
the flow control mechanism. Flow rate of Castor oil is controlled through the motion
of the reciprocating piston into the channel. (b) shows the schematic of an initially
spherical drop undergoing deformation due to the external flow. L is the semi-major
length of the deformed drop and b is the semi-minor length (also referred to as the
width).
The fluids used in this study were Castor oil (procured from Bulk Apothecary)
as the bulk fluid and Silicone oil (procured from MicroLubrol?) as the drop. The
viscosity of Castor oil was 0.72 kg m?1 s?1 and density was 961 kg m?3. The viscosity
of Silicone oil was 0.20 kg m?1 s?1 and density was 970 kg m?3. The viscosity
ratio for the system was thus 0.28 and the interfacial tension between the two
phases was approximately 0.004 N m?1. The main experiment involved studying
the deformation of the Silicone oil drop in Castor oil flow through the channel.
11
After injecting the drop into the static Castor oil in the channel, the piston was
slowly moved so that the drop appeared in the field of view of the camera. At this
position, the piston motion was stopped, and the drop was allowed to stabilize into
a sphere. For all experiments, the initial location of the drop was chosen as (30 mm,
0, 0) as per the coordinate system indicated in figure 2.1. The motion induced by
the small difference in density between the two fluids was insignificant compared
to the flow rate imposed and it allowed us to neglect any effect of gravity on the
experiment. The Arduino was then programmed to supply a definite amount of
voltage to the motor which set the flow rate in the channel. Using the initial radius
of the drop and the extensional rate [16], the initial capillary number (Cai) of the
drop was calculated using equation 2.3. A camera (Casio Exilim Pro EX-F1) was
used to capture images of the drop as it translated and deformed in the flow field.
Due to the symmetry of the channel geometry, the drop was restricted to flow along
the centerline. Figure 2.2(b) shows a schematic of drop transition from initial state
to a deformed state due to the influence of external flow.
Using different combinations of initial radius and flow rate in the channel,
we imposed different capillary numbers on the drop. The initial radius of drops
ranged from 1.0 mm to 2.2 mm. Reynolds number for all experiments, calculated
using the drop radius and properties, never exceeded 1.00. For each experiment, the
recorded images of drop deformation in the flow were analysed using a custom-built
MATLAB program which extracted the end-to-end length of the drop and its width
at the center of mass location. The half-width was used in equation to calculate
Casm for every transient drop shape. Note that Casm = Cai for the initial shape.
12
2.4 Results and Discussion
Central to our experimental methods was the ability of the flow to impose
a near constant extensional rate along the drop trajectory. We confirmed this by
evaluating the x component of velocity along the centerline of the channel using
lubrication theory. The velocity was validated experimentally, and with numerical
simulations [16]. Armed with the relationship between extensional rate and the flow
rate of the single-phase flow, we injected drops into the flow and subjected them to
various capillary numbers (Cai).
In this section, we first present drop behavior in two domains ? below and
above the critical capillary number ? and show the significance of the new measure
of Casm. Then we use data from literature and present the critical Casm at different
viscosity ratios and discuss its significance.
2.4.1 Drop deformation at Cai < Cacrit
The critical capillary number (Cacrit) for pure extension of drops of viscosity
ratio 0.28 is approximately 0.15 [2]. When the imposed Cai was less than Cacrit,
the viscous stress on the drop caused the drop to change its curvature at every
point according to equation 2.1. Since the flow was associated with a pressure
gradient in the streamwise direction, pressure outside the drop near its leading end
in the flow was larger and caused the drop curvature at that point to increase.
This in turn caused a reduction in width of the drop at its center of mass due to
conservation of drop volume ? resulting in a decrease in curvature at that point
13
in the xy plane. When the imposed capillary number was low, small changes in
curvature were sufficient for the interfacial stress to fully balance the viscous stress
imposed by the outside fluid and the drop attained a stable shape. We subjected the
drop to many such steady states at Cai < Cacrit. Figure 2.3 shows an experiment
with Cai = 0.12.
10 ?? (?) (?) (?) (?) (?) (?)
Figure 2.3: Drop deformation at Cai = 0.12. (a) is the initial state where the flow is
initiated. (b) to (f) shows subsequent deformation. The drop deforms and quickly
attains a steady prolate spheroidal shape.
After the flow onset, the drop transformation from a sphere to a prolate
spheroidal shape was quantified by measuring the drop L/b, where L is the semi-
major length and b is the semi-minor length of the spheroid. Attainment of steady
states for other Cai < Cacrit in time is shown in figure 2.4. It can also be seen from
figure 2.4 that around Cai = 0.19, L/b continuously increased in the flow consistent
with the behavior slightly beyond the critical capillary number.
14
Figure 2.4: Drop deformation at various Cai versus ??t where ?? is the extensional rate
and t is time. Note the attainment of steady states for low Cai and the continuous
deformation for Cai > Cacrit. Sample error bar is shown for Cai = 0.14.
Since a prolate spheroid is associated with curvature along two planes at every
location along its major axis, the decrease in width of the drop at the center of mass
location had an unintended consequence. As the width reduced, the curvature in
the yz plane ? visualized in figure 2.2(b) using a dotted line on the deformed shape
? increased and caused the pressure associated with interfacial tension to increase.
This increase in pressure on the inside of the drop prevented further reduction in
width. Thus, as the Cai increased, the degree of reduction in width reduced. A
state was reached where further increase in extensional rate on the drop only caused
a smaller decrease in width ? compared to a larger reduction in width at lower Cai.
Thus, the product of extensional rate and instantaneous semi-minor dimension was
rendered constant for Cai larger than this critical value. This implied that the
15
capillary number associated with the semi-minor dimension (Casm), as calculated
from equation 2.4, was also constant for these cases.
The attainment of a constant value of Casm for various steady states was better
visualized by tracking Casm for every Cai. Figure 2.5 shows other Cai < Cacrit
where all drops attained steady shapes that corresponded to the same critical value
of Casm ? 0.11. Note that each curve represents an experiment starting from an
initial state of a spherical drop with L/b = 1 and ending at deformed shape where
L/b > 1. Since Cai < Cacrit all yielded steady shapes, note that all data points
become concurrent after a modest degree of drop deformation.
Figure 2.5: Tracking the L/b of deforming drops as a function of Casm. Each curve
represents a single experiment carried out at a fixed Cai. Drops start from an
initial state of L/b = 1 where Cai = Casm. Subsequent deformation is tracked till
drops reached a steady state for Cai < Cacrit, or till drops exit from the flow for
Cai > Cacrit. The vertical dashed line is the lower bound (and a unique value for a
given viscosity ratio) of Casm for all Cai. Sample error bar is shown for Cai = 0.14.
Further evidence for the existence of a constant Casm at Cai < Cacrit was
16
found by analysis of data from [2]. In their comprehensive study of drop behavior
at various initial capillary numbers, the critical capillary number was determined
as the Cai state where a stable drop did not exist. However, their definition of Cai
was based on a length scale equal to the initial radius of the spherical drop. From
a volume conservation calculation, we determined the instantaneous width of the
drop at every steady state and used it in equation 2.4 to calculate Casm. Figure 2.6
shows a plot of Casm versus Cai for three viscosity ratios from [2].
Figure 2.6: Data extracted from [2]. The plot shows Casm at steady states calculated
for various Cai. The three curves represent different viscosity ratios (?). Note that
as Cai is increased, Casm becomes independent of Cai and is referred to as the
critical Casm. The largest Cai for each viscosity ratio is the Cacrit.
For Cai << Cacrit, note that the Casm increases with increase in Cai. This is
consistent with drop behavior at small degrees of deformation where the curvature
in the yz plane is not sufficient to oppose further reduction in width. However, after
a certain value of Cai was exceeded, further increase in Cai rendered Casm constant
17
(a value we refer to as the critical Casm) as evident from figure 2.6. We obtained a
critical Casm = 0.11 for our system with viscosity ratio of 0.28. Our next goal was
to discern the role of critical Casm when the imposed capillary number was large.
2.4.2 Drop deformation at Cai > Cacrit
In this parametric space, the viscous stress on the drop was much larger than
the interfacial stress that held the drop together. However, the dynamics of drop
deformation explained in section 2.4.1 still applied. Drops at large Cai reduced in
width by a large amount compared to their steady state counterparts at low Cai.
As long as the drops were subject to flow in the channel, they continued to deform
and attained long shapes. Figure 2.7 shows an experiment at Cai = 0.60 where the
drop never stopped deforming in the flow.
10 ?? (?) (?) (?) (?) (?) (?)
Figure 2.7: Drop deformation at Cai = 0.60. (a) is the initial state where the flow is
initiated. (b) to (f) shows subsequent deformation. The drop deforms continuously
in the flow and no steady state is observed.
18
As we subjected drops to many Cai > Cacrit and plotted the change in shape
over time, we discovered a self-similar behavior. As seen in figure 2.8, every Cai
followed the same trajectory for L/b vs ??t where ?? is the extensional rate and t is
time.
Figure 2.8: Self-similar behavior of drops at Cai > Cacrit.
Since the drop deformation process at large Cai was self-similar, an upper
bound for L/b existed regardless of Cai > Cacrit. The reason for existence of an
upper bound was due to the diminishing duration of time spent by the drop in the
flow with increase in flow rate. Notwithstanding this, the self-similar curve in figure
2.8 uniquely characterized the drop behavior at large Cai.
Further, figure 2.9 shows that the behavior at large Cai nearly follows the
transformation from a sphere to a prolate spheroid. In figure 2.9, L/b ? b??3
represents a volume conservation between a sphere and a prolate spheroid, and
19
L/b ? b??2 represents an area conservation between a circle and an ellipse. For
similar L/b, the latter transformation causes larger reduction in width compared to
the former. It can be seen that the deforming drop deviated from both the curves.
The deviation from L/b ? b??3 observed for long shapes was due to the planar nature
of the flow where the drop narrowed more in the xy plane than the xz plane. Since
the semi-minor dimension was only tracked in the xy plane, the drop transition
deviated from the L/b ? b??3 behavior in figure 2.9. Despite the larger reduction in
xy dimension, the drop behavior was not consistent with L/b ? b??2 and the reason
will be explained ahead.
Figure 2.9: Kinematics of drop deformation from initial state (L/b = 1) to deformed
state. b? = b/a where a is the initial radius of the drop.
We also tracked Casm of deforming drops at Cai > Cacrit already shown in
figure 2.5. Since the extensional rate on drops was large, the curvature at the leading
end increased more than the case with Cai < Cacrit. The semi-minor dimension
20
continued to reduce as L/b ? b??3 but the process suffered with an increase in yz
curvature at the center of mass location. Increasing curvature prevented a large
reduction in width (L/b ? b??2) and thus the drop shape transition did not remain
consistent with the green solid line in figure 2.9. The larger slope of the experimental
data in figure 2.9 compared with the L/b ? b??2 lead to the conclusion that the semi-
minor dimension of the drop nearly stopped changing as the semi-major length of
the drop continued to increase. This is possible for slender drops since a small
reduction in width can cause a large increase in length. We extracted the cross-
sectional profile of a deforming drop from the experimental data at Cai = 0.44
and is shown in figure 2.10. Note the attainment of a state towards the end where
the semi-minor dimension b? at the center of mass location of the drop nearly stops
changing while the semi-major dimension L? continues to increase.
21
Figure 2.10: Instantaneous cross sections in the xy plane of a deforming drop at
Cai = 0.44. b? = b/a and L? = L/a where a is the initial radius of the drop. (a) is the
initial state and (b) to (j) shows the subsequent deformation in time. Note that b? at
the center of mass location changes by a small amount with increasing deformation.
Despite the large deforming viscous forces on the drop, the semi-minor
dimension cannot reduce to zero. Before this stage is reached, daughter drops
pinch-off from the parent slender drop in a process called elongative end pinching,
but the study is beyond the scope of this chapter. However, it was natural to expect
a lower bound for Casm based on the dynamics of drop deformation within the
explored parametric space. From the discussion in section 2.4.1, the obvious choice
for the lower bound was the critical Casm as interpreted in that parametric space.
If the drops spent longer time in the flow, the experimental data for all drops at
Cai > Cacrit in figure 2.5 would asymptotically reach the critical Casm.
22
2.4.3 Significance of semi-minor length b in extensional deformation
of drops
All of the literature on extensional deformation of drops quantify the capillary
number Cai using the initial radius of the drop (a) as the length scale. The
theoretical basis for this is that both small deformation and slender body theories
that predict drop deformation express critical parameters in terms of the initial
radius. For example, in small deformation theory, the slight deviation from
sphericity of the drop is expressed as a deviation from its initial radius a. Similarly,
the slender body theory expresses the slenderness of a drop as L/a. Using a as
a length scale is convenient because it is a truly independent parameter in the
system that can be determined a priori. From an industrial application perspective,
it is also convenient to quantify dispersion metrics in terms of a parameter ? the
initial capillary number ? that can be established before commencing the mixing
operation. However, Stegeman et al. [6] have pointed out certain issues with relying
on the initial capillary number to predict drop sizes after the mixing process. In
particular, the drop size after the mixing process at large capillary numbers were
smaller than the prediction by the Grace curve. The Grace curve quantifies the
critical capillary number Cacrit based on initial radius and provides no information
about the transient sizes of drops undergoing deformation.
Thus, there is a clear need to devise a new metric that considers the kinematics
of drop deformation at any capillary number. As seen in previous sections, using
the semi-minor length b of the drop accomplishes it. Calculating a capillary number
23
using b as the length scale revealed unique trends in behavior both above and below
the critical capillary number.
2.4.4 Critical Casm at different viscosity ratios
Since we established that deforming drops at any extensional rate had a lower
bound for Casm, it was of interest to plot the critical Casm and compare it with
Cacrit that were obtained from other experiments [2, 5]. To accomplish this, we
used data from [2] who studied drop behavior over a wide range of viscosity ratios
(0.001 to 88.6). It was relatively easy to calculate the semi-minor dimension of the
drop at the critical capillary number using a volume conservation calculation. Using
the semi-minor dimension, we calculated the critical Casm as per equation 2.4 and
plotted it over different viscosity ratios in figure 2.11.
Figure 2.11: Cacrit and critical Casm at different viscosity ratios (?) using
data extracted from [2]. The red diamond represents critical Casm obtained
experimentally in this study for ? = 0.28.
24
The critical Casm at every viscosity ratio is lower than the Cacrit. This
was expected because the Cacrit used the radius of an undeformed drop a as the
length scale while Casm is based on the instantaneous semi-minor dimension b of a
deforming drop. Thus, at a given viscosity ratio, critical Casm < Cacrit.
2.5 Conclusion
The aim of this work was to investigate drop deformation at small and large
capillary numbers compared with the critical capillary number to find a universal
metric that could characterize drop behavior in the entire parametric space of
capillary number. While the critical capillary number is useful in classifying drop
behavior into stable and unstable regimes, it does not provide any insight into drop
dynamics at super critical capillary numbers. Our work addressed this by devising a
new measure called the semi-minor capillary number. The measure took advantage
of a universal drop behavior ? exerting steady extensional strain on drops causes a
monotonic reduction in their semi-minor dimension. We showed that drops at both
small and large capillary numbers all attained the same critical value of semi-minor
capillary number.
The flow channel used in the study was able to impose small and large
extensional rates by simple flow control. The unique design imposed a near constant
extensional rate on suspended drops. Such non-stagnant extensional flows produced
in the channel are commonly used for extensional mixing of fluids in industries
and figure 2.8, which shows the degree of drop deformation as a function of net
25
extensional strain, is useful data to design such mixers.
Finally, figure 2.11 which shows the critical semi-minor capillary number over
several viscosity ratios is significant in many ways. While the critical capillary
number curve ? commonly referred to as the Grace curve ? is useful to find the
critical initial size of drops in a flow, it provides no information about the transient
shapes that those deforming drops attain. Further, if a mixing process involves
super critical capillary numbers, the Grace curve can only be used to estimate the
daughter drop sizes after breakup of the slender thread. If estimates are needed
for the minimum drop size attainable at any capillary number before breakup, the
critical semi-minor capillary number curve is the only useful metric.
26
Chapter 3: Velocity of suspended fluid particles in a low Reynolds
number converging flow
3.1 Abstract
We studied a pressure driven, low Reynolds number fluid flow through a planar
channel whose spanwise width along the flow varied inversely as the streamwise
coordinate, such that the extensional rate on the centerline was near constant. The
effect of the near constant extensional rate on an immiscible droplet of Silicone
oil was studied by tracking its deformation. The droplet rapidly deformed into
an ellipsoid and displayed a consistent lag velocity compared to the single-phase
background flow at the same point. The observations were attributed to flow
induced deformation of the immiscible droplet which was a function of the magnitude
of the initial Capillary number. The streamwise component of the single phase
velocity along the centerline of the converging flow was also estimated to leading
order using lubrication theory. The estimated velocity compared favorably with
numerical simulations; and validation with experimental measurement of the flow
of Castor oil through the channel by tracking tracer particles. The accuracy of
the determination of the velocity field by the lubrication theory allowed for the
27
careful measurement of the velocity difference between the drop and suspended fluid
velocities. This research validated lubrication theory predictions of flow velocity
through a converging channel, and provided an experimental insight on behavior of
a suspended phase.
3.2 Introduction
Fluid motion at low Reynolds number through channels with narrow
geometries can be described by an approximation of the Navier Stokes equation,
called the lubrication approximation. The lubrication approximation assumes a
narrow channel gap where the distance along the direction transverse to the flow
is much smaller than the characteristic distance along the flow direction [8]. The
study of the effect of such flows - which is associated with large velocity gradients
along certain directions - on a suspended phase is motivated by a unique application
in polymer based additive manufacturing, where polymer flow through nozzles can
be described by the lubrication approximation [45].
Narrow channels are ubiquitous in applications that utilize changes in
geometry to manipulate fluid motion. These changes include the use of converging-
diverging sections [44, 46, 47, 48], corrugated channel walls [49, 50], serpentine
channels [51, 52, 53] and deformable channels [54, 55, 56]. Confined flows such as
flow around a corner [57, 58] have been treated using asymptotic techniques and have
revealed the existence of certain unique flow features that arise due to the geometry
of the flow. The single phase flow of a fluid through a channel undergoing geometric
28
changes - like varying cross section - can usually be estimated using a known solution
for a closely related simple geometry [59, 60, 61] or solved using lubrication theory
[7, 62, 63]. The latter is a useful tool which uses asymptotic techniques and provides
reliable results even at low orders of approximation. Lubrication theory has also
been extended to higher orders of approximation [64], widening it?s applicability to
a larger class of single phase flows.
On the other hand, the study of multiphase confined flows is motivated by the
ability of the bulk phase to affect the motion of the dispersed phase. Of particular
interest is the case where the interface between the two phases is deformable.
Examples include the motion of a deformable capsule in a corner flow [65], migration
of particles towards walls of a deformable channel [66], and away from a flexible
membrane suspended in the flow [67]. Particle motion in the vicinity of the walls
of the channel can be studied using lubrication approximation [68] in the context
of the varying gap between a wall and the curvature of the solid as it approaches
the wall. Motion of solids with zero local curvature at the minimum gap to the
wall has also been studied [69]. Related to the problem, another class of multiphase
flows that is studied using lubrication theory is the transport of a densely filled solid
system [70] where the approximation can be applied due to the narrow gap between
the particles. However, when the particle concentration in the fluid is sparse, the
motion of the particle is affected by one way coupling between the fluid and particle,
and has been studied in converging-diverging flows [46]. Such flow channels can also
impose velocity gradients which can be used to deform a suspended particle.
Unique, and controllable, velocity gradients can be engineered by designing
29
bounding walls of the channel based on requirement. For a symmetric, converging
channel, an extensional rate can be imposed along the centerline in the streamwise
direction which can lead to elongation of a suspended, deformable interface [38, 39].
Multiple studies [42, 43, 71, 72] have shown that the extensional rate can be near
constant if the spanwise width varies inversely as the streamwise coordinate. The
near constant extensional rate can be used to stretch a small but deformable particle
suspended in it. Neglecting inertial effects at low Reynolds number, the degree of
deformation of an immiscible droplet with radius a depends on the initial Capillary
number (Ca) defined as
a???
Ca = (3.1)
?
where ?? is the near constant extensional rate imposed by the outer fluid for a fixed
flow rate, ? is the viscosity of the outer fluid, and ? is the interfacial tension between
the two phases. The droplet is assumed to be of sufficiently small volume fraction
so that it?s behavior in the flow solely depends on the streamwise velocity gradient
imposed by the bulk phase. For Capillary number below a critical value, the droplet
attains a shape that is stable in the imposed velocity gradient. However, exceeding
the critical Capillary number will cause the droplet to continuously elongate and
lead to eventual breakup. Classical experiments [1, 4, 5] have characterized the
critical Capillary number as a function of viscosity ratio in extensional flows.
The extensional rate imposed by the bulk phase can be studied using
lubrication theory. An analysis by Lauga et al.[7] used lubrication theory to describe
the low Reynolds number flow of a fluid through a narrow channel with rectangular
30
cross section where two bounding planes were parallel, and the other two bounding
planes were slowly varying. They derived the equations for flow and showed that it
is fully three dimensional even at leading order.
In the present work, we study a low Reynolds number flow through such
a planar converging channel. The width of the channel varies as inverse of the
streamwise distance which imposes a near constant extensional rate along the
centerline. The lubrication approximation used by Lauga et al. [7] is used to predict
the velocity field. We use numerical simulations of the low Reynolds number flow to
validate the predictions of the lubrication theory. The velocity field is also measured
experimentally by suspending a small solid particle in the flow which acts as a tracer
to measure fluid velocity.
Conversely, when the size of the particle is larger, Faxen?s law predicts a
relative translational velocity between the particle and the single phase velocity at
the same point. The relative velocity arises due to the force exerted by the bulk flow
on the particle. In this study, we explore the effect of the viscous forces exerted by
the flow on a deformable particle where we present experiments performed on the
deformation of a single immiscible droplet suspended at the center of the channel.
While the centerline of the flow is not associated with any transverse velocity due
to geometrical symmetry, the existence of the constant streamwise velocity gradient
- the constant extensional rate - will cause the droplet to stretch. These problems
have applications in polymer based additive manufacturing [73] where the viscous
flow of the polymer through narrow die geometries can be described by lubrication
theory, and the effect of the flow on a suspended phase [74] needs to be understood.
31
Figure 3.1: (a) Front view and (b) isometric view of a planar channel with flow
streamwise along x. The spanwise width reduces inversely as the streamwise
coordinate. All dimensions are in mm. The origin of the coordinate system is
equidistant from the two planar bounding walls. Half-width of the channel at the
outlet (where x = 160 mm) is 3.53 mm. We use an axial length scale ? = 160 mm,
and a transverse length scale h = 10 mm.
3.3 Lubrication analysis for a planar converging flow
We consider a pressure driven, low Reynolds number flow through a planar
converging channel shown in Fig. 3.1. The flow is streamwise along x direction and
the converging section has a wall profile based on the equation f(x) = 600/(x+ 10).
The channel is oriented with the streamwise direction along gravity.
Lauga et al. [7] posit the use of such planar channels, having streamwise
32
variations in cross sectional area, for mixing at low Reynolds numbers. Using
lubrication theory, they showed that the velocity field is fully three-dimensional
even at leading order which is advantageous for geometry induced mixing of species.
We validate their analysis by studying a single phase flow in the channel shown
in Fig. 3.1, and extend their study by investigating a two phase flow effect. As
per the analysis of Lauga et al. [7], the governing equations of fluid flow are the
incompressible stokes equation and mass conservation ?
?p = ??2u, (3.2)
? ? u = 0. (3.3)
In equations 3.2 and 3.3, p is the pressure, u is the velocity field, and ? is the
viscosity of the fluid. If ? is an axial length scale and h is a transverse length
scale, then  = (h/?) 1 means that dimension along transverse direction is much
smaller than the dimension along the flow direction. This is a common assumption
for lubrication theory to be applicable. Using these scales, the x, y and z lengths
were non dimensionalized by Lauga et al. [7] as
(x, y, z) = (?x?, hy?, hz?). (3.4)
Further, u and p, the velocity component along x and pressure respectively were
non dimensionalized as
Q
u = u? (3.5)
h2
33
??Q
p = p? (3.6)
h4
where Q is the flow rate in the channel. Using the techniques of asymptotic analysis,
Lauga et al. [7] solved for the leading order non dimensional u component of velocity
as
{
1 dp? ?0 4(?1)n cosh
u?0(x?, y?, z?) = (z?
2 ? 1) + ( ( ) }kny? ) cos knz? , (3.7)
2 dx? k3
? n cosh kn 0 nf(x?)
where p?0 is the leading order solution for the pressure and k = (n +
1
n )?.2
Further, the pressure gradient along the x direction was obtained as
{ ? ( ) }tanh k f(x?) ?1dp?0 3 6 n
= ? 1 . (3.8)
dx? 4f(x?) f(x?) k5
n?0 n
In this study, we focus on the centerline of the flow (y?, z? = 0) in the flow
channel shown in Fig. 3.1. The wall profile in terms of non dimensional variables
was f(x?) = 6/(16x?+ 1). Substituting this in Equation 3.7, we get
{ }
1 dp? ? n0
u?0(x?) = ?
4(?1) 1
1 + 6 (3.9)2 dx? k3
? n cosh kn 0 n 16x?+1
where dp?0/dx? is obtained by substituting f(x?) = 6/(16x?+1) in equation 3.8. dp?0/dx?
and u? = u?0(x?) were numerically truncated at n = 50 (same as Lauga et al. [7]). The
solid line in Fig. 3.2 shows a plot of u? vs x? and the inset plot shows the variation
of ?u?/?x? along x?.
The plot indicates that the leading order streamwise component of velocity
34
Figure 3.2: Non dimensional velocity (u? = u/(Q/h2)) vs. Non dimensional
streamwise coordinate (x? = x/?) along the centerline of the flow obtained using
lubrication approximation (solid line) and numerical simulation (dashed line). Error
bars on experimental data represent standard deviation from the mean value. Inset
graph plots the slope of the lubrication approximation and is equivalent to the
variation of extensional rate.
35
increases approximately linearly in the flow along the centerline. This implies that
the extensional rate (or ?u?/?x?) is near constant. We used numerical simulations of
the flow to find the velocity along the centerline, and experimentally measured it
by using tracer particles in the flow.
3.3.1 Numerical simulations
Numerical simulations are commonly used to visualise and derive certain
physical quantities associated with fluid flows. We first discretized the three
dimensional geometry of flow using hexahedral elements. Appropriate boundary
conditions were applied and the conservation of mass and momentum equations were
solved using a numerical method. In this study, we used a commercially available
software - ANSYS FLUENT - for the entire operation. The supplementary material
details domain discretization, numerical scheme, and verification of results. A major
theme of this study is to show the effect of the flow, which is initiated from rest,
on a suspended fluid particle. Transient effects of such a flow can be neglected and
our rationale is elaborated in the supplemental material. The dashed line in Fig.
3.2 shows the velocity obtained using numerical simulations and validates the flow
velocity along the centerline obtained from the lubrication theory. A further level of
validation is obtained by experimentally measuring the velocity along the centerline
using tracer particles.
36
3.3.2 Experimental measurement of velocity
Fig. 3.3a shows the experimental setup with the flow channel and the linear
actuator used for flow control. The flow channel was constructed using a transparent
block of acrylic where the flow domain shown in Fig. 3.1 was machined. The channel
was suspended using an Aluminum frame. A linear actuator, programmed to move
at specific speeds, was used to impose specific flow rate of the bulk fluid. The
bulk fluid used in this study was castor oil (Viscosity = 0.72 kg m?1 s?1, density
= 961 kg m?3). At the outlet, a pipe was used to channel the flow into a tank
maintained at the same height as the experimental apparatus. The static pressure
head ensured that gravity did not cause unintended outflow from the flow channel.
A single Low Density Polyethylene (LDPE) pellet was suspended as a tracer
particle into the flow of castor oil through the channel. Density of LDPE is
approximately equal to castor oil and the particle was cylindrical shaped with an
average diameter of 2 mm and length 3 mm. After carefully suspending the single
particle at the center of the channel, the flow was initiated and a constant flow
rate was imposed. Images of the particle in the flow were taken using a camera
(Casio Exilim Pro EX-F1) at 30 frames per second to measure the translation
velocity. The experiment was repeated for three different flow rates with constant
inlet speeds of 2.4 mm s?1, 3.2 mm s?1, and 4.8 mm s?1. This translated to flow rates
of 5760 mm3 s?1, 7680 mm3 s?1, and 11 520 mm3 s?1 respectively. Fig. 3.2 shows the
measured velocity of the tracer particle which approximated the background flow.
The validation exercise showed that lubrication theory could approximate the
37
flow in the channel along the centerline and the equations presented by Lauga et
al.[7] could be used to gain further insights on velocity gradients. It is a useful tool
to model the flow in the channel and resolve their designs. In the next section, we
study the effect of such a flow on suspended immiscible droplets.
3.4 Effect of flow on suspended immiscible droplets
We injected a droplet of silicone oil (Viscosity = 0.2 kg m?1 s?1, Density =
970 kg m?3) with radius around 2 mm into castor oil (Viscosity ratio ? 0.28) using
the droplet injection port. We then actuated the flow of castor oil and moved the
droplet to the approximate location (30 mm, 0, 0) of channel shown in Fig. 3.1 and
the flow actuation was stopped. At this initial location of the experiment, the droplet
was allowed to relax into a sphere. Density difference between the two phases was
small enough to assume no effect of gravity on the experiment and the interfacial
tension was approximately 0.004 N m?1. Initial location of the droplet was chosen
as (30 mm, 0, 0) to avoid subjecting the droplet to the initial spike in extensional
rate seen in the inset plot of Fig. 3.2. The initial radius (a) was measured and the
initial Capillary number was calculated using equation 3.1 before commencing the
experiment. Reynolds number,
?V a
Re = , (3.10)
?
calculated with ? = density of outer fluid (961 kg m?3), ? = viscosity of outer fluid
(0.72 kg m?1 s?1), velocity scale V equal to the inlet velocity, and length scale a equal
to initial radius of the droplet was around 0.04 to 0.16 over different experiments
38
and inlet speeds. At such low Reynolds number, the flow develops to the steady
state value almost instantaneously. An experimental run consisted of subjecting the
droplet to the flow at fixed flow rate. We only subjected the droplet to the flow
from an initial location of x? ? 0.20 to x? ? 0.60 where the extensional rate was
most steady, and avoided the variation near the inlet and the outlet of the channel.
Images of the droplet in motion were captured at 10 frames per second for lower
flow rates and 30 frames per second for higher flow rates. The Capillary number
on the droplet was varied over different experimental runs by keeping the droplet
size constant and varying the flow rate. However, for experiments at the largest
Capillary number, we used slightly larger drop sizes due to experimental constraint
of producing large flow rates. We varied the Capillary number from 0.12 to 0.77 to
illuminate the effect of the initial Capillary number on droplet behavior. Fig. 3.3
shows two sets of experiments at initial Capillary number of 0.15 and 0.60.
The droplet with initial Capillary number of 0.15 in Fig. 3.3 deformed and
quickly assumed a steady elongated shape and translated in the flow field. Capillary
number of 0.15 is thus lower than the critical Capillary number which refers to
condition of continuous deformation under the fixed extensional rate imposed by the
outer fluid. For the viscosity ratio of the fluids we used, the critical Capillary number
for extensional deformation of droplets is approximately 0.25 [1, 5]. However, the
droplet behavior was different for large Capillary numbers. For an initial Capillary
number of 0.60 the droplet continuously stretched in the imposed flow and this
behavior is typical under conditions beyond the critical Capillary number. However,
droplet breakup could not be observed since it quickly left the field of view. The
39
Figure 3.3: (a) The experimental apparatus to measure droplet deformation in the
flow channel. Flow rate of Castor oil is controlled through the motion of the linear
actuator into the channel along the downward direction. The motion of a Silicone
oil droplet (highlighted black circle) in the region of interest (black dotted line) is
captured using high speed photography. Viscosity ratio is around 0.28. Fig. b1 and
c1 shows the initial state for Capillary number equal to 0.15 and 0.60 respectively.
Fig. b2 to b6 and c2 to c6 shows the subsequent deformation induced by the motion.
40
Figure 3.4: L/b vs x? for different initial Capillary numbers (Ca). The x? refers to
the instantaneous center of mass location of the droplet. Note the attainment of
a steady elongated shape for low initial Capillary numbers (hollow symbols) and
continuous elongation for high initial Capillary numbers (solid symbols).
time spent by the droplet in the channel was inversely proportional to the inlet flow
rate. Fig. 3.4 shows the change in L/b, where L is the half length (major axis) and
b is the half width (minor axis) of the ellipsoidal droplet, for four different initial
Capillary numbers.
We also measured the velocity of the droplet by tracking the center of mass.
Fig. 3.5 shows the velocity of droplets at different initial Capillary numbers
compared with the lubrication approximation of the single phase background flow.
For the smaller values of initial Capillary number (Ca = 0.12 and Ca = 0.15),
due to the small particle size and negligible deformation, droplets closely tracked
the background flow at their steady state. However, every droplet lagged the
background flow with the magnitude of lag increasing with initial Capillary number.
41
We explain this observation as a consequence of flow induced deformation which
causes a deforming object to lag the background flow.
Figure 3.5: Comparison of droplet translational velocity and the background flow
velocity. Ca refers to initial Capillary number and x? refers to the instantaneous
center of mass location of the droplet. Note the large initial slope for low initial
Capillary numbers (hollow symbols) compared with the high initial Capillary
numbers (solid symbols).
When the flow was initiated, the extent of droplet deformation was a function
of the initial Capillary number relative to the critical Capillary number. In the case
of small initial Capillary number (Ca = 0.12 and Ca = 0.15) the droplet deformed
quickly to a steady-state shape that was consistent with the balance between viscous
and interfacial effects. The viscous stress on the droplet was the product of the
viscosity of the external fluid and the extensional rate imposed by it, and was
an O(0.1) quantity. However, the interfacial stress that held the shape together
was the ratio of interfacial tension and the initial size of the drop and was an O(1)
42
quantity. Droplets at low initial Capillary numbers thus reached a steady state since
the viscous stress from the outside fluid could not overcome the interfacial stress.
However, in the case of a large initial Capillary number (Ca = 0.60 and Ca = 0.77),
where the external viscous stress was much greater than the interfacial stress, the
minor axis of the droplet (b) continuously reduced until it reached O(bcrit), where
bcrit is the droplet radius associated with the critical Capillary number at that flow
rate. At this point, the minor axis of the droplet asymptotically stopped reducing
and the primary change in dimension of the droplet from this point onwards was
mostly the elongation of the major axis (L). L/b continued to thus increase without
any bound.
In all the cases, as seen in Fig. 3.5, the droplets experienced both a change
in shape and velocity. In the case of small initial Capillary number (Ca = 0.12 and
Ca = 0.15) which is lower than the critical Capillary number, the droplet assumed
a slightly deformed ellipsoidal steady shape. The droplets initially demonstrated
a rapid increase in velocity u? with minimal translation x?. Since the velocity of
the droplet was determined by the location of the center of the droplet, much of
the change of velocity was related to the change of shape. The change of shape is
minimal and the droplet quickly achieved a steady-state. The droplet then closely
lagged the background fluid velocity by assuming a slightly deformed ellipsoidal
steady shape. This effect can be seen more profoundly for a slightly larger initial
Capillary number of 0.17 which is still lower than the critical Capillary number.
Fig. 3.6 shows droplet behavior at Ca = 0.17 and shows the correlation of steady
state attainment with reduction in the magnitude of lag - which is defined as the
43
difference between the background flow velocity and the droplet velocity.
Figure 3.6: Motion induced deformation of droplet at initial Capillary number of
0.17. Crosses represent variation of L/b and circles represent variation of u?. Solid
line is the lubrication approximation of velocity of the single phase background flow.
The droplet attains a steady shape at x? ? 0.3. The x? refers to instantaneous center
of mass location of the droplet.
The droplet behavior beyond the critical Capillary number was similar but
required more interpretation. When the initial Capillary number was larger than
the critical Capillary number, the droplets were subjected to higher viscous forces
creating an affine stretching where the drop deformed from a sphere to an ellipsoid
and then to a cylinder. Droplet behavior at a large initial Capillary number of
Ca = 0.67 is shown in Fig. 3.7. As evident from the plot, the droplet continued
to deform with the minor axis (b) asymptotically reducing, while the length (L)
continued to grow - causing L/b to increase. Since the largest change in shape
occurred initially, the droplet took longer to achieve the background flow velocity
44
and the region of rapid increase in velocity with small translation along x? was
extended. This caused the magnitude of lag to be higher than for droplets at low
initial Capillary numbers. Furthermore, Fig. 3.7 shows that the lag magnitude was
greater initially and slowly reduced as the droplet progressively reduced in width.
Figure 3.7: Motion induced deformation of droplet at initial Capillary number of
0.67. Squares represent variation of L/b and circles represent variation of u?. Solid
line is the lubrication approximation of velocity of the single phase background flow.
The droplet continuously deforms in the domain with asymptotic reduction in b after
reaching bcrit. The x? refers to instantaneous center of mass location of the droplet.
3.5 Conclusion
We used the lubrication approximation of the velocity field to compute a low
Reynolds number flow through a converging channel. The leading order lubrication
approximation of the velocity was found to agree to a high degree with numerical
simulation and experimental measurement of the flow. The linear variation of the
45
streamwise velocity along the flow rendered the extensional rate near constant along
the centerline. The flow channel can thus be used in applications where a constant
extensional rate is desired in a low Reynolds number flow.
When immiscible fluid particles were suspended in the flow, we observed a lag
in velocity of the particles compared to the single phase velocity at the same point.
The occurrence of a relative translational velocity is consistent with the behavior of
a particle suspended in an unconfined flow undergoing motion far from it. We report
two distinct domains for lag velocity based on initial Capillary number. First, for
Capillary number smaller than the critical Capillary number, the lag velocity rapidly
approached a near constant value and closely lagged the background flow. This is
due to the rapid approach to a steady-state shape balancing the external viscous
and interfacial tension forces. Second, for Capillary number much greater than the
critical Capillary number, where the droplet undergoes a large change of shape from
a sphere to an ellipsoid for a longer time. The subsequent asymptotic lag velocity
was greater in the second domain than the first. The magnitude of lag increased
with the initial Capillary number and this non-intuitive behavior of droplets in low
Reynolds number confined flows is a consequence of the finite size of the particles
and the deformation induced by the flow.
Our study can be extended by using the lubrication approximation of the flow
to predict the velocity gradients along the transverse direction to the flow. When
a droplet is suspended offset of the centerline and flow initiated, it experiences
transverse velocity gradients and undergoes shear deformation. Lubrication theory
can help predict the behavior of droplets at these locations. We envision an
46
application of our study in Polymer Based Additive Manufacturing (PBAM) which
routinely involve low Reynolds number flows through narrow nozzles. A new
development in this area is the introduction of an immiscible fluid into the bulk
flow which deforms the introduced phase due to viscous effects. Controlling the
geometry of the channel and the flow rate provides a means to control the size
and shape of the immiscible fluid as it is deposited on the build plate of the PBAM
machine. Unique parts can thus be printed which have a predetermined distribution
of a second immiscible phase in the bulk polymer.
47
Chapter 4: Modelling shear-influenced deformation of drops in a
converging channel using lubrication theory
4.1 Abstract
Low Reynolds number flow through planar converging channels imposes pure
extensional strain on suspended particles along the centerline of the flow. However,
when the particles are near the wall, the no-slip condition for the velocity imposes
an additional shear component to the total strain. If the suspended particles are
deformable ? like drops ? their shape and size can be tuned by simply controlling the
flow rate. In this study we apply concepts from lubrication theory and kinematics of
mixing to describe the behavior of drops undergoing shear-influenced deformation.
We theoretically model the motion of a drop suspended in the lubrication flow and
predict its deformation by considering its initial location and time spent in the flow.
We present a space-time phase space that predicts all possible particle trajectories
and the procedure to calculate the drop deformation from the phase space. We
then experimentally validate the model by tracking the deformation of suspended
Silicone oil drops in a flow of Castor oil through a converging channel. Our model
accurately predicts the net drop deformation for different combinations of initial
48
drop locations and time spent in the flow by the drop. We believe that our model
can be conveniently used to calculate drop deformation in flows where the single
phase flow solution is available.
4.2 Introduction
Confined flows are ubiquitous in applications across several length scales ?
ranging from water pipes of a household plumbing system to the blood vessels of
a human circulatory system. While some confined flows have inertial effects, many
applications in microfluidics [75, 76], polymer processing [45], and 3D printing [77]
involve flows where all inertial effects can be neglected. In each of these applications,
the absence of a free surface results in a unique flow profile due to the no-slip
boundary condition at the walls. The profile is parabolic with streamwise velocity
maximum at the center of the flow and zero at the walls ? resulting in a spanwise
velocity gradient. These gradients are constant in steady flows in channels with
non-varying walls. However, the presence of slowly-varying walls leads to varying
degree of these gradients along the flow. These velocity gradients can be harnessed
to deform suspended particles ? like drops [78, 79] ? enabling new applications in
the aforementioned areas. For example, suspending immiscible drops in a polymer
flow allows for 3D printing of unique geometries with a controllable distribution of
drop sizes within the printed geometry [80]. In this chapter, we show how to model
the deformation of such a suspended drop in a low Reynolds number converging
flow and validate the model using experiments.
49
Lubrication theory is a useful tool to study low Reynolds number flows through
narrow channels [8, 81, 82]. The theory takes advantage of the presence of a
small parameter in the flow geometry ? usually the spanwise depth of the channel
compared with the overall streamwise length ? and uses it to perturb the Stokes
equations of motion. The resulting series solution of the flow is accurate even at
low orders of approximation. A previous study [7] used the technique to solve
for the flow in a planar channel with slowly varying walls. Due to manufacturing
constraints associated with axisymmetric channels, planar channels are ubiquitous
in low Reynolds number microfluidic flows that can be fabricated in a single
lithographic step. While single phase microfluidic flows have many well-established
applications in biotechnology, multiphase microfluidic flows is an emerging research
area. For example, some studies have explored the use of microfluidic channels as
flow focusing devices for a suspended fluid. The geometry of the channel ? which is
typically narrow and has a rectangular cross section ? enables passive control over
the process. While scaling analysis [23, 83] has been broadly used to relate the flow
rate through the channel and the morphology of the suspended phase, there have
been no attempts to use detailed solutions of the flow in channels to predict their
effect on the suspended drops [84].
When the solution for a flow is available, it is relatively easy to predict
the trajectories of particles suspended in the flow. Applying equations of mixing
kinematics allows for Lagrangian tracking of such particles in a steady state velocity
field [85]. In this context, substituting solutions from lubrication theory into
trajectory predicting equations of the theory of mixing should allow for determining
50
the behavior of suspended particles in the flow. To demonstrate the intended
application of lubrication theory, we study a low Reynolds number flow through
a planar converging channel shown in Figure 4.1.
120 20
?????
70
? ?
? ?
600
160 ? = ? + 10
(?) (?) (?) ??????
Figure 4.1: (a) Front view, (b) side view, and (c) isometric view of a planar
channel with flow streamwise along x. All dimensions are in mm. The origin of
the coordinate system is equidistant from the two planar bounding walls.
The channel has a fixed depth of 20 mm and the converging portion has a
length of 160 mm. The converging portion of the channel has a wall profile based
on a hyperbolic equation which renders the extensional rate of the single phase low
Reynolds number flow near constant along the centerline. A previous study [16]
used lubrication theory to study the flow along the centerline while this study will
focus on locations near the wall that are shear-influenced.
This chapter is organized as follows. In section 4.3 we present the theoretical
model, including assumptions, that predicts drop deformation in the flow. In section
51
4.4 we present the experiments performed to validate the model. In section 4.5,
we present our results in two parts. First we present a phase space that depicts
trajectories of particles in space-time. Then we illustrate how the phase space can
be used to determine drop deformation in the flow and present model validation
results. Finally, we provide some potential applications of our results.
4.3 Theoretical model
We consider an isothermal, Newtonian, and low Reynolds number flow of a
fluid with viscosity ?o and density ?o through the channel shown in Figure 4.1.
Following the steps outlined in the previous study by Lauga et al. [7] to find the
single-phase flow solution, we non-dimensionalize all variables using the following
equations. Note that all capped variables are non-dimensional.
(x, y, z) = (?x?, hy?, hz?). (4.1)
Q
u = u? (4.2)
h2
??oQ
p = p? (4.3)
h4
In equation 4.1, ? = 160 mm, h = 10 mm, u is the x component of velocity, Q
is the flow rate, and p is the pressure. Following Lauga et al. [7], we find the leading
order solution for the x component of the velocity as
52
{ ? ( ) }1 dp? n0 4(?1) cosh kny?
u?0(x?, y?, z?) = (z?
2 ? 1) + ( ) cos knz? , (4.4)
2 dx? k3
? n cosh knf(x?)n 0
where p? is the leading order solution for the pressure and k = (n+ 10 n )?. The2
wall profile is f(x?) = 6/(16x?+ 1). The axial pressure gradient is given by
{ ? ( ) }dp?0 3 6 tanh knf(x?) ?1
= ? 1 . (4.5)
dx? 4f(x?) f(x?) k5
n?0 n
A single drop of a second fluid with viscosity ?i and density ?i is injected into
the flow at (y? = 0, z? = (z?1 + z?2)/2). The initial configuration and the subsequent
deformation of the drop induced by the flow is depicted in Figure 4.2.
Initially, the spherical drop is located at x?0 and is bound between z?1 and
z?1. Due to the motion of the surrounding fluid as indicated in equation 4.4, the
drop undergoes shear-induced deformation. By implementing a simple Lagrangian
tracking of two points on the drop, we can track the deformation in time. From
the theory of mixing kinematics, a target point tracks the flow field in which it is
suspended and its trajectory is governed by the equation
dx
= v(x, t), (4.6)
dt
where v(x, t) is the Eulerian velocity field. The particle path is given by the
solution of equation with x = X at t = 0. We assume that the y and z components of
velocity is zero and any point in the flow moves only along the x direction. Equation
53
????
? ? , ??? ???? ? ? , ??? ????
???2
???1
???
??????????
? ? ?0 ?1 ?2 ?3 ?4
? ? , ??? ???? ? ? , ??? ????
????
Figure 4.2: Side view of channel (as shown in Figure 4.1(b)) shown with a drop
initially suspended at (x? = x?0, y? = 0, z? = (z?1 + z?2)/2). When the flow is actuated,
the drop undergoes shear-influenced deformation but is restricted between z?1 and
z?1.
54
??????
4.6 becomes
dx
= u(x, t) (4.7)
dt
We convert equation 4.7 to non-dimensional form using equation 4.2 for
velocity and ? = 160 mm as the length scale and obtain
dx? QT
= u? , (4.8)
dt? Lh2
where t? is the dimensionless time and T is the time scale. Since a length scale
and velocity scale was already defined in equations 4.1 and 4.2, we assume their
ratio as the time scale T . Thus, T = Lh2/Q and we integrate equation 4.8 from
0 to t?f where t?f is the dimensionless time spent by the particle in the flow. The
equation that predicts the particle trajectory is thus given by
? t?f
x? = u?dt? (4.9)
0
We are now in a position to apply this concept to study the behavior of a
suspended drop in the flow. In a small time interval ?t, the point (x?0, z?2) translates
a distance equal to u?0(x?0, z?2)?t and reaches (x?1, z?2). The point (x?0, z?1) translates
a distance equal to u?0(x?0, z?1)?t and reaches (x?2, z?1). Since z?2 is closer to the wall
of the channel, u?0(x?0, z?2) is smaller than u?0(x?0, z?1). This causes the drop to shear
and it elongates to attain a length equal to |x?1 ? x?2| as seen in the xy plane. The
process continues in time and the drop becomes longer. Flow history seen by the
55
two points is important to take into account while integrating equation 4.9. This is
because the drop is moving through a gradient in the x component of velocity.
Finally, we make the following assumptions for the model.
1. The flow is low Reynolds number and thus develops near-instantaneously to
the steady state everywhere as indicated in equation 4.4. This allows us to
ignore any transients in the flow as it develops to the form shown in equation
4.4.
2. The imposed capillary number on the drop ? which compares the magnitude
of external viscous force and the interfacial force ? at the flow onset is ? 10.
The magnitude of viscous force on the drop increases further along the flow.
Large capillary numbers allow us to ignore the effect of the interfacial force
which resists deformation.
3. Related to the previous point, when drops exhibit large deformation the
difference in viscosity between the drop and the surrounding can be neglected.
This is an important assumption since viscosity ratio does indeed affect the
behavior of suspended drops at low capillary numbers. Since we test the model
at large capillary numbers, the drop is expected to track the flow in equation
4.4.
4. A previous analysis by Lauga et al. [7] showed that there is a z component of
velocity in the flow which has the tendency to migrate the drop towards the
centerline. However, we ignore the z velocity component and assume that the
56
drop is restricted to flow between z?1 and z?1. Since the drop is always along
y? = 0, the y component of velocity is also zero.
To illustrate an example of the predictive power of equation 4.9, we numerically
solve equation 4.9 for t?f = 1. We discretize total time t?f into 100 steps and offset
z? into 50 units. We then implement a simple time marching scheme and obtain x?
of all points in the flow at every time interval. This generates a space-time phase
space of trajectories of all points offset of the centerline (z? = 0 to z? = 1) and will
be discussed in section 4.5.
4.4 Experiments
To test the model, we injected a drop of Silicone oil (viscosity = 0.35 kg m?1 s?1
and density = 970 kg m?3) into the flow of Castor oil (viscosity = 0.72 kg m?1 s?1
and density = 961 kg m?3) through the channel. The initial volume of the drop was
50 ?L and the flow rate was approximately 4600 mm3s?1. The slight difference in
density did not affect the experiment since the imposed flow rate was much larger
than the motion due to gravity effect. A previous study [16] has provided a detailed
description of the apparatus and experimental procedure. Images of the flow were
taken at 6 frames per second and Figure 4.3 shows the motion and deformation of
the drop in the flow injected at z? = 0.5.
57
10 mm (a) (b) (c) (d) (e) (f) (g)
Figure 4.3: Shear-influenced deformation of a Silicone oil drop in Castor oil flow.
(a) shows the initial configuration at rest where the diameter of the drop is
approximately 4.5 mm and the drop is located at z? = 0.5. (b) to (g) show the
subsequent motion induced deformation. See Figure 4.2 for a kinematic description
of the drop behavior from the side view. In (g) the drop has attained a length of
approximately 44 mm. Experiment was repeated for different z? initial drop locations.
Initially, the drop and the background fluid were at rest as shown in figure
(a). After flow actuation, the drop underwent shear-induced deformation and the
photography was stopped before the leading end of the drop exited the field of view
of the camera. We measured the final length of the drop length at this position. To
validate our theoretical model, we performed three separate experiments at the same
flow rate by injecting drops at z? = 0.4, 0.5, 0.6. We did this to illustrate the effect
of increasing shear on the drop with increasing offset distance z? from the centerline.
For every experiment, the final drop length attained was measured and compared
with the prediction from the model.
58
4.5 Results and discussion
4.5.1 Spatial-temporal phase space
Substituting equation 4.4 in 4.9 and integrating for t? = 1, we obtain a phase
space for x? as a function of z? and t?. The phase space is a representation of the
Lagrangian tracking of points seeded in the flow at x? = 0. Figure 4.4 shows the
space where z? = 0 is the centerline of the flow and z? = 1 is the wall of the channel.
Figure 4.4: Space-time phase space of trajectories of points in flow described by
equation 4.4 and following notation shown in Figure 4.1. At t? = 0, x? = 0. As time
progresses, particle paths follow the background velocity field. Points at the center
of the channel (z? = 0) travel the furthest while points at the wall do not move.
The phase space has several interesting features worth elaborating. Figure 4.4
shows that any point at the wall (z? = 1) does not undergo any motion in time and
the point at the center of the flow (z? = 0) undergoes the maximum translation.
This is consistent with the boundary condition of no-slip at the wall and a parabolic
59
u? velocity profile where the magnitude is maximum at the center of the channel.
The part of the phase space along z? = 0 is also consistent with the results from a
previous study [16] which experimentally validated equation 4.4. The phase space
can also be extended to longer t? = 1 depending on the flow time.
4.5.2 Comparison with experiments
The phase space in Figure 4.4 can be used to predict the final length attained
by drops in the flow. We illustrate this by considering an example where a drop
was initially injected into the flow at x? = 0 and suspended between z?1 = 0.4 and
z?1 = 0.6. By tracking the two points on the drop as explained in section 4.3, we
obtain the trajectory of the points in time as shown in Figure 4.5. At t? = 1, the
final length attained by the drop is the difference in x?.
60
Figure 4.5: Trajectories for z?1 = 0.4 and z?1 = 0.6 extracted from Figure 4.4. The
two points separate from each other along the plotted trajectories. Difference in x?
at any time instant provides a measure of their separation.
Finally, we compare the model the with the experimental results for drops
injected at z? = 0.4, 0.5, 0.6. Figure 4.6 shows the final lengths predicted by the
model and the experimentally obtained results. Given the assumptions in section,
the model gives reasonably good predictions of the final lengths of the drops.
61
Figure 4.6: Comparison of model prediction of drop lengths |x?1?x?2| vs experimental
measurements.
Figure 4.6 shows that drops injected closer to the channel walls deformed
the largest for nearly the same time spent in the flow. Our results show that the
influence of shear close to the channel walls imposes larger strain on the drop than
a previous study [16] which deformed drops under pure extensional flow at the same
flow rate.
4.6 Conclusion
In conclusion, we presented a model that described the deformation of drops
suspended in a low Reynolds number flow. We chose a planar converging flow ?
commonly found in nozzles of additive manufacturing machines and extensional
mixers in emulsion industries ? and first described a low Reynolds number single
phase flow using lubrication theory. While previous work [16] had successfully
62
validated the theory for pure extensional flow along the centerline, we studied other
locations in the flow that were shear-influenced. As shown in section 4.5, such flows
reliably elongate drops to lengths that are at least an order of magnitude larger than
the initial drop size.
In the flow, we tracked the trajectory of points following the flow and
presented a simple method to calculate the net deformation of suspended drops
? specifically the final length attained after shear-influenced deformation. We
performed experiments that validated our model for several combinations of
processing conditions.
In summary we have shown in this work that, under specific assumptions,
concepts from lubrication theory and mixing kinematics can be combined to predict
drop deformation. We believe that our model will be useful to industries where drop
dispersion in a well-defined mixer is a key process.
63
Chapter 5: Effect of inertia on extensional deformation and
capillary-driven breakup of drops
5.1 Abstract
We investigated extensional deformation of drops suspended in stagnant and
non-stagnant extensional flows, and the subsequent capillary-driven breakup after
flow stoppage. While the low Reynolds number approximation of the problem
has received experimental and theoretical attention over the years, the influence
of inertia on both phenomena is not well studied. First, using direct numerical
simulations we validated our scheme with previous experimental results and showed
that the influence of inertia on elongating drops resulted in slightly longer stable
shapes. Next, we considered a slender drop suspended in a quiescent fluid ?
representing external flow stoppage condition ? and systematically studied the
capillary-driven breakup of the drop in a previously unexplored part of the
parametric space. Both inertial and viscous effects of the outside fluid were
considered. In the new space, we showed how the structure of the flow field was
modified upon addition of inertia and presented a stability diagram for slender drops.
64
5.2 Introduction
A fundamental trait of single-phase viscous fluid flow is the presence of spatial
or temporal velocity gradients in the flow. These velocity gradients can be used to
deform suspended particles ? providing a way to produce dispersions in immiscible
liquid-liquid systems [8, 86]. The fluid dynamics problem of drop deformation and
breakup is of interest to several industries, like petroleum and pharmaceutical [87],
all aiming to improve efficiency of their mixing devices. Flow in such mixing devices
are typically associated with inertia, either in the laminar or turbulent regime,
where the suspended drop first deforms in the flow and subsequently undergoes
either flow-driven or capillary-driven breakup. While the drop deformation in low
Reynolds number flows is well studied [22, 24, 88], the role of inertia in such systems
? especially in the capillary-driven breakup regime ? has received less attention. In
this study, we investigate the role of inertia of the external flow in deforming drops
and its subsequent influence on capillary-driven breakup.
The problem of drop deformation in another fluid undergoing motion is a
classical fluid dynamics problem and has been well researched over the past century
[4, 5]. Determining the shape of a drop in a flow is non-trivial because of the presence
of the free boundary of the droplet whose shape has to be determined as part of the
solution [8]. Thus, studying the problem experimentally necessitates simplification
of other flow variables so that the problem is easier to solve. One such simplification
is the assumption of a linear flow field external to the drop, far from it. Implicit in
this assumption is that the size of the drop is small compared to a length scale of
65
the bulk flow [23]. Such linear flow fields generate a spatially homogeneous velocity
gradient which deforms the drop. Using this assumption, drop deformation in the
low Reynolds number regime has been investigated using experimental [2, 4, 33],
theoretical [9, 26, 89], and numerical methods [90, 91, 91, 92, 93] ? leading to the
determination of the stability criterion for drops in such flows. A natural question
that arises is: what happens to drops when inertial effects are considered in such
systems?
While the problem of drop deformation with inertial effects is considerably
harder to solve with classical methods, there have been some attempts to extend
the small-deformation and slender body theories to the inertial domain [22, 94]. The
presence of inertia in the external flow provides additional aid to the viscous force
that deforms the drop. Thus, longer steady drop shapes that are stabilized by the
larger dynamic pressure of drops in the flow is expected in this regime. Several
numerical studies have shown this effect [95, 96, 97, 98] and we use that data to
validate our numerical scheme in this study. While it is natural to assume that it
would be easier to breakup drops in an inertial flow, it has been shown that the
external flow ? with or without inertia ? continuously postpones breakup. Thus,
another framework is necessary to breakup drops in such multiphase systems.
One way to achieve breakup of slender drops is by stretching them to the
desired slenderness and stopping the external flow. This onsets a capillary flow
in the drop which can induce breakup [1]. Another possibility is retraction of the
drop back to the original shape if the slenderness is inadequate. Regardless of the
outcome, since the capillary flow problem involves two fluids, the problem can be
66
studied in different parts of a parametric space that comprises viscosity ratio ?
which is the ratio of viscosity of inner fluid to the outer fluid ? and the magnitude
of Reynolds number. We now review the drop behavior in different parts of this
parametric space and hypothesize the behavior in an unexplored part of the space.
When the viscosity ratio is very large, the viscosity of the external fluid can
be neglected. At large Reynolds number, the behavior in this regime has been
widely studied both experimentally [12, 13, 15] and theoretically [10, 14]. From an
initial steady state, the two ends of the drop retract towards each other minimizing
the interfacial area. A recent study has shown that capillary waves emerge from
both ends on the surface of the drop and interact with each other leading to their
interference [15]. Depending on the initial slenderness, the drop either retracts into
a single sphere or undergoes end-pinching. Inertial effects in this regime also cause
non-monotonic pinching of the ends especially at large Reynolds number. As the
Reynolds number is reduced, increasing viscous effects lead to suppression of all
capillary waves and at the limit Re? 0, a slender drop cannot breakup regardless
of slenderness [11].
The same problem of retraction of a slender drop at different Reynolds number
can also be studied in finite viscosity ratio systems. In this part of the parametric
space, the slender drop is surrounded by a another viscous fluid. Depending on
the slenderness of the drop, at the limit Re ? 0, it either undergoes retractive
end-pinching [1, 89], where the ends of the drop pinch-off successively, or undergoes
breakup by the mechanism of capillary wave instability where a large number of
daughter drops are formed at the same time. This begets the question: can the
67
dynamics of capillary wave interaction and non-monotonic end pinching seen in
systems with infinite viscosity ratio be reproduced in the current space? To answer
this question, we investigated the dynamics in the new part of the parametric space
by adding inertial effects to the finite viscosity ratio problem.
This chapter is organized as follows. We first begin with a description of the
problem statement and the numerical method we use to investigate it. Next, we
present our findings in two parts. The first part reveals the effect of inertia on drops
in an extensional flow field. In the second part, we remove the external flow field and
allow an elongated drop to retract under the influence of interfacial tension. Finally,
we summarize the effect of inertia on both phases and discuss the significance of our
findings.
5.3 Methods
5.3.1 Problem setup
In this study, we first investigated the extensional behavior of immiscible drops
in an externally imposed linear flow field at finite Reynolds number. The flow field
was defined with x and y velocity components as u and v respectively. They were
specified as
u = ??x (5.1)
and
v = ???y (5.2)
68
where ?? was the magnitude of the strain rate. This flow field produced a stagnation
point at (x, y) = 0 and when a drop was suspended at this point, it underwent
extension and the degree of extension depended on the magnitude of ??. The
schematic representation of the domain is shown in figure 5.1.
Figure 5.1: Domain for the stagnant extensional flow problem. The drop is shown
suspended at the center of a square 8 X 8 domain. All dimensions are non-
dimensionalized using a length scale equal to the initial radius of the drop.
The drop with viscosity ?1 and density ?1 is shown suspended in another fluid
with viscosity ?2 and density ?2 at the center of a square domain with ?1 = ?2. The
interfacial tension between the two fluids was ?. Using the initial radius (a) of the
drop as the length scale and velocity magnitude (V ) of the external flow at the edge
center of the domain, we studied the drop elongation at various Reynolds numbers
69
(Res) and Weber numbers (Wes) defined as
?2V a
Res = (5.3)
?2
and
?2V
2a
Wes = (5.4)
?
respectively.
To study non-stagnant extension of drops, a converging channel configuration
shown in figure 5.2 was used. The walls of the channel was based on a hyperbolic
equation and the flow was capable of imposing a constant extension rate along the
centerline.
Reynolds number (Rens) and Weber number (Wens) was redefined using length
scale equal to the inlet width (Lin) and velocity scale equal to the inlet x component
of velocity (Vin) at the origin in figure 5.2.
?2VinLin
Rens = (5.5)
?2
and
? V 22 Lin
Wens =
in (5.6)
?
As drops elongated in the flow for various values of Rens and Wens, they
attained elongated shapes consistent with the balance between viscous stress and
interfacial stress. However, to study the effect of inertial and viscous forces on the
drop when the forcing flow was removed, a simpler configuration of a slender drop
70
Figure 5.2: Domain for the non-stagnant extensional flow problem. The drop is
shown suspended in a converging channel. All dimensions are non-dimensionalized
using a length scale equal to the inlet width of the channel (120 mm).
with hemispherical ends ? consistent with previous studies [3, 10, 12, 15] ? was
chosen. The schematic representation of a quarter of the domain is shown in figure
5.3.
The long, slender drop is shown suspended in the external fluid which is now
quiescent. Due to the large curvature of the ends, a net force due to interfacial
tension acted ? leading to retraction of the drop. From the initial state, the slender
drop was allowed to evolve under the influence of this interfacial tension and the
phenomenon was studied for different Reynolds numbers (Rec). Initial radius of the
end cap (rc) of the slender drop was the length scale for the problem. Since there
71
Figure 5.3: Quarter domain for the capillary-driven flow problem. A long slender
drop is shown suspended in another quiescent fluid. The drop has an aspect ratio =
30. Such slender drops are unstable at moderate Reynolds numbers when the effect
of the surrounding fluid is ignored.
was no imposed velocity scale like the previous ca?se of an externally-driven flow, the
velocity scale (Vc) for the problem was equal to ?/?1rc. This was equaivalent to
setting the Weber number (Wec) equal to 1. Reynolds numbers (Rec) was defined
as
?1Vcrc
Rec = (5.7)
?1
5.3.2 Governing equations
The governing equations for the problems were the mass conservation equation
and the momentum conservation equation where both fluids were assumed to be
incompressible and isothermal. The equations were
? ? u = 0 (5.8)
and [ ]
?u [ ( )]
? + (u ? ?)u = ??p+? ? ? ?u +?uT + ?g (5.9)
?t
72
where u is the velocity vector, ? is density, ? is the dynamic viscosity, p is pressure,
t is time, and g is the gravity vector. These equations in the context of the current
two-phase flow problem were valid within both fluids. However, the presence of an
immiscible boundary between the two fluids needed separate treatment. We used a
level-set function ?(x, t) to capture the evolution of the immiscible boundary and the
ghost-fluid method [99] to capture the interfacial dynamics. The level-set function
was an implicitly defined function with magnitude equal to zero on the immiscible
boundary, negative within the drop, and positive outside the drop. See figures 5.1 or
5.2 for definitions. The level-set function was advected using the local fluid velocity
u by the equation
??(x, t)
+ u ? ??(x, t) = 0. (5.10)
?t
This provided a way to capture the evolution of the interface between the two fluids
using the solution of the flow from equation 5.9. Since the same equation was
numerically solved in both the phases, ? and ? were specified as a function of ?(x, t)
so that a single equation for ? and ? could be used in the entire domain. The
equations for ? and ? were:
? = ?1[1? I(?)] + ?2I(?) (5.11)
and
? = ?1[1? I(?)] + ?2I(?). (5.12)
73
In equations 5.11 and 5.12, I(?) is a smooth function that allows for the variables
to smoothly change across the interface where a sudden jump in ? and ? occurs. We
allowed a smooth transition of ? and ? across a distance  equal to twice the grid
resolution ?. Thus, I(?) was given by
( )
1 ? 1
I(?) = ? erf + . (5.13)
2 ? 2
where erf ? [?1, 1] was the error function. Since the problems we investigated
had interfaces associated with interfacial tension, the pressure across the interface
underwent a jump in magnitude [p]I depending on the degree of the interfacial
curvature ?. Thus, [p]I was given by
[p]I = ??. (5.14)
The curvature ? of the interface was estimated using ? as
??
? = ? ? (5.15)
|??|
5.3.3 Numerical scheme and boundary conditions
The domains shown in figures 5.1, 5.2, and 5.3 were discretized into staggered
grids with square cells where the x component of velocity was computed at the
center of east edge and y component of velocity was computed at the center of the
north edge. Scalar variables like pressure, density, viscosity, and ? were computed
74
at the cell centers. A depiction of a staggered cell is shown in figure 5.4.
Figure 5.4: Discretization scheme for the numerical method. In the shown staggered
cell with grid spacing ?x = ?y, the vector variables are defined on the cell faces
and scalar variables are defined on the cell centers.
For the initial study of drop elongation in the external flow given by equation
5.1 and 5.2, we discretized the domain shown in figure into 128 X 128 cells. For the
non-stagnant extension, we discretized the domain into 640 X 480 cells. Finally, for
the study of capillary-driven breakup of the slender drop, we discretized the domain
shown in figure into 980 X 70 cells. Grid independence was particularly important for
the capillary-driven breakup problem and was verified at different levels of domain
discretization. The governing equations were solved using the projection method.
For the first stage of our study where we explored drop elongation in stagnant
and non-stagnant extensional flows, we solved the equation in a two-dimensional
domain to accurately capture the influence of a planar external flow. For the
subsequent stage of our study where we explored capillary flow in slender drops,
it was important to capture both the radii of curvature that characterized the
75
interface. We thus solved the governing equations in an axisymmetric domain for the
second state of our study. We used a fifth-order weighted essentially nonoscillatory
(WENO) scheme to discretize equation 5.10 and the ghost-fluid method to capture
the pressure jump across the interface. Finally, after every time step, we reinitialized
the interface by reconstructing ? as the minimum distance from the grid to the
interface. This allowed for the projection of interface topology onto the cartesian
grid without modifying the underlaying scalar variables [100]. The numerical scheme
was validated in previous studies for different applications [101, 102, 103].
Boundary conditions for the stagnant extensional flow problem in figure 5.1
were as follows. We applied a Neumann boundary condition for pressure and
imposed the constant velocity defined in equations 5.1 and 5.2 on the four walls of
the domain. For the non-stagnant extensional flow problem in figure 5.2, Dirichlet
boundary condition for pressure was applied on wall BC while Neumann boundary
condition was used for the other three walls. At the inlet AD, a parabolic velocity
profile ? corresponding to a fully developed flow upstream ? was defined with zero
velocity at A and D. The solid cells along the walls of the channel had no-slip velocity
condition. Fully developed flow profile for a single phase low Reynolds number flow
was initialized throughout the domain.
For the axisymmetric problem in figure 5.3, the boundary conditions were
slightly different. Dirichlet boundary condition for pressure was imposed on wall
BC and the while Neumann boundary condition for pressure was imposed on walls
AB and CD. Since CD was the axis of symmetry, we imposed a symmetric boundary
condition for velocity while no-slip and no-flux was imposed on wall AB. On the wall
76
BC, we imposed a symmetric boundary condition for u and slip boundary condition
for v allowing the fluid to freely flow out of the domain. The boundary conditions
for pressure and velocity on the wall symmetric to BC (at x = ?70) about the y
axis was the same as the conditions on BC. At the initial state, we set v, p = 0 and
allowed the capillary flow to develop.
Computations were perfomed using UMD supercomputing resources. The
domain in all three cases was split into 4 zones and the governing equations were
solved using a parallel computing framework.
5.3.4 Parametric space for numerical experiments
For problem defined in figure 5.1, we were interested in exploring the effect of
inertia on extensional deformation of the drop. We first set viscosity ratio ? = 1.00,
Res = 1.00, and systematically varied the Wes from 0.05 to 0.11. These conditions
were consistent with the previous study by Ramaswamy et al. [96] who explored
uniaxial elongation of drops in an axisymmetric external flow. We used the same
parameters in our simulation for validation purpose. For every condition we allowed
the drop to elongate till it attained a steady state in the flow.
Similarly, for the non-stagnant extensional flow, we studied conditions that
caused stable and unstable drop behavior. A previous study by Sangli and Bigio [16]
had classified drop behavior in non-stagnant extensional flows at different capillary
numbers (Ca). We chose a condition from their study corresponding to unstable
behavior of the drop at Ca = 0.65. Given our choice of velocity and length scales,
77
Wang et al. [3]
100 Hoepffner and 
Eggers and 
Pare [13] Fontelos [10]
Conto et al. [14]
Current study Stone et al. [1]0.1 to 10 Stone and Leal [8]
0.005 to 0.02 10 to 100
1
?? ??
Figure 5.5: The parametric space for investigation of capillary-driven breakup of
slender drops. The space comprises viscosity ratio (?) and Ohnesorge number (Oh).
In this study, both inertial and viscous effects of the outside fluid are considered.
these conditions corresponded to Rens = 2.53,Wens = 7.99 respectively.
For the subsequent study of capillary-driven flow, the parametric space
described in section 5.2 is succinctly represented in figure 5.5. We chose three
different viscosity ratios ? = 0.01, 0.10, 1.00 and four different Reynolds numbers
Rec = 25, 50, 100, 200. Ohnesorge number (Oh) was defined as
?
Wec 1
Oh = = (5.16)
Rec Rec
78
5.4 Results and discussion
5.4.1 Inertial effects on drop deformation in stagnant extensional flow
In this section, we report the drop elongation behavior for both the viscosity
ratios studied at Res = 1.00. Figure 5.6 shows the steady elongated states attained
by drops at various conditions.
Figure 5.6: (a) to (d) shows the steady states of drops at Wes = 0.05, Wes = 0.07,
Wes = 0.09, and Wes = 0.11 respectively. Reynolds number (Res) for all cases is
equal to 1.
We defined a non-dimensional deformation metric for an elongated drop shown
in figure 5.7 as
L? b
Df = (5.17)
L+ b
and compared it with the measurements by Ramaswamy et al. [96] in figure 5.8.
Comparing the steady state Df at Res = 1.00 with creeping flow Res ? 0, we
observed that steady shapes tended to be longer. This observation was consistent
79
Figure 5.7: Semi-major length L and semi-minor length b of a deformed drop.
with the previous study in this area by Ramaswamy et al. [96]. Note that in creeping
flows, capillary number captures the effect of viscous deforming force (instead of
inertial deforming force) over the interfacial force. The Weber number is related to
the capillary number as Wes = Res ?Ca where Ca is the capillary number. At ? =
1.00, longer steady shapes at the same Weber number indicated that the presence of
slight inertia in the flow allowed the associated dynamic pressure to stabilize longer
drops. The last Weber number studied was the critical Weber number beyond which
no stable shape was observed.
Figure 5.8: Effect of slight inertia at Res = 1 on drops undergoing stagnant
extension.
80
Figure 5.9: (a) to (f) represent transient deformation of drops in a non-stagnant
extensional flow at Rens = 2.53 and Wens = 7.99. The left panel in every image is
the experimental observation and the right panel is the computational result.
5.4.2 Inertial effects on drop deformation in non-stagnant extensional
flow
Figure 5.9 shows the transient drop deformation at Rens = 2.53,Wens = 7.99.
Note that the drop deforms and continuously elongates as long as it is present in
the flow field. The flow is strain-limited because of the finite length of the channel.
Thus, there is an upper bound for total strain on the drop. Figure 5.10 shows the
comparison with experiments at the same condition.
81
Figure 5.10: Comparison of simulation and experimental results for non-stagnant
extension of a drop at Rens = 2.53 and Wens = 7.99.
5.4.3 Inertial effects on capillary-driven breakup
In the previous sections, we studied how an external flow can cause drops to
elongate and become unsteady after the critical Weber number is exceeded. As the
deformation progresses, the drop becomes slender. However, breakup can be hard
to achieve with this mode since the external flow stabilizes any perturbation on
the surface whose growth can lead to breakup. For slender drops, the magnitude
of interfacial force at the ends is inversely proportional to the radius of curvature.
Thus, the large interfacial force can be taken advantage of to cause drop break-up.
When the external flow is removed, the slender drop finds itself overstretched in the
quiescent field. Interfacial force dictates the dynamics and tends to minimize the
interfacial area. It achieves this by either retracting into a single drop or breaking
82
up the interface to create multiple smaller drops.
5.4.3.1 Validation with previous experiments
Before studying the capillary-driven breakup in the new parametric space, we
validated the simulation scheme with previous experimental results. Wang et al.
[3] studied retraction of water drops in air under inertia and presented a stability
criterion for slender drops of different aspect ratios. We chose an experimental
result reported at Oh = 0.0057, setup a numerical simulation as per sections 5.3.2
and 5.3.3, and compared the evolution of drop shape in the capillary-driven flow.
Figure 5.11 shows the comparison between experimental results and the simulation.
Wang et al. [3] report the experimental result in figure 5.11 for a slender drop
with initial aspect ratio equal to 12. However, we observed good agreement with
the experimental result for an aspect ratio equal to 14. This slight discrepancy can
be attributed to a combination of the following reasons.
1. Difference in numerical scheme between the two studies.
2. Experimental error in measurement of the initial drop shape in figure 5.11.
3. Difference in shape of the end and neglecting the initial impulse that resulted
in the creation of the slender shape.
Since our numerical scheme was capable of providing insights into capillary-
driven breakup, we chose to proceed with studying the dynamics in the parametric
space of figure 5.5.
83
(a)
(b)
Figure 5.11: Comparison of (a) experimental results [3] and (b) simulations of
capillary-driven breakup at Oh = 0.0057.
84
Figure 5.12: Capillary driven flow at Rec = 25 and ? = 1.00. The drop is
conditionally unstable ? breaking up into two daughter drops ? but asymptotically
stable.
5.4.3.2 Capillary-driven breakup in new parametric space
Figure 5.12 shows the interfacial force driven flow for a sample case of Rec = 25
at a viscosity ratio ? = 1.00.
The slender drop initially retracts under the effect of the interfacial force and
the radius at the ends become larger. In time, it can be seen that the drops pinch off
and yet re-coalesce to produce a single drop. Thus, the instability is conditional and
the system is asymptotically stable. The behavior can be explained as follows. The
bulb at the end of the slender drop traverses through the outer fluid of the same
viscosity. Inertia in the system provided enough momentum for the bulbs from
85
Figure 5.13: Capillary driven flow at Re = 25 and ? = 0.10. The drop is
asymptotically unstable and results in two daughter drops with a small satellite
drop suspended between them.
both ends to collapse into each other. However, after breakup, there was enough
momentum available in the bulbs ? which is basically an inertial effect ? to cause re-
coalescence. This observation can be used to tune the behavior of the slender drop
to generate multiple drops after breakup by increasing the viscosity of the outside
fluid. Figure 5.13 shows the drop behavior at viscosity ratio ? = 0.10.
In this case with the same amount of inertia in the system, the motion of the
bulb through the higher viscosity fluid was hindered. This resulted in pinch off of
the end of the drop and the bulbs were held in place in the high viscosity fluid.
Viscous effect had a damping influence on the bulb momentum and the system
asymptotically generated two drops. Continuing this trend by further increasing
86
Figure 5.14: Capillary driven flow at Rec = 25 and ? = 0.01. This drop is
asymptotically unstable ? resulting in three daughter drops with satellite drops
between them.
the viscosity of the outside fluid (? = 0.01), we see behavior shown in figure 5.14.
Three drops are asymptotically produced in this case. The system thus
undergoes a transition from asymptotically stable to asymptotically unstable state
between ? = 1.00 and ? = 0.10. We investigated this transition point over different
Reynolds numbers. Figure 5.15 shows the asymptotic end states of the drops in the
parametric space.
It can be seen in figure 5.15 that the transition point is inversely proportional
to the Reynolds number. Since Reynolds number represents a balance between
inertial and viscous forces, increasing Reynolds number equips the bulb with enough
momentum to support coalescence over breakup. Thus, even when the surrounding
87
1.00
0.10
0.01
25 50 100 200
??
Figure 5.15: Asymptotic fates of the slender drop for different combinations of
viscosity ratio (?) and Reynolds number (Rec).
fluid is highly viscous (? = 0.01), inertia overcomes the damping effect of the outside
fluid and reduces the number of daughter drops after breakup.
5.4.3.3 Effect of viscosity ratio on vorticity diffusion
Recent studies [3, 14, 15] have shown that the fate of a slender drop suspended
in air (? ? ?), retracting under the effect of surface tension, depends on the
interaction of capillary waves emerging from the two ends. Examination of vorticity
inside the drop revealed the formation of a vortex ring and a detached-jet type of flow
which caused the neck ? the point of minimum radius on the stem ? to reopen. In
such systems, the pinch off was non-monotonic and was postponed to later time since
the neck reopened. While the capillary wave interaction model can explain the fate
88
of drops suspended in air, the presence of the outside fluid complicates the associated
dynamics. Our parametric space comprised of ? ? 1.00 where viscous diffusion into
the surrounding fluid could be an important factor affecting the dynamics. This
is especially true at ? = 0.01 where, similar to breakup in systems with no inertia
[1], the neck pinch-off was monotonic and no neck reopening effect was observed.
Thus, a combination of signature effects of both parts of the parametric space ? 1)
Rec ? 0 with finite ? and 2) Finite Rec with ??? ? were observed in our study
leading to the behavior seen in figure 5.15.
A closer examination of the capillary flow allowed for a detailed description
of the neck reopening phenomenon reported in previous studies. Figures 5.16, 5.17,
and 5.18 shows the evolution of the flow at three snapshots in time ? before neck
reopening, at minimum neck radius, and after neck reopening ? at Rec = 100 and
? = 1.00.
89
Figure 5.16: Vorticity distribution in the system at Rec = 100 and ? = 1.00 before
attainment of minimum neck radius. Regions of opposite signed vortices can be
seen.
90
Figure 5.17: Vorticity distribution in the system at Rec = 100 and ? = 1.00 at
minimum neck radius. Onset of the separation of the jet flow from the neck to the
bulb can be observed.
91
Figure 5.18: Vorticity distribution in the system at Rec = 100 and ? = 1.00 after
reopening of the neck. A prominent vortex ring can be seen persisting in the bulb.
A simple description of the dynamics as explained by Hoepffner and Pare? [14]
for a drop suspended in air can be applied to study the system. As the bulb retracted
and neck radius reduced, fluid accelerated from the central portion of the slender
drop into the bulb through the neck. As it accelerated, it did not recover all of the
upstream pressure and the flow separated from the wall of the bulb downstream.
Since the pressure was not fully recovered, the bulb was unable to withstand the
capillary pressure and fluid was expelled back into the neck ? causing neck to reopen.
Examining the vorticity inside the bulb in figure 5.18, we observed a vortex ring
whose formation ? as reported by Hoepffner and Pare? [14] ? was critical to the neck
92
reopening phenomenon.
In figures 5.16 and 5.17, we also observed a narrow band of vorticity persisting
at the interface with the opposite sign as the main vortex ring. This region of
vorticity supported the formation of the vortex ring which in turn contributed to the
neck reopening phenomenon. We realized that the diffusion of this region of vorticity
would then hinder the neck reopening and allow for the recovery of the monotonic
pinching. Indeed, examining a monotonic pinching at Rec = 100 and ? = 0.01, we
observe that raising the viscosity of the outside fluid at the same Reynolds number
diffuses the vorticity in the system. Figure 5.19 shows the vorticity distribution
associated with monotonic end pinching.
93
Figure 5.19: Vorticity distribution in the system just before monotonic pinch-off
of the neck. Note that vorticity is diffused and no concentrated vortex rings are
observed.
It is evident from figure 5.19 that the large viscosity of the surrounding fluid
helped diffuse the vorticity in the system. When the viscosity of the external fluid
was large, vorticity was able to diffuse across the interface leading to a diminished
and almost non-existent vortex ring (see [104, 105] for a description of vorticity
transport across a flexible interface). This did not allow the neck to reopen and the
neck radius monotonically reduced to zero. The effect of vorticity diffusion was also
implicit in the fact that the bulb motion through the high viscous fluid was difficult.
94
5.5 Conclusion
This study was an attempt to uncover the dynamics behind drop deformation
and breakup in inertial systems. Most mixing processes in industries occur at
large Reynolds number and this study unveiled the behavior of drops in such
systems. Specifically, we investigated inertial effects on deforming drops using direct
numerical simulations and validated it with previous results from both stagnant and
non-stagnant extensional flows.
Subsequently, we studied the capillary-driven breakup of slender drops
suspended in a quiescent fluid. This problem was a natural extension of the previous
study where breakup of elongated drops was desired. While interfacial tension
can be used to drive the flow that breaks up drops, it is unclear how the effect
works in conjunction with viscous and inertial forces in the system. To address this
issue, we investigated capillary-driven breakup in a previously unexplored part of
the parametric space where viscosity of the outer fluid either supported or hindered
breakup of drops. Such problems are also common in industries where transient
capillary flows are used to dispersively mix drops within a fluid. In this study, we
thoroughly explained the effect of surrounding fluid at various Reynolds numbers
and rationalized the role of the three competing forces in the system. We presented
a stability diagram for drops in the new parametric space which can be useful guide
to industries during mixing process design. Experimental evidence will lead more
weightage to our findings and is a natural area for future study.
95
Appendix A: Supplementary material to chapter 3
A.1 Numerical simulation in ANSYS FLUENT
The geometry for the simulation consisted of the three dimensional flow
channel depicted in Fig. A.1 where fluid flow is streamwise along x in the direction
of gravity. We discretized the entire three dimensional domain using 179,080
hexahedral elements.
Figure A.1: (a) Front view and (b) isometric view of a planar channel with flow
streamwise along x. The flow domain was discretized into 179,080 hexahedral
elements.
96
Since the objective was to replicate experimental conditions, we performed the
numerical simulation of the flow for the highest achievable inlet speed of 8 mm s?1.
A small gauge pressure at the outlet was applied to prevent outflow of fluid under
stationary conditions. The pertinent boundary conditions for the flow were as
follows. At the inlet, a fixed velocity of 8 mm s?1 was applied. At the outlet, a small
guage pressure of 1764 Pa was maintained to replicate the experimental condition.
No-slip boundary condition for velocity was imposed along the channel walls.
A.1.1 Specifications of numerical solver
On the discretized geometry, we used a pressure-based solver to the integral
form of the time dependent governing equations of the incompressible flow available
as an option in ANSYS FLUENT. The pressure-velocity coupled scheme and spatial
discretization of pressure at second order was chosen. We used the second order
upwinding for momentum and an absolute convergence criteria of 1? 10?4 for all
residuals. The time step size was 5? 10?4 s for the transient simulation.
A.1.2 Results and verification
The transient simulation solved the flow at every time step and we tracked the
x component of velocity along the line joining (0, 0, 0) and (160 mm, 0, 0). We refer
to this line as the centerline of the flow where, due to the symmetry in the channel,
the y and z components of velocity are zero. Fig. A.2 shows the evolution of the non
dimensional velocity u? along the centerline x? at various time steps starting from an
97
initial state of no motion. Velocity and streamwise distance are non dimensionalized
as
Q
u = u? (A.1)
h2
and
x = ?x? (A.2)
out of which a non dimensional time scale arises. Q is the flow rate, h = 10 mm,
and ? = 160 mm. Fig. A.2 shows that the flow becomes fully developed in a very
short time scale that is much smaller than the time scale where a suspended phase
can feel the effect of the outer flow and start translating. Since the simulation was
performed at the highest achievable inlet speed in our experiments, we concluded
that the Reynolds number for lower inlet speeds will be much smaller and the flow
will develop almost instantaneously. Hence we chose to perform, and verify, steady
state simulations for the flow with the assumption that transient effects on suspended
particles were negligible.
To verify the results from the simulation, we performed a grid independence
study by decreasing the number of elements to 23,544 since the original number
of elements in the domain - 179,080 - was deemed too high. Fig. A.3 shows the
identical results from the grid independence study for a steady state flow at different
levels of discretization which verifies our original choice.
98
Figure A.2: Non dimensional velocity of the flow u? along the centerline x? at various
non dimensional timesteps. The time is non dimensionalized using the velocity and
length scale. Flow rapidly develops to the steady state.
Figure A.3: Streamwise velocity along the centerline at two levels of flow domain
discretization. Solid line represents 23,544 elements and dashed line represents
179,080 elements.
99
Figure A.4: Measurement of speed of tracer particle in a flow with inlet speed of
2.4 mm s?1
A.2 Raw experimental data
Experimental data for lubrication theory validation is shown in Fig. A.4 to
Fig. A.6 and measurements of droplet speeds at various Capillary numbers is shown
in Fig. A.7 to Fig. A.10. All plots use the coordinate system shown in Fig. A.1.
The non-dimensional version of the data is presented in the main paper.
100
Figure A.5: Non-normalized data for speed of tracer particle in a flow with inlet
speed of 3.2 mm s?1
Figure A.6: Non-normalized data for speed of tracer particle in a flow with inlet
speed of 4.8 mm s?1
101
Figure A.7: Speed of droplet with initial Capillary number of 0.12 compared with
the lubrication theory prediction.
Figure A.8: Speed of droplet with initial Capillary number of 0.15 compared with
the lubrication theory prediction.
102
Figure A.9: Speed of droplet with initial Capillary number of 0.60 compared with
the lubrication theory prediction.
Figure A.10: Speed of droplet with initial Capillary number of 0.77 compared with
the lubrication theory prediction.
103
Appendix B: Supplementary material to chapter 5
B.1 Grid independence study
For the capillary-driven breakup problem, verifying grid independence was
important to ensure validity of the numerical results. A sample case is illustrated
here to establish the validity of the results. We considered a case (Rec = 50 and
? = 0.10) where the capillary-driven flow produced an asymptotically unstable state.
Two levels of domain discretization ? 980 X 70 and 1176 X 84 ? was performed and
the RMS velocity of the flow was tracked. The latter discretization represents the
resolution used for all results reported in this study. Figure B.1 establishes grid
independence where nearly the same RMS velocity was observed for both levels of
discretization.
B.2 Inertial effect on breakup of slender drop in [1]
To provide some context to the results discussed in chapter 5, results from
Stone et al. [1] can be explored in the new part of the parametric space by varying
the level of inertia in the system. Figure B.2 shows the initial configuration from
the study by Stone et al. [1]. To this system, various levels of inertia is added and
104
Figure B.1: RMS velocity in the domain at Rec = 50 and ? = 0.10 for two levels of
domain discretization. Red is at 1176 X 84 resolution and green is at 980 X 70.
105
Figure B.2: Initial configuration for the capillary-driven flow problem in [1]
the drop behavior is tracked.
At Rec ? 0 and ? = 0.05, Stone et al. [1] studied the retraction of the slender
drop formed in a four-roll mill. When the external flow was stopped, the bulb at
the ends started to increase in radius and traverse towards the center of the slender
drop. However, this motion of the bulb was hindered because the outside fluid was
highly viscous. Since the ends of the drop could not retract, a neck ? defined as
the section with minimum radius ? was formed and resulted in pinch-off. Figure
B.3 shows the development of the capillary flow that resulted in the creation of
six daughter drops. Each daughter drop was formed successively in a phenomenon
known as retractive end pinching. In this section, we report several attributes of this
capillary flow by considering the effect of inertia which has largely been neglected in
previous studies. The development of the capillary flow for the same slender drop
at Rec = 100 is shown in figure B.4.
The two ends of the slender drops shown in figure B.4 were initially drawn
towards each other by a strong capillary flow that tried to minimize the interfacial
106
Figure B.3: Capillary-driven breakup in a slender drop at Rc ? 0 and ? = 0.05.
Figure is reproduced from Stone et al. [1] with permission.
area. The radius of the bulb at the end increased and a neck was formed. Pinch-off
was defined as the instant where the neck radius became zero. However, the radius
of the neck did not reach zero monotonically. After attaining a local minimum, the
neck reopened and the radius at the neck increased. Pinch-off was postponed to a
later time which resulted in the formation of daughter drops. From figure B.4, it
can be seen that at Rec = 100, five daughter drops were formed. The reopening
of the neck happened twice in the case of Rec = 250 and Rec = 500. However,
the consequence of the neck reopening with increasing Reynolds numbers resulted
in completely different final distributions of daughter drops as shown in figures B.5
and B.6.
107
Figure B.4: (a) to (k) show the capillary-driven breakup in a slender drop at Rec =
100 and ? = 0.05. Comparing with figure at Rec ? 0, only five daughter drops were
obtained at steady state (k).
108
Figure B.5: Steady state after capillary-driven breakup at Rec = 250 and ? = 0.05
producing four daughter drops.
Figure B.6: Steady state after capillary-driven breakup at Rec = 500 and ? = 0.05
producing three daughter drops.
B.3 Influence of the end-shape on capillary-driven flow
The shape of the ends of the slender drop has an effect on the capillary-driven
flow. The effect can be rationalized with a simple description. When the radius
of the end of the drop is large, the curvature is small and thus the magnitude of
interfacial force driving the flow is small. Large bulbs thus weakly drive the capillary
flow and bulb pinch-off becomes more likely. The opposite effect occurs when the
radius of the bulb is small. The flow is strongly dominated by interfacial force
leading to the system preferring coalescence instead of end-pinch.
Consider the case of the capillary-driven flow in a slender drop with Rec = 50
and ? = 0.10. In chapter 5, it was shown that this condition leads to an
asymptotically unstable end state where two daughter drops were formed. In this
section, we will show how changing the shape of the end for the same flow parameters
will result in a completely different configuration at the end.
To this end, consider an artificially imposed shape of the end as shown in figure
109
Figure B.7: Transient flow in a slender drop with a large radius of curvature near
the tips.
B.7. The bulb has a radius equal 2.5 times than the slender stem. Analyzing the
transient flow in figure B.7 reveals that the heavy bulb pinches off from the stem
very early and the steady state configuration yielded three daughter drops.
B.4 Non-trivial breakup modes under large inertia
Inertial effects can result in non-trivial behavior of slender drops that is
completely different from behavior at low Reynolds number. In this section we
report two such observations and discuss the mechanism.
110
B.4.1 Entrapment of surrounding fluid
Consider a slender drop with aspect ratio = 10 in a system with ? = 1.00.
When inertial effects are considered ? i.e. setting Rec = 50 ? in the capillary-
driven flow, we observed that the drop was conditionally unstable and asymptotically
stable. The drop underwent breakage at the middle and the two bulbs merged
together to form a stable drop. The mechanism was discussed in chapter 5 but
here we report a non-trivial behavior where the surrounding phase gets entrapped
into the drop. Figure B.8 shows the transient behavior of the drop and the phase
entrapment.
B.4.2 Torus formation
The parametric space shown in chapter 5 can be extended for ? > 1.00. In
this part of the parametric space, viscous effects from the surrounding fluid are less
important than the inertia associated with the capillary-driven flow. This causes an
interesting behavior where tori start to form in the transient flow. We investigated
this at ? = 100 and Rec = 100 for a slender drop with aspect ratio = 30. Figure
B.9 shows the formation of a torus after the bulbs pinch off.
111
Figure B.8: Inertial capillary flow in a slender drop with aspect ratio = 10 resulting
in entrapment of the surrounding phase.
112
Figure B.9: Formation of a torus structure in the transient flow. In the bottom
frame, the torus is surrounded by two daughter drops. The initial slender drop had
an aspect ratio = 30. For the inertial-capillary flow, Rec = 100 and ? = 100.
113
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