ABSTRACT
Title of dissertation: MAGNETORHEOLOGICAL FLUID
DYNAMICS FOR HIGH SPEED
ENERGY ABSORBERS
Stephen G. Sherman
Doctor of Philosophy, 2016
Dissertation directed by: Professor Norman M. Wereley
Department of Aerospace Engineering
Fluids with a controllable yield stress allow rapid variations in viscous force
in response to an externally applied field. These fluids are used in adaptive energy
dissipating devices, which have a controllable force response, reducing shock and
vibration loads on occupants and structures. This thesis investigates the physics
of these fluids at high speeds and shear rates, through particle modeling and fluid
dynamics. The focus is on the experimentally observed reduction in controllable force
at high speeds seen in magnetorheological (MR) fluid, a suspension of magnetizable
particles that develop a yield stress when a magnetic field is applied.
After ruling out particle dynamic effects, this dissertation takes the first
rigorous look at the fluid dynamics of a controllable yield stress fluid entering an
active region. A simplified model of the flow is developed and, using computational
fluid dynamics to inform a control volume analysis, we show that the reduction
in high speed controllable force is caused by fluid dynamics. The control volume
analysis provides a rigorous criteria for the onset of high speed force effects, based
purely on nondimensional fluid quantities. Fits for pressure loss in the simplified flow
are constructed, allowing yield force prediction in arbitrary flow mode geometries.
The fits are experimentally validated by accurately predicting yield force in all of
the known high speed devices. These results should enable the design of a next
generation of high performance adaptive energy absorbers.
MAGNETORHEOLOGICAL FLUID DYNAMICS WITH
APPLICATION TO HIGH SPEED ENERGY ABSORBERS
by
Stephen G. Sherman
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
2016
Advisory Committee:
Professor Norman M. Wereley, Chair/Advisor
Professor James Baeder
Professor Howard Elman
Professor Alison Flatau
Professor Derek Paley
c© Copyright by
Stephen G. Sherman
2016
Acknowledgments
This thesis would not have been possible without an advisor who encouraged
and enabled me to do the best work I could. For this, and so much more, I thank
my advisor Prof. Norman Wereley. For putting me on this path, all the credit, or
perhaps blame, belongs to Prof. Derek Paley, my undergraduate advisor. If I had
not been assigned a new professor who needed bodies for his new lab, I doubt I ever
would have gone down this path.
I must also thank my other committee members, Prof. James Baeder, Prof.
Alison Flatau and Prof. Howard Elman, for their advice and support in writing this
thesis. And of course, none of this would have been possible without Prof. Wereley,
Paley and Baeder writing the letters of recommendation that won the fellowship
that supported this work.
I also would have been unable to complete this without my labmates and
friends. I must thank Andrew Becnel, Ami Powell and Harinder Singh, who passed
on their knowledge of MR fluids. The final section of this thesis is impossible to
imagine without Dr. Young-Tai Choi, whose counsel was invaluable. To my group
mates, Tom Pillsbury, John Chambers, Caitlin King, Mingfu Wen, Robert Vocke,
Ryan Robinson, Brian Powers, John Chambers, John Kunkel, Lei Xie, Erica Hocking
and Xian-Xu Bai, whose patience I will never be able to repay. To Shane, Will, Joe
and Elizabeth, for always being willing to get lunch.
Finally, to my parents and sister, for supporting me in all my adventures and
for encouraging me to pursue this crazy goal.
ii
This material is based upon work supported by the National Science Foundation
Graduate Research Fellowship Program under Grant No. DGE1322106. Any opinions,
findings, and conclusions or recommendations expressed in this material are those
of the author(s) and do not necessarily reflect the views of the National Science
Foundation.
iii
Table of Contents
List of Tables vii
List of Figures viii
List of Abbreviations xii
1 Introduction 1
1.1 Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 The equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Improving particle model accuracy . . . . . . . . . . . . . . . . . . . 11
1.4 Particle model analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.6 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Flow of a controllable yield stress fluid into an active region 20
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.1 Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.2 1D Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.3 Quasi 1D flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.4 2D flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.4.1 Transition zone effects . . . . . . . . . . . . . . . . . 40
2.2.4.2 Momentum flux effects . . . . . . . . . . . . . . . . . 41
2.2.4.3 Flow features of interest . . . . . . . . . . . . . . . . 43
2.2.5 Correction Factor . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2.5.1 An analytical estimate for change in active length . . 47
2.2.6 Computational Fluid Dynamics . . . . . . . . . . . . . . . . . 50
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3.1 Numerical validation . . . . . . . . . . . . . . . . . . . . . . . 51
2.3.2 Activation flow . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3.2.1 Basic Flow visualization . . . . . . . . . . . . . . . . 55
2.3.2.2 Graetz coordinate and fits . . . . . . . . . . . . . . . 63
iv
2.3.2.3 Force correction factor . . . . . . . . . . . . . . . . . 68
2.3.3 Deactivation Flow . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3 Experimental validation 75
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.2.1 Off-state forces . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.2.2 Yield Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.2.3 Contraction flow . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2.4 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.2.5 Time domain effects . . . . . . . . . . . . . . . . . . . . . . . 91
3.3 Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.3.1 Goncalves’ slit rheometer . . . . . . . . . . . . . . . . . . . . . 95
3.3.1.1 Temperature effects . . . . . . . . . . . . . . . . . . 96
3.3.1.2 Velocity squared effects . . . . . . . . . . . . . . . . 98
3.3.1.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . . 99
3.3.1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.3.2 Min Mao’s double ended rod damper . . . . . . . . . . . . . . 104
3.3.3 Harinder Singh’s large damper . . . . . . . . . . . . . . . . . . 109
3.3.4 Choi and Robinson . . . . . . . . . . . . . . . . . . . . . . . . 111
3.3.5 Sanwa Tekki seismic dampers . . . . . . . . . . . . . . . . . . 113
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4 Conclusions 119
4.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
A Flow Images 124
B Correction factors 135
C Temperature dependence of MR fluids 144
C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
C.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
C.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
C.3.1 Simple MR Fluids . . . . . . . . . . . . . . . . . . . . . . . . . 150
C.3.2 Commercial MR Fluids . . . . . . . . . . . . . . . . . . . . . . 154
C.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
D Sheet formation at large scales 157
D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
D.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
D.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
D.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
v
E The effect of particle size distribution on chain structures in MR fluids 168
E.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
E.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
E.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
E.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
E.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
F The Bingham and Mason numbers 182
F.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
F.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
F.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
F.3.1 The Bingham Number . . . . . . . . . . . . . . . . . . . . . . 185
F.3.2 Mason Number . . . . . . . . . . . . . . . . . . . . . . . . . . 187
F.4 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
F.4.1 Normalized Yield Stress . . . . . . . . . . . . . . . . . . . . . 193
F.4.2 Normalized Viscosity . . . . . . . . . . . . . . . . . . . . . . . 195
F.5 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
F.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
F.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
F.8 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
G The high shear rate chain limit 207
G.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
G.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
G.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
G.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
H Chain dynamics response time 217
H.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
H.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
H.3 Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
H.4 Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
H.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
vi
List of Tables
3.1 Fluid dynamic properties of potential high speed magnetorheological
dampers. We compare our model to the devices marked with an
asterisk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.2 Key parameters for the high speed devices in Table 3.1. . . . . . . . 94
B.1 np for LA/Dh = 13.97 . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
B.2 nv for LA/Dh = 13.97 . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
B.3 np for LA/Dh = 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
B.4 nv for LA/Dh = 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
B.5 np for LA/Dh = 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
B.6 nv for LA/Dh = 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
B.7 np for LA/Dh = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
B.8 nv for LA/Dh = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
C.1 Custom Fluid Properties . . . . . . . . . . . . . . . . . . . . . . . . . 152
C.2 Lord Fluid Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
vii
List of Figures
1.1 Diagram of two particles in a shearing fluid. . . . . . . . . . . . . . . 6
2.1 Rheological models. Solid is Bingham plastic, dashed is biviscous,
dash-dot is Newtonian. . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 The basic steady-state 1-D flow profiles, parabolic flow and plug flow. 32
2.3 A simplified model of Goncalves’ experiment. The active region is
shaded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4 Simulation model of the end of Goncalves duct. . . . . . . . . . . . . 39
2.5 Profile momentum, β vs. Bingham number for various values of η∗. . 41
2.6 Pressure active length correction from momentum flux effects, np(β),
Eq. (2.33). The straight pink lines are the limiting values for np =
−0.25 from Equation 2.34 and Equation 2.35. . . . . . . . . . . . . . 48
2.7 Two dimensional periodic pipe error as a function of N , the number
of elements to the midline. . . . . . . . . . . . . . . . . . . . . . . . 52
2.8 Activation flow solution as a function of mesh size at Re = 500 and
Bi = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.9 Solution as a function of η∗ at Re = 2000, Bi = 100. . . . . . . . . . 54
2.10 Normalized velocity at Bi = 10. The active region starts at x = 0. . 56
2.11 Plug visualization through viscosity at Bi = 10. The active region
starts at x = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.12 Normalized velocity and wall shear stress as a function of Re for
Bi = 10 and LA/Dh = 7. Dashed lines, – –, is the fully developed
solution, Equation 2.17 or 2.19, and – · is parabolic wall shear stress,
Equation 2.20 and · · · is the 90% entrance length convergence criteria. 59
2.13 Comparison of velocity and wall shear stress across various active
lengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.14 Velocity entrance length xe,u for La/Dh = 14. Dotted black lines are
the bounds of sampled space. The pink dashed line is Cτy = 0.4. . . 61
2.15 Pressure loss as a function of position at Re=2000.0, Bi=10.0. Fully
developed losses are from Equation 2.21, and parabolic losses from
Equation 2.22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
viii
2.16 Velocity and wall shear stress in terms of Graetz coordinate. The
thick dashed black lines are entrance length based fits. Dashed lines a
fully developed value, dotted line is the entrance length criteria, and
dash-dot is the parabolic wall shear stress value. . . . . . . . . . . . . 64
2.17 Graetz scaled entrance length. Dashed lines are fits of xe,u and xe,τw . 65
2.18 Inertia correction in various forms. The dotted lines are the upper and
lower bounds on sampled space and the pink dashed line is Cτy = 0.4.
The slid contours are obtained directly from CFD. The light grey
circles in Fig. 2.18b are the CFD test points. Dashed contours in
Fig. 2.18c are Eq. 2.45, and the brown dashed line in Fig. 2.18d is the
transition to turbulence. . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.19 Active length correction, np from He = 10
3 − 106. Dashed lines are
fits of np. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.20 Active length viscous correction, nv. Dashed lines are fits. . . . . . . 70
2.21 Apparent yield force, normalized by parabolic yield force using np.
Solid lines are CFD, dashed lines are fits, dash dot is the fully developed
model, and dotted is the parabolic model. . . . . . . . . . . . . . . . 71
2.22 Deactivation correction factors for LA/Dh = 4 and LN/Dh = 13.97.
The box is the test window of Goncalves. . . . . . . . . . . . . . . . 72
3.1 Fluid dynamic properties of MR dampers in both raw units and
nondimensional form. See Table 3.1 for details. . . . . . . . . . . . . 78
3.2 Schematic of a step contraction. . . . . . . . . . . . . . . . . . . . . 85
3.3 Response time for Bingham plastic channel start up flow. . . . . . . . 92
3.4 As reported. For LA = 25.4 mm, η∞ = 0.116 Pa s, K = 0.01, and for
LA = 6.35 mm, η∞ = 0.137 Pa s, K = 0.02. Solid lines are activation
flow based fits, dotted lines the parabolic model, dash-dot plug flow.
Pink dashed line is Cτy = 0.4. . . . . . . . . . . . . . . . . . . . . . . 100
3.5 Temperature increase modeled by ∆T ∝ v∆P , and the ASTM D341
viscosity-temperature correction for the carrier fluid. For LA =
25.4 mm, η∞ = 0.110 Pa s, K = 1.07, and for LA = 6.35 mm, η∞ =
0.131 Pa s, K = 1.22. Pink dashed line is Cτy = 0.4. . . . . . . . . . . 103
3.6 Min Mao’s SSMREA1 [14]. Solid lines are activation flow based fits,
dotted lines the parabolic model, dash-dot plug flow. The x symbols
correspond to sinusoidal testing data, circles are drop stand data. In
the yield force diagrams, pink dash dot is τy,app/0.5ρv
2
gap = 0.4, and
dashed brown is the onset of turbulence for a Bingham plastic. . . . . 106
3.7 Mao’s SSMREA2 [14]. Solid lines are activation flow based fits,
dotted lines the parabolic model, dash-dot plug flow. The x symbols
correspond to sinusoidal testing data, circles are drop stand data. In
the yield force diagrams, pink dash dot is τy,app/0.5ρv
2
gap = 0.4, and
dashed brown is the onset of turbulence for a Bingham plastic. . . . . 107
ix
3.8 Singh’s large damper [109]. Solid lines are activation flow based fits,
dotted lines the parabolic model, dash-dot plug flow. In the yield force
diagram, pink dash dot is τy,app/0.5ρv
2
gap = 0.4, and dashed brown is
the onset of turbulence for a Bingham plastic. . . . . . . . . . . . . . 110
3.9 Choi’s landing gear damper [107]. Solid lines are activation flow based
fits, dotted lines the parabolic model, dash-dot plug flow. In the yield
force diagram, pink dash dot is τy,app/0.5ρv
2
gap = 0.4, and dashed
brown is the onset of turbulence for a Bingham plastic. . . . . . . . . 112
3.10 Sodeyama’s 300 kN seismic damper [15]. Solid lines are activation flow
based fits, dotted lines the parabolic model, dash-dot plug flow. In
the yield force diagram, pink dash dot is τy,app/0.5ρv
2
gap = 0.4 . . . . . 115
A.1 Normalized velocity at Bi = 1. The active region starts at x = 0. . . 125
A.2 Normalized velocity at Bi = 3. The active region starts at x = 0. . . 126
A.3 Normalized velocity at Bi = 10. The active region starts at x = 0. . 127
A.4 Normalized velocity at Bi = 30. The active region starts at x = 0. . 128
A.5 Normalized velocity at Bi = 100. The active region starts at x = 0. . 129
A.6 Normalized viscosity at Bi = 1. The active region starts at x = 0. . . 130
A.7 Normalized viscosity at Bi = 3. The active region starts at x = 0. . . 131
A.8 Normalized viscosity at Bi = 10. The active region starts at x = 0. . 132
A.9 Normalized viscosity at Bi = 30. The active region starts at x = 0. . 133
A.10 Normalized viscosity at Bi = 100. The active region starts at x = 0. 134
C.1 Particle magnetization in the PlainFE30 fluid as a function of applied
field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
C.2 Viscosity vs. temperature for the simple fluids, including the magne-
tized sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
C.3 Offstate yield stress for the simple fluids. FE30-MCR corresponds
to samples magnetized by the rheometer and FE30-Nd for samples
magnetized by a neodymium magnet. . . . . . . . . . . . . . . . . . 153
C.4 Viscosity for temperature for a series of commercially available fluids
and their extracted carrier fluid. . . . . . . . . . . . . . . . . . . . . . 155
D.1 Structural pattern of ER fluid under 3 conditions: (a) no electric field
or shear, showing a random distribution of particles, (b) electric field
is applied, causing chain formation and (c) with both electric field
and shear flow, showing lamellar sheet formation. . . . . . . . . . . . 158
D.2 Schematic of simulation volume. . . . . . . . . . . . . . . . . . . . . . 160
D.3 Graphical output of the simulation code, at shear γ = 8 with M =
300 kA/m and γ˙ = 500 s−1, looking in the direction of shear. . . . . . 161
D.4 Incremental number density of particles at H0 = 100 kA/m and
γ˙ = 250 s−1 in a portion of a h = 0.5 mm volume as γ increases. . . . 162
D.5 Nondimensional chain parameters versus shear at h = 0.5 mm. Blue
and red lines correspond to M = 300 and 600 kA/m respectively. . . . 166
x
D.6 Nondimensional chain parameters versus shear at M = 300 kA/m.
Blue and red lines correspond to γ˙ = 500 and 2000 s−1 respectively. . 167
E.1 Particle size distribution. . . . . . . . . . . . . . . . . . . . . . . . . . 171
E.2 Simulation volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
E.3 Stress versus Mason number. . . . . . . . . . . . . . . . . . . . . . . . 176
E.4 Particle structure at γ = 4 and Mn = 2.4× 10−4 with a volume height
of 0.35 mm. Colors correspond to force on the particles. . . . . . . . 177
E.5 Stress versus distribution size for various Mason numbers. . . . . . . 179
E.6 Cluster size versus distribution size for various Mason numbers. . . . 179
E.7 Connectivity versus distribution size for various Mason numbers. . . . 179
F.1 Idealized rheogram or shear stress vs. shear rate diagram for an MR
fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
F.2 Diagram of two particles in a shearing fluid. . . . . . . . . . . . . . . 188
F.3 Normalized apparent viscosity for Lord MRF-140CG . . . . . . . . . 198
F.4 Normalized stress vs Mason number, with high Mason number data
as inset. The black line corresponds to a linear least squares fit, and
shows how the low Mason based fit agrees with the inset high Mason
number data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
G.1 Reference frames for the particle system. . . . . . . . . . . . . . . . . 209
G.2 Static stress with lines indicating the limiting Mason number for S=2,
3 and 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
G.3 Dynamic stress with lines indicating Mn∗ for S=2, 3 and 4. . . . . . 214
G.4 Steady state chain length with dashed vertical lines indicating Mn∗
for S = 2, 3 and 4. The solid line indicates the predicted value of
cluster size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
H.1 Simulated dimensionless response time as a function of volume fraction. 226
H.2 Effects of simulation parameters on γr. . . . . . . . . . . . . . . . . . 227
xi
List of Symbols and Abbreviations
Fluid quantities
τ Shear stress
τy Yield stress
τw Wall shear stress
γ Shear strain
γ˙ Shear rate
p Local pressure
P Average cross section pressure
u Local velocity
v0 Reference velocity
vgap Active gap mean velocity, typical choice for v0
vp Piston velocity
dp
dx
Pressure gradient in the flow direction
N Number of grid elements in the crosss flow direction
η Viscosity
η∞ Plastic viscosity or viscosity at high shear rate
η0 Low shear rate viscosity
ηc Carrier fluid viscosity
ρ Density
K Hydraulic loss coefficient
Geometric quantities
h Active gap height
Dh Active gap hydraulic diameter
L Length
LA Active region length
LN Passive region length
A Area
Ap Piston cross section area
Ares Fluid reservoir cross section area
Agap Active gap cross section area
Force model
β Profile momentum factor
np Pressure active length correction
nv Viscous loss active length correction
∆P Duct pressure loss
x′ (x/Dh)/Re
xe,τw Wall shear stress entrance length
xe,u Velocity entrance length
xii
Particle quantities
a Radius
σ Characteristic diameter
x Position
k Collision stiffness parameter
F Force
N Number of particles
m Dipole moment
M Particle mass
Magnetics
H H field
B B field
M Magnetization
µ0 Vacuum permeability
µr Relative permeability
χ Susceptibility
Nondimensional numbers
Re Reynolds number Inertial to viscous
ρvgapDh
η∞
He Hedstrom number Dimensionless yield stress
ρτyD
2
h
η2∞
Bi Bingham number Yield to viscous
τy h
η∞vgap
Cτy Yield stress coefficient Yield to inertial
τy
1
2
v2gap
Mn Mason number Particle viscous to magnetic 144
ηcγ˙
µ0M
2
δ Dimensionless plug thickness Yield stress to wall stress τy/τw
η∗ Regularization parameter Post-yield to pre-yield viscosity η∞/η0
xiii
Superscripts and subscripts
B Bingham
b biviscous
N Newtonian
0 Reference
y Yield
c Transition to turbulence or carrier fluid
p Parabolic
w Wall
e Entrance
gap Active gap
res Reservoir
rep Repulsion
fit Curve fit
act Activation flow
∞ high shear rate
Abbreviations
MR Magnetorheological
ER Electrorheological
MREA MR energy absorber
SAOS Small amplitude oscillatory shear
GPU Graphics processing unit
CFD Computational fluid dynamics
xiv
Chapter 1: Introduction
Mitigating vibration and impact loads is an essential task for designing safe
and reliable systems. Devices designed to prevent these unwanted loads fall into
three categories: passive, semi-active and active, defined by the level of control
each device gives [1]–[4]. Passive devices resist motion with a fixed force, and are
fixed systems. Classic examples are elastomeric isolators, viscous energy absorbers,
frictional dampers and deformable structures. Active devices can apply a variable
force that resists or encourages motion, but are necessarily more complicated, with
typical devices being actuated by hydraulic, pneumatic or piezoelectric mechanisms.
Semi-active devices can apply a variable force, but only one that resists, and offer a
balance of complexity and controllable force.
The benefit of a controllable energy absorption device is that closed-loop control
can be used to reduce loads more effectively across a wider range of disturbances
than with a fixed device. Semi-active dampers have become popular because the
balance between controllability and complexity allows improved performance in
small volumes at affordable prices. Some variants of semi-active dampers include
variable orifice dampers and viscous dampers with controllable yield stress fluids.
This dissertation focuses on controllable yield stress fluid devices, the most popular
1
type of semi-active damper [5].
Controllable yield stress fluids are fluids that upon the application of an external
electric/magnetic field, develop a controllable yield stress, and can thus be used to
create a controllable force. There are two main types of controllable yield stress fluids,
electrorheological (ER) and magnetorheological (MR) fluids, named for their change
in viscosity (rheological response) upon application of an electrical or magnetic
field respectively. Both fluids consist of micron scale particles in a carrier oil, and
upon application of an electric/magnetic field, develop chain-like microstructures
and a yield stress whose strength is dependent on the strength of the applied field.
Electrorheological fluids are suspensions of dielectric particles, typically silica, in a
non-conducting oil, invented by Winslow [6], have yield stresses of about 5 kPa, and
operate at electric fields of roughly 1 kV/mm. Magnetorheological fluids, invented
by Rabinow [7] are suspensions of magnetically permeable particles, typically iron,
in oil and have yield stresses of around 50 kPa. Magnetorheological fluids are more
popular than ER fluids, as they offer larger yield stresses, operate at lower voltages,
and are insensitive to the presence of impurities. However, the physics of ER and
MR fluids are very similar, and insights from one fluid can be applied to the other.
While the focus of this dissertation is on MR fluids, all of the models and results in
this dissertation should be equally applicable to ER fluids.
Devices using ER/MR fluids work in some combination of three modes: shear,
flow and squeeze. This dissertation focuses on flow mode devices, and in flow mode,
2
the Bingham plastic model is the classic rheological model for the fluid:
τ = τy + η∞γ˙ (1.1)
where stress τ is composed of a field-dependent yield stress τy and a field independent
plastic viscosity η∞. Flow mode devices work by pushing lots of fluid through a
small gap, generating viscous pressure losses. For controllable yield stress fluids,
part of the gap, the active gap, has an electric/magnetic field running through it,
generating a controllable yield stress, altering the pressure loss in the gap, and thus
providing a controllable viscous force. The the controllable component of the force
output is known as the yield force, and is the difference in force between when the
applied field is on and when the field is off , Fy = Fon − Foff .
Magnetorheological devices have found wide application in the automotive
industry, particularly as adaptive suspensions [5], [8]. Recently, interest has grown
into using MR devices to absorb high-speed impacts for improved automotive and
helicopter crash safety [9], [10]. However tests at impact speeds have shown a
disappearing yield force, potentially preventing these devices from being used in
impact absorbers.
In particular two sets of experiments come to mind. Goncalves [11] and
Goncalves et al. [12] tested a capillary rheometer capable of reaching 37 m/s in the
active gap, and sees a drop in controllable force when the fluid spends less than one
millisecond in the active region. In my group, Mao [13] and Mao et al. [14] tested the
impact performance of dampers at peak piston velocities up to 5 m/s, corresponding
3
to fluid speeds over 40 m/s in the active gap and saw a smaller reduction in yield
force. Perhaps surprisingly, in the course of this research we realized that some
seismic energy absorbers also operate at high speeds, despite piston velocities below
0.5 m/s, as large piston to gap area rations mean gap velocities can be in excess of
25 m/s [15]. In these devices, yield force decreases as fluid velocity increases, and we
call this reduction high speed yield force falloff. Our problem is that there are no
models for yield force that produce the experimentally observed high speed yield
force falloff.
But for devices operating at conventional speeds, if we know the yield stress,
the Bingham plastic model accurately predicts the yield force. But since existing
Bingham plastic yield force models cannot predict high speed yield force falloff, nor
can they predict phenomenon like squeeze-shear strengthening of MR fluids [16],
it is clear that we need an improved understanding of the physics. Since the field
induced microstructure causes the yield stress, it seems reasonable to investigate the
microstructure. To do this, we turn to particle modeling.
1.1 Particles
Particle modeling is popular because a basic model will easily reproduce
the basic features of experimentally observed microstructure. With the aid of a
microscope, the microstructure is revealed to be thick columnar structures spanning
the active gap, breaking and reforming under shear. These columnar structures are
often idealized as particle chains, and chains form the classic explanation for why
4
the yield force exists. However, nearly all good images of ER/MR fluids under shear
are of suspensions with a particle volume fraction, φ, less than 2% [17], [18], whereas
a typical engineering scenario uses fluids that are totally opaque, as the suspension
has a high solids loading, φ > 20%.
For dense fluids, the preferred method for understanding the chain structures
is to numerically simulate them. There has been masses of work in this area, but
the general goal is nearly always the same: to understand the chains in order to
understand the yield stress. But in all these simulations, yield stress is underpredicted
by a factor of three or greater for conventional MR fluids at large magnetic fields
[18]–[30]. However, the changes in simulated yield stress and particle structure
correlate reasonably well to experimental results, making them useful in designing
novel fluids [29], [30].
Since my group was designing devices that were experiencing counter-intuitive
experimental results, investigating the chain structures in real devices seemed like it
might provide useful insight into the problem. However, devices are large, and most
papers in the literature look at small, often 2D, control volumes, typically using
≈ 103 particles. At the start of my graduate career, there existed a potential solution
to this problem: the ability to run highly parallel code up to 100× faster by using
the computers graphics processing unit (GPUs) instead of its CPU [31]. So I wrote
a custom code capable of simulating the behavior of 106 particles in an MR fluid
on a single GPU [19], an increase of roughly 100× over the literature. With typical
8 µm particles, this number is sufficient to simulate a cubic fluid element with height
h = 0.5 mm (typical parallel-plate rheometer measurements have a gap height of
5
Figure 1.1: Diagram of two particles in a shearing fluid.
0.5 mm), thereby meeting the goal of simulating at experimental length scales.
But increasing simulation scale had no effect (when properly scaled) on the
particle chain structure or the yield stress [19]. The lack of change in chain structure
showed directly that the experimental phenomena were not length scale related, and
the lack of change in yield stress that simulating at device scales would not resolve
the yield force inaccuracy. However, we had developed a good tool, and the set of
physics used in the particle dynamics has obvious limitations, so I hoped to improve
the accuracy by improving the physics. In order to understand my attempts at
improving the physics, the next section goes through the construction of the typical
model, as well as serving as a literature review of how others have approached this
same problem.
1.2 The equations of motion
The particles in ER/MR fluids are electrically/magnetically polarizable par-
ticles, typically spherical in shape, and placed in between the poles of a large
6
electric/magnetic circuit. In this, case the classic treatment is to model the particles
as uniform spheres interacting in a uniform external magnetic field. Since linearly
polarizable spheres in a uniform field develop a pure dipole moment, the standard
approximation is to treat the particles as point dipoles. Similarly, so will particles in
a magnetic field large enough to fully saturate the particle. Specializing for MR fluids
here on out, for particle i with position xi, shown in Fig. 1.1, with radius ai, vector
magnetization M, and mean magnetization M , the dipole moment is mi = 4pia
3
iM/3.
We also adopt the chemical engineering notation, where the reference diameter is σ.
The magnetic field from dipole j at xi is
Bij =
µ0
4pi
3rˆij(mj · rˆij)−mj
r3ij
, (1.2)
where rij = xi − xj. The force on dipole i in an arbitrary B field is Fi = ∇(mi ·B).
Then the force on dipole i from the magnetic field of dipole j is [32]
Fmij =
3µ0mimj
4pir4
(rˆij(mˆi · mˆj + mˆi(rˆij · mˆj) + mˆj(rˆij · mˆi)− 5rˆij(rˆij · mˆi)(rˆij · mˆj)) ,
where superscript m denotes magnetic forces.
Next we specialize for the case where the external applied field is large and the
the dipole moments are aligned. Using θij as defined in Fig. 1.1, the magnetic force
is
Fmij = F
m
ij
(3 cos2 θij − 1)er + sin 2θijeθ
(rij/(ai + aj))4
(1.3)
7
with the dipole force magnitude,
Fmij =
3µ0mimj
4pi(ai + aj)4
and reference force magnitude
F0 =
pi
48
µ0M
2σ2.
Multipole moments or mutual dipole models can be used to increase the fidelity
of the magnetic models at intermediate field strengths [27], [33], reducing off-axis
repulsion forces. However, as discussed below these models have a minimal effect on
predicted yield stresses when compared at an equivalent magnetization.
For particle contact repulsion forces, we choose the standard Klingenberg style
model [34],
Frepij = −2Fmij exp
[
−k
(rij
σ
− 1
)]
er (1.4)
Alternate repulsion force models exist [20], [26], including ones more physically
realistic, but these generally have minimal impact, as k > 50 is sufficiently stiff for
most problems of interest [20].
The other major force on the particle is the viscous drag force. Since the
particles are very small spheres, the Reynolds number is nearly zero, viscous drag
forces can be represented by Stokes drag for an isolated sphere,
Fd = −6piηcai(x˙i − vc) = −Di(x˙i − vc),
8
with with particle i drag factor Di = 6piηcai, reference drag factor D0 = 3piηcσ, and
carrier fluid velocity vc. To capture the effects of shearing, the carrier fluid velocity is
assumed to be the shear flow velocity profile, vc = γ˙yex. While assuming an isolated
sphere is problematic at large volume fractions, simulations employing smoothed
particle hydrodynamics and Lattice Boltzmann methods showed no major changes
to the observed yield stress [21], [22], [26].
The next observation is that, in the dynamics, the accelerations of the particles
are negligible [25], [34]. Starting with the equations of motion for a single particle,
4piρa3i
3
x¨i = −Di(x˙i − γ˙y ex) +
N∑
j=1
j 6=i
(
Fmij + F
rep
ij
)
. (1.5)
Note that for computational ease, force interactions are cut off when rij ≥ 4σ, or at
0.5% of the particles-in-contact maximum.
Nondimensionalizing through force F0, time scale D0σ/F0 = 144ηc/µ0M
2, and
length scale σ, using Mi for mass, and dividing by F0 yields:
MiF0
D20σ
x¨∗i = −
Di
D0
(x˙∗i −
D0σγ˙
F0
y∗ ex) +
N∑
j=1
j 6=i
(
Fmij
F0
+
Frepij
F0
)
(1.6)
where x¨∗i , x˙
∗
i and y
∗ are dimensionless acceleration, velocity and y coordinate re-
spectively. The leading term is here fairly small, ≈ 10−2 but perhaps not truly
negligible. However, results show that models including acceleration produce yield
stresses equivalent to those that do not [21]. This finding is unsurprising, as the
yield stress is essentially a static phenomenon. Dropping acceleration, we obtain the
9
simplified model
Di
D0
x˙∗i =
Di
D0
Mn y∗ ex +
N∑
j=1
j 6=i
(
Fmij
F0
+
Fcij
F0
)
. (1.7)
This expression introduces Mason number,
Mn =
D0σγ˙
F0
= 144
ηcγ˙
µ0M2
, (1.8)
the ratio of particle shear forces to particle magnetic forces. This definition has the
benefit of using a consistent length scale used in the derivation, unlike in [35], which
mixes length scales, producing a factor of 9/2 in the front. The Mason number is
the most useful concept that comes out of particle dynamics, because if viscosity is
plotted against Mason number, measurements at different magnetic fields collapse
onto the same master curve. This works best with Mason number in terms of M , not
H, as τy/M
2 is approximately constant for any applied field, due to the measured
magnetic properties accounting for material nonlinearities and self interaction [35],
[36]. Becnel et al. [37], [38] used this to demonstrate that high shear-rate shear-mode
devices exhibit the same Mason number scaling as low speed devices, demonstrating
that the physics at low and high shear rates are unchanged.
The advantages of using M2 extends to modeling the particle interactions.
For magnetic force models using nonlinear material properties and magnetically
interacting particles, the particles will mutually magnetize, increasing M , and causing
stronger interactions when compared to magnetization by the reference H field alone.
10
However, finite element calculations and mutual dipoles models show that these
interactions are stronger only when compared at an equivalent H [35], [36], as at
an equivalent M forces are nearly unchanged. Therefore, developing more refined
models of interacting spheres will have minimal benefit, as we know at saturation the
particles are definitely dipoles, and from experiments using Mason number analysis,
that the fluid behavior at saturation magnetization is similar to the behavior at
lower field strengths.
There are other concerns, such as thermal effects from Brownian motion, but
these are expected from theory to be negligible for typical high solids-loading MR
fluids at large magnetic fields [39]. To demonstrate this, I performed a rheological
study on a series of typical MR fluids combined with a nondimensional analysis
of remnant magnetization effects and demonstrated that the observed changes in
suspension viscosity due to temperature were equivalent the temperature changes in
viscosity of the carrier fluid [40], which is contained in Appendix C.
1.3 Improving particle model accuracy
But as stated before, this typical model of MR fluid has a substantial limitation:
it does not predict yield stress accurately. More problematically, it also seems that
refining the existing force terms produces little benefit, something that I learned
after investigating various improvements.
First, in Sherman et al. [19], Appendix D, we explored the effects of volume
size on the chain structures in the fluid, leveraging our high performance code. We
11
also made the observation that our simulation code was reproducing the lamellar
sheet phenomenon seen experimentally. However, during later work it became clear
that the sheet formations are strongly influenced by the force cut off, and in fact are
likely a numerical artifact. Similar large scale work [20], saw similar sheet structures,
but doesn’t draw the conclusion that we have, that they are a numerical artifact.
However, the force cut is placed far enough away that these sheets have no effect on
yield force.
Later, in Sherman and Wereley [41], Appendix E we improved particle model
accuracy by including a realistic particle size distribution. We observed significant
changes in the particle structure during chain and sheet formation, as well as some
changes to stress during γ = 4–8. However, this averaging period was based on
uniform sized results, but with a size distribution some transients can still be seen.
Later, a much more thorough paper [42], which included both thermal and inertial
effects, investigated the same concerns. They found that the size distribution affected
the structure, smoothing the yielding process, but that steady state stress was
essentially unaffected. While the conclusions between the two vary slightly, I think
both papers agree that adding a size distribution changes the particle structure and
the transients of the stress response.
The most direct approach is to refine the existing force terms, as discussed
in the previous section. In particular, Hanna Lagger used a model which refined
all the existing force terms simultaneously [20]–[22], improving magnetic, contact
and hydrodynamic force model accuracy, but still produced yield stresses roughly
one third of those seen experimentally. Since the equations of motion used by all
12
authors underestimate the yield stress by such a large factor, and improving accuracy
of individual elements produces no major change in yield force, I believe that a
fundamental force is missing.
Since the error in yield force scales with magnetic field, it follows that we need
to add a force that also scales with magnetic field but is not a magnetic force, as
magnetic forces have already been investigated. The only force that I can think of
that satisfies this criteria is friction, where the magnetic attraction forces will cause
large normal forces, and thus large amounts of friction that depend on magnetic field
strength. I believe the lack of friction to be the most viable explanation as to why
most simulations fail to produce accurate yield stresses.
Including friction also explains a series of experimental results. Vereda et
al. [43] looked at spherical particles with different amounts of surface roughness,
and concluded that friction is needed to accurately model the particle dynamics.
They show that particle surface roughness has a significant impact on experimental
measurements of yield stress, but demonstrate via FEM that surface roughness does
not effect the magnetic particle forces at equivalent M2. From this, the authors
conclude that friction must be responsible for the enhancement in yield stress caused
by increasing the surface roughness. Similarly, for slender rod particles, several
authors have concluded that friction is needed to accurately model the interactions
between rods [44]–[46]. Many authors also report that precise yield stress values
are dependent on the testing instrument surface roughness and material [47], [48],
with magnetic wall materials increasing yield stress, a result consistent with particles
interacting with boundary-image particles and resulting normal and friction forces.
13
Friction is also the most viable explanation for squeeze-shear strengthening, where
an applied normal force increases shear mode shear stresses [49].
However, properly implementing static friction is complicated, as it can easily
drive the timestep to zero, and advanced techniques are not available for the acceler-
ation free models used here. Simulations using friction have been performed particle
chains under small amplitude oscillatory strain and found including friction improved
accuracy [50]. However, to my knowledge, there is also no way to estimate friction
ahead of time in a way that will produce sufficiently accurate (5–10% error) forces.
Estimating friction, as demonstrated in the previous papers, is strongly dependent
on the particle shape, surface roughness, and any fluid additives, and thus requires
tribological models for each of those parameters. These are details we normally
neglect, but when attempting to construct an accurate friction model, cannot. It is
not coincidental that the design of MR fluids focuses heavily on particle shape, size,
and surface coating, as these are properties with significant impact on yield stress,
as well as other important concerns like settling time, offstate viscosity and lifetime
absorbed energy [46], [51], [52].
Then to develop an accurate simulation model that includes friction, we need
to know the fine details of the particle morphology and surface coating thickness and
chemistry, develop correction factors, and then use them as inputs to a high fidelity
code for predicting yield stress. While developing and experimentally validating
such a model is feasible, the challenge is substantial, as it will require a tremendous
number of measurements and an expert of knowledge of tribology. Such a test
campaign would have benefits beyond obtaining an accurate yield stress, as it would
14
also resolve questions about those other secondary properties such as settling time,
offstate viscosity and lifetime absorbed energy.
While particle models are the only way forward for predicting fluid properties,
the focus of this dissertation is different: predicting device yield force. Our most
basic assumption about yield force is that it comes from the yield stress via some
established relationship. Since this assumption holds for most devices, predicting
yield force boils down to knowing the yield stress. There are two ways to accurately
know the yield stress: 1) measure the yield stress, or 2) create and validate a model
of friction at the particle level. Since measuring yield stress of a fluid takes about two
hours, and developing a friction model will take years, measuring yield stress sounds
preferable when possible. However, this is not a valid approach for devices that
lack an accurate yield force model, such as in high speed and squeeze-strengthened
devices.
In these situations, we need to develop new yield force models. We can do this
in two ways: by developing a highly accurate friction model or by correcting the
Bingham plastic model. Since the Bingham plastic model is so simple, I claim it is
preferable to investigate and fix the failures of existing Bingham plastic yield force
models, rather than attempt to understand the true nature of interparticle friction.
1.4 Particle model analysis
However, the inability to predict yield stress accurately does not prevent us
from using particle models to understand the physics of these fluids, as the relative
15
trends are widely agreed to be accurate.
First, we realized that we wanted to link insight developed at the particle
scale to the bulk scale. Sherman et al. [53], Appendix F, does this by exploring the
relationship between the Bingham and Mason numbers, the nondimensional numbers
governing the macroscopic (fluid) and microscopic (particle) scales respectively. The
core observation is that Bi ∝ Mn−1. This simple result allows us to translate trends
that occur at different scales. As an example it allows us to neglect high Mason
number (Mn > 0.1) theoretical results, as we know that they are experimentally
irrelevant, as Bingham number is small, and thus controllable force is low. This same
thinking also helped us when analyzing a series high shear experiments, as we were
able to show that the high shear rate experiments fall onto the same Mason number
master curve as the low shear rate data, experimentally demonstrating that the yield
stress persists at large shear rates [37], [38]. When thinking about high speed yield
force falloff, the focus of this thesis, Mason number analysis gives us confidence that
the yield stress can exist at the high shear rates associated with high speeds.
Sherman and Wereley [54], Appendix G, explores a result from [55], where it
was shown that there is an upper Mason number limit on the existence of stable
chains. This could indicate a point at which the yield stress should disappear, and
could serve as an explanation for high speed yield force falloff. Our simulation work
demonstrated that this only happens at extremely low volume fractions, φ < 0.02,
where the particles actually form single width chains instead of the typical thick
column and sheet structures. We can also apply the Bi-Mn relationship to determine
that this effect would only occur when controllable force is low, and is not a viable
16
explanation for high speed force falloff.
Finally, Appendix H examines the particle dynamics and showed the existence
of two primary time scales, the particle dynamics time scale and the shear rate time
scale. We then used our simulation code to show that the response time scales with
the shear rate, giving a response time of a constant shear. We show that this fixed
response shear is consistent with experimental measurements of response time, and
is small enough that assuming a zero yield stress response time is reasonable. A fixed
shear response time implies that response time is speed independent, and would
manifest as a fixed apparent shortening of the active region.
1.5 Fluid Dynamics
So if we want to develop a model for high speed yield force falloff by fixing
existing models, we need to identify what needs improvement. I believe the best way
to improve current yield force models is to look at the problem through the lens of
fluid dynamics, and supporting my view are the following observations:
1. The fluid is moving at large velocities vgap > 10 m/s and the Reynolds number
is relatively large, Re > 1000;
2. ‘High speed’ in the context of controllable yield stress fluids is not defined;
3. Existing yield force models assume Re = 0 or Re =∞;
4. Fluid dynamics does interesting things when the Reynolds number is not zero,
and nobody has performed a rigorous exploration of 2D flows of controllable
17
yield stress fluids with a large Reynolds number.
This dissertation is an examination of the fluid dynamics of controllable yield stress
fluids and shows that high speed yield force falloff can be described with Bingham
plastic rheology and a yield force model that accounts for Re > 0. Chapter 2 is an
analysis of a simplified version of Goncalves’ experiment, defines ‘high speed’ and
produces a set of fits that describe the simplified flow. Chapter 3 shows that these
fits accurately predict high speed yield force falloff in a series of practical devices,
with conclusions contained in Chapter 4. These results will enable device designers
to accurately predict yield force at high speeds.
1.6 Contributions
The main contributions of this dissertation are:
1. A rigorous nondimensional analysis of MR fluids. We demonstrated that the
Reynolds and Hedstrom numbers are the most convenient descriptors of MR
fluid dynamics, while Reynolds and Bingham are the most fundamental. We
also found that yield stress coefficient, Cτy = τy/0.5ρv
2
0, can be used to estimate
the onset of high speed effects.
2. A simplified model of controllable yield stress fluids entering an active region
in flow mode, which we call ‘activation flow’. This model is the first rigorous
generalizable model of the fluid dynamics of active region flow.
3. A simplified control volume analysis which lets us determine when speed effects
18
will become apparent, contained in Eq. 2.34 and 2.35. These limits can be
turned into a simple criteria for when quasi-static models can be used safely:
Cτy > 0.4 and Re < 120 for a typical device with LA/Dh = 5 wishing to stay
within 5% of the fully developed solution.
4. A set of rigorous control volume based fits to activation flow that are accurate
from 0 < Re < 5000, 103 < He < 106, 2 < LA/Dh < 14. These fits are
contained in Eq. 3.1 to 3.8, and should accurately predict yield force in flow
mode devices.
5. An experimental validation of the claim that activation flow describes the fluid
dynamics of high speed devices by accurately predicting yield forces for all of
the known high speed MR fluid devices.
6. A demonstration that fluid dynamics is a useful way to think about MR fluids.
This includes both classic control volume analysis and computational fluid
dynamics.
19
Chapter 2: Flow of a controllable yield stress fluid into an active
region
2.1 Introduction
High speed yield force falloff is a phenomenon seen in flow mode magnetorhe-
ological energy absorbers operating at large fluid velocities, where the yield force
Fy = Fon − Foff , decreases at high speeds. This would place an upper limit on
device operating speeds, limiting the design space. In particular, Goncalves [11] and
Goncalves et al. [12] tested a capillary rheometer with a 37 m/s maximum flow rate,
and observed a 90% reduction in apparent yield stress at high speeds in short active
regions. Later, Mao [13] and Mao et al. [14] tested a double-rod damper with a
maximum active region fluid velocity of 40 m/s and saw an up to 50% reduction in
yield force at maximum speed. Perhaps surprisingly, some seismic dampers operate at
large gap velocities and can see high speed force falloff [15], [56]. A second intriguing
category of evidence is that as device engineers, we change yield force models at high
speeds. At low speeds, a fully developed Bingham plastic flow model is typically
used [8], and at high speeds, switching to a parabolic profile with superposed yield
stress at high speeds [14], an approach which can reduce yield force by up to 33%.
20
This paper investigates high speed yield force falloff, and concludes it is the result of
fluid dynamics, and develops a model capable of predicting yield force in all of the
previously mentioned experiments.
In explaining the high speed yield force falloff effect, the simplest explanation
is shear thinning at high shear rates. In high speed flow mode devices, the shear
rates may be extremely large, γ˙  100 000 s−1 and there is a paucity rheological test
data in this region. The most direct form of rheological measurement is conducted
in shear mode, but typical shear mode parallel plate or cone and plate rheomters
are limited to 1000 s−1 or even less. However there are other shear mode geometries
that allow for higher shear rates. Laun et al. [57] develop an enclosed parallel plate
test cell, creating a double gap test cell with a maximum shear rate γ˙ = 10 000 s−1,
and report stresses in agreement with traditional low speed geometries. Becnel et
al. [38] tested a concentric cylinder and a rotary vane device [37], both reaching
up to 25 000 s−1, and found that both devices produced measurements that agreed
with conventional test results. Gu¨th and Maas [58] and Gu¨th et al. [59] tested
a Taylor-Couette cell up to 34 000 s−1, and observed a 10% reduction in force at
high speeds. These experiment also showed a strong sensitivity to inertial effects,
with Taylor number instabilities causing a large increase in device force output. In
reviewing these results, the yield stress apparently persists at high shear rates, and
shear thinning is on the order of that observed at low shear rates, so that shear
thinning is unlikely to explain high speed force falloff.
Flow mode rheometers allow higher shear rates, with wall shear rates in a typical
slit type capillary rheometer easily exceeding 20 000 s−1. Wang and Gordaninejad
21
[60] tested a flow mode rheometer with 1 mm gap at 1 m/s mean velocity, and
observed an apparent increase in shear thinning compared to parallel plate rheometer
measurements. Gabriel and Laun [61] tested a slit rheometer with 0.34 mm gap and
1.2 m/s mean velocity and reported reduced viscosity at high speed when compared
compared to parallel plate measurements. Both studies quote maximum shear rates
on the order of 50 000 s−1, and have results similar to other high shear rate shear
mode studies [38], [57]–[59].
However, capillary rheometer experiments at high speeds introduces two new
concerns: viscous heat production effects and inertial effects. Viscous heat production
causes the fluid temperature to increase, which in turn decreases viscosity, and will
appear as an apparent shear thinning. We believe thermal–viscous effects are a major
contributor to the shear thinning seen in the previous papers. Capillary rheometer
analysis assumes there are no inertial effects, but there is no known criteria for
inertial effects, nor any description of what those effects would be. This study
investigates inertial effects in a rheometer type flow, and provides both a prediction
for the onset and effects of inertia. Our results show that no inertial effects occurred
in the previous papers.
However, there have been very high speed capillary rheometer tests where
inertia does have an effect. The best and only example of this is in Goncalves [11]
and Goncalves et al. [12], where a slit device with variable length active region and
a gap height of 1 mm is tested at mean flow speeds up to 37 m/s. Goncalves treats
the device as a rheology experiment, and observes that when the fluid residence
time in the active region drops below 1 ms, the device yield force falls by up to 90%.
22
In Goncalves [11], the authors attribute this effect to a yield stress response time,
which would pose a major challenge to designing high speed MR fluid devices, as it
would place an upper bound on device speed for compact geometries. However, the
authors report a significant 12 ◦C temperature rise, and critically, their analysis uses
a quasi-static model to determine yield stresses, which, we will show causes erroneous
conclusions. While Goncalves’ analysis is limited by the quasi-static perspective, it
remains the only proper high-speed flow-mode experiment, and is the primary focus
of this work.
The idea of the yield stress taking time to develop is not unprecedented. There
have been extensive tests determining the time it takes for yield stress to develop
after the magnetic field switched on. Koo et al. [62] investigate the response time
of a commercial MR damper at low speeds , finding a 10 ms field rise time, and a
20 ms force rise time, but emphasize the impact that system compliance can have
on such measurements. Sahin et al. [63] measured the pressure rise in flow mode
devices after a step current is commanded , and find a flow mode response time of 10
to 20 ms. Laun and Gabriel [64] measure shear mode response times at shear rates
of 100 s−1 under a sinusoidal magnetic field, and find response times of 2 to 3 ms.
For device engineering, these fluid response times are short enough that they are
dwarfed by magnetic circuit response times, and thus not of major concern. While
these time domain results could be extended to the analysis of steady flow of fluid
entering an active region, time dependent fluid rheology is a substantial complication
we would like to avoid. If we can show that high speed yield force falloff is a fluid
dynamic effects, it is a preferable explanation over time dependent fluid properties,
23
as it introduces no new physics.
Another explanation is that at very high shear rates, the yield stress disappears
completely. Martin and Anderson [55] develop an analytical chain model that suggests
there are no stable particle chains above a critical Mason number, and particle chains
are the standard explanation for the cause of the MR effect. We developed and ran
a particle simulation of this regime [54], and our results indicated that there is no
yield stress falloff for typical fluid volume fractions, even beyond the no-chain limit,
as the particles are sufficiently close to be interacting magnetically even when the
chains break down. Checking the critical Mason number against these high speed
flow results of interest, the Mason number is close to, but below the critical Mason
number, casting doubt on this shear rate dependent explanation. In our recent work,
we observed that force rise time corresponded to a fixed shear angle, as expected
from theory [65]. Thus, we expect that increasing shear rates would reduce response
time, and since high speed flow experiments operate at extremely large shear rates,
this would push the response time towards zero.
We hypothesize that the loss of yield force at high speed, or high speed
yield force falloff is in a fact a fluid dynamic effect, and we investigate this idea
through computational fluid dynamics (CFD) and control volume analysis. While
our goal is to investigate high speed yield force falloff, 2D flow in a duct also a
problem of basic interest, as there have been few examinations of 2D inertial flows
of variable yield stress fluids. We approach the problem by developing a simplified
model of magnetorheological fluid entering an active region, designed to replicate
the experiment of Goncalves [11] while assuming a zero response time fluid. We
24
demonstrate that this model captures the fundamental behaviors and successfully
describes high speed yield force falloff.
Most computational fluid dynamics work on fluids with yield stress has focused
on creeping flow, but here we look at the effects of non-zero Reynolds number. Abdali
et al. [66] looks at creeping entry and exit flows of Bingham plastics, and tabulates
the Bagley correction as a function of Bingham number, an approach we mimic.
Mitsoulis et al. [67] look at pressure-driven creeping Bingham flow over a cavity, and
uses a continuation scheme to reduce computational effort. Taylor [68] computed the
fully-developed cross-sectional velocity profiles of Bingham plastics in rectangular
ducts, and concludes that ducts of 10:1 cross section aspect ratio can be effectively
treated as parallel plates. For inertial Bingham plastic flows, the primary focus has
been on entrance region flows for thermal transport. Min et al. [69] and Vradis
et al. [70] both look at entrance flows, and conclude that the flow is strongly two
dimensional, and that increasing Bingham number reduces entrance length, but do
not collect any force correction factors.
Previous work on using CFD for active magnetorheological fluid is fairly limited,
especially for pressure driven flows. Park et al. [71] used the fluid dynamics and
magnetism solver from ANSYS to get the varying magnetic field, and thus yield
stress, in a shear mode brake. Bompos and Nikolakopoulos [72] successfully simulated
an eccentric shear mode magnetorheological journal bearing using ANSYS. Gedik
et al. [73] looked at a problem similar to ours, but the MR fluid is modeled using
a magnetohydrodynamics code, which gives plug-like velocity profiles, but returns
pressure losses several orders of magnitude below experimentally expected values.
25
Parlak and Engin [74] used ANSYS CFX to simulate the performance of a low speed
MR damper, and their results compare reasonably well to experiment, but has a
coarse grid and a possible lack of convergence. Bullough et al. [75] uses ANSYS
Fluent to look at pressure driven and shear flow in a Couette cell, and finds good
agreement with analytical models. In these papers, velocities are low and yield
stresses large, so inertia and momentum effects are small, which ensures that no
interesting fluid dynamics are observed.
However, we are not the first to identify a high speed regime. Goldasz and
Sapinski [76] identify the existence of a high speed regime when performing a
nondimensional analysis of the pre-yield fluid dynamics of MR fluids. In the high
speed regime, Go ldasz and Sapin´ski [77] used CFD to develop qualitative insight
into MR fluid flow, and take an approach extremely similar to the one used in this
study. They examine several model problems of fluid entering a region of yield stress,
and observe large regions of not fully developed flow, and a decreasing yield force at
high speeds. We see the same things, but we rigorously quantify our results using
control volume analysis.
In this study, we perform an analysis that provides force correction factors
applicable to essentially all flow mode devices, and show through control volume
and numerical analysis that high speed force falloff is a fluid dynamic effect. We
develop a simplified model of Goncalves’ experiment, ‘activation flow’, where fully
developed Newtonian flow instantly develops a yield stress in a 2D duct. This
model matches the analysis used in Goncalves, but with the addition of 2D flow and
inertial effects. First, we show that including the momentum flux terms in a 1D
26
(a) Shear stress (b) Viscosity
Figure 2.1: Rheological models. Solid is Bingham plastic, dashed is biviscous,
dash-dot is Newtonian.
pressure loss model will produce high speed yield force falloff. We then examine 2D
flow, modifying an OpenFOAM CFD solver to handle a fluid with spatially varying
viscosity (one that has no response time), and validate it to carefully define the region
of solver accuracy. We then run our CFD model over a wide span of non-dimensional
operating conditions, and observe, as expected, reduced pressure losses compared
to classic fully developed models. From this, we conclude that the apparent loss of
yield stress seen in Goncalves is not a response time effect, but, instead, an inertia
and momentum dependent fluid dynamic effect. We then develop fits for pressure
loss that should be applicable to a large class of devices.
27
2.2 Background
2.2.1 Rheology
The standard rheological model for MR fluids is as a Bingham plastic
τ = τy + η∞γ˙, (2.1)
where τy is the yield stress, η∞ is the plastic viscosity and γ˙ is the shear rate. This
is shown in Fig. 2.1. Another alternative to the Bingham plastic model is to look
at shear thinning models. Two popular choices are the Herschel-Bulkley model,
τ = τy +kγ˙
n and the Casson plastic model,
√
τ =
√
τy +
√
ηγ˙ both of which can offer
accuracy improvements. We do not use these shear thinning models, as we would
like to see if the high speed force falloff can be explained using only the Bingham
plastic model. However, we do make the standard simplifying assumption that the
plastic viscosity is equal to the field-off viscosity. From the perspective of shear
thinning, this assumption reduces shear stress, as field-on measurements typically
have a slightly larger plastic viscosity.
However, when numerically modeling fluids with yield stress, viscosity, η = τ/γ˙,
is infinite at zero shear rate, an unacceptable feature. The standard fix is to regularize
the pre-yield behavior through a finite viscosity, covered thoroughly in a review
by Mitsoulis [78]. The classic solution is to put a limiter on viscosity, creating a
28
piecewise continuous stress curve called the biviscous model,
τ =

τy + η∞γ˙, if τy + η∞γ˙ < η0γ˙
η0γ˙, otherwise
(2.2)
where η0 is the low shear viscosity, and is also shown in Fig. 2.1. To quantify the error
of the Bingham plastic approximation, the viscosity ratio can be defined, η∗ = η∞/η0,
the ratio of high shear rate to low shear rate viscosity, with η∗ = 10−3 as the standard
choice. An alternative model is to use a smooth approximation [79],
τ = η∞γ˙ + τy (1− exp [−mγ˙]) (2.3)
with stiffness parameter m. This exponential model can improve solver performance,
but for equivalent values of η∗, the smooth model is a much poorer approximation of
the Bingham plastic solution. Mitsoulis makes a compelling case the that exponential
model, with sufficient stiffness, offers the best mix of performance and accuracy.
Here, we choose the biviscous model, as it is more familiar to device engineering and
offers straightforward analytical solutions, which we wanted for verification.
Nondimensional description of Bingham plastic behavior is typically done
through the Bingham number and the Hedstrom number. The Bingham number, is
defined as the ratio of yield stress forces to viscous stress forces,
Bi =
τyh
η∞v0
, (2.4)
29
where v0 is the mean velocity, and h is the gap height. The Bingham number is the
most fundamental description of Bingham plastics, as it solely determines velocity
profile shape [80]. However, the Bingham number is dependent on both yield stress
and velocity simultaneously, so it is awkward to work with.
Hedstrom number comes from the chemical engineering literature [81], [82],
He =
ρτyD
2
h
η2∞
(2.5)
where ρ is fluid density, and Dh is the hydraulic diameter. Hedstrom number is an
essential description of Bingham plastics, as it can be used to nondimensionalize
yield stress. Hedstrom number has appeared in the ER fluids literature [75], [83], as
a way of reporting yield stresses, but has not caught on in the MR fluids world. For
comparing MR fluid experiments (variable τy) to traditional Bingham plastics (fixed
τy), Hedstrom number is invaluable, as the chemical engineering literature reports
flow data in terms of He and Re [82]. Critically, this allows us to use extensive
chemical engineering literature for the Bingham plastic transition to turbulence, a
concern for high speed flow experiments.
Reynolds number, the essential Newtonian nondimensional number, is the ratio
of inertial to viscous forces, where for a Bingham plastic,
Re =
ρv0Dh
η∞
. (2.6)
For traditional Newtonian fluids, Reynolds number is the governing parameter. Here
30
Re is used to nondimensionalize velocity.
We also make extensive use of yield stress over dynamic pressure, which we
call yield stress coefficient,
Cτy =
τy
1
2
ρv20
(2.7)
the ratio of yield stress to inertial forces. Uses of this ratio in the literature are minimal.
Hedstro¨m [81] examined the inverse, and found no particular application. Alexandrou
et al. [84] uses it great effect when characterizing the flow structure of Bingham
plastics filling a cavity. Bresch et al. [85] uses this term when modeling rapidly
moving avalanches as Bingham plastics. We show that this ratio is fundamental in
estimating the onset of dynamic behavior in a magnetorheological fluid. Note that
only two nondimensional numbers are needed to describe the flow,
Cτy = 2He/Re
2 = 4Bi/Re (2.8)
allowing Cτy to be determined from any other pair.
We use these different nondimensionalizations of yield stress for different
purposes. When solving the fluid equations, we will operate in Bingham-Reynolds
space, as these are the two independent variables for the flow velocity profile, with Bi
controlling steady state flow profile and Re controlling the length of the transition zone.
When comparing our results to experimental results, we will work with Hedstrom
number instead of Bingham number, as a constant He number is equivalent to an
experiment occurring at fixed magnetic field. Finally, when discussing entrance
31
(a) Parabola (b) Plug
Figure 2.2: The basic steady-state 1-D flow profiles, parabolic flow and plug flow.
lengths and other inertial effects, we use Cτy and Re, as it clarifies limiting trends.
2.2.2 1D Flow
Since we are modeling the activation of MR fluid in a rectangular duct, the fully
developed velocity profile and wall shear stress for both Newtonian and biviscous
fluids provide means to analyze and verify the CFD output. We will denote Newtonian
solutions with subscript N , fully developed Bingham plastic plug flow with B, fully
developed biviscous solutions with subscript b, and biviscous fluids with parabolic
profile with subscript p. Steady (∂u/∂t = 0), fully developed (∂u/∂x = 0) flow
through a straight duct is the solution to the equation:
dτ
dy
=
dp
dx
(2.9)
By substituting in a rheological model for τ , an equation in u is obtained. Through
control volume analysis, this can be rewritten as dp/dx = −2τw/h, where τw is
32
wall shear stress, and h is gap height. For a Newtonian fluid, τ = η∞∂u/∂y, with
mean flow v0 and no-slip boundary conditions, the steady-state velocity profile is a
parabola
uN = 6v0
(
y
h
− y
2
h2
)
(2.10)
with corresponding pressure gradient
(
dp
dx
)
N
= −12η∞v0
h2
(2.11)
and associated wall shear stress τwN = 6η∞v0/h. An image is shown in Fig. 2.2a.
There are multiple approaches for determining pressure losses in a fully devel-
oped Bingham plastic, where the central core of the fluid moves of as solid. This
is often referred to as ‘plug’ flow, and is shown in Fig. 2.2b. Goncalves uses the
steady-state Bingham plastic pressure loss equations from Phillips [86],
P3 − (1 + 3T )P2 + 4T 3 = 0 (2.12)
where P is the ratio of Bingham plug flow pressure gradient to Newtonian flow
pressure gradient,
P = −dp
dx
h2
12v0η∞
(2.13)
and T is a nondimensionalized yield stress,
T = τyh
12v0η∞
. (2.14)
33
Note T = Bi/12. Goncalves uses this method in reverse, taking observed pressure
losses and uses the polynomial to obtain an apparent yield stress. However, this
assumes the fully developed velocity profile, an assumption which does not hold at
large velocities.
For biviscous fluids, Wereley et al. [87] provides a polynomial for δ, the fraction
of unyielded (η = η0) fluid in the gap, which can in turn be used to obtain pressure
loss. Here, we modify the equation to use a Bingham number based on dynamic
yield stress, not static yield stress, obtaining
[
1
2
δ3 −
(
3
2
+
6
Bi
)
δ
]
(1− η∗) + 1 = 0 (2.15)
where δ is found from the physically sensible root, 0 ≤ δ ≤ 1. When a sensible solution
does not exist, the fluid is completely preyield, corresponding to Bi > 6(1− η∗)/η∗,
and acts as a Newtonian fluid with viscosity η0. Pressure loss in biviscous plug flow
is (
dp
dx
)
b
= − 2τy
(1− η∗)δh (2.16)
When η∗ = 0, (2.15) will reduce to the Bingham plastic solution and can be arranged
such that it is equivalent to (2.12). Plug thickness can also be used to obtain the
34
velocity profile,
uab =
v0 Bi
δ(1− η∗) (2.17)
ub = u
a
b {[1− (1− η∗)δ]y¯ − y¯2} 0 < y¯ <
1
2
(1− δ)
ub = u
a
b [η
∗(y¯ − y¯2) + 0.25(1− η∗)(1− δ)2] 1
2
(1− δ) < y¯ < 1
2
(1 + δ)
ub = u
a
b {[1− (1− η∗)δ] (1− y¯)− (1− y¯)2}
1
2
(1 + δ) < y¯ < 1
where y¯ = y/h is the normalized gap coordinate. Note that u/v0 is solely a function
of Bingham number. For η∗ = 10−3, midline velocity for a biviscous fluid differs by
only 0.5% at Bi = 103.
We can also consider the case of Bingham plastic with a parabolic profile, which
is equivalent to superimposing a yield stress on the Newtonian solution (2.10). For a
Bingham plastic, the parabolic flow pressure loss is
(
dp
dx
)
p
= −
(
12v0η∞
h2
+
2τy
h
)
, (2.18)
and corresponding wall shear stress is τwp = τwN + τy. In a parabolic profile, the
biviscous model perfectly approximates the Bingham plastic model when Bi <
6(1− η∗)/η∗. In this work, this condition is always true.
In a controllable fluid, the controllable force range is a key quantity. This can
be captured by the wall shear stress ratio, τw/τwN , the relative change in viscous
stress caused by the magnetic field at a given point. These shear stress ratios are
especially useful when describing the development of viscous losses in 2D flow, as in
35
2D flow, ∂p/∂x 6= 2τw/h and ∂p/∂x is no longer constant in the y direction. A fully
developed biviscous fluid has wall shear stress ratio,
τwb
τwN
=
Bi
6δ
1
1− η∗ (2.19)
When the flow is fully developed and η∗ = 0, by construction, τwb/τwN = P. For a
biviscous fluid with parabolic profile and Bi < 6(1− η∗)/η∗,
τwp
τwN
= 1 +
Bi
6
. (2.20)
In a flow transitioning from parabolic flow to plug flow, we can use these values as
bounds for the local flow behavior.
2.2.3 Quasi 1D flow
Before attempting full solutions of the Navier Stokes equations, it is tempting
to try and reduce the problem to a quasi 1D duct. In these models, an axial
velocity profile form is assumed, and using conservation of mass, momentum, and an
additional cross-flow pressure gradient relationship, a one dimensional differential
equation for flow development can be created. This nonlinear differential equation is
marched down the duct, providing entrance length and pressure losses.
For Bingham plastics, these methods have been primarily used to analyze
entrance region flows, where uniform flow at the inlet develops into plug flow, typically
at low Reynolds and Bingham numbers. Chen et al. [88] looks at this problem through
36
various analytical boundary layer methods, with a spatially dependent plug flow
thickness, and finds that a 1
2
ρv2 type correction factor is dependent only on steady-
state plug thickness. Shah and Soto [89] look at a similar scheme, where the flow
profile is discretized, and finds near perfect agreement with Chen’s second method.
However, when compared to full 2D solutions of the entrance flow problem,
these solutions fall short. Vradis et al. [70] finds that the quasi 1D solutions are
‘fundamentally different’, missing a key adverse pressure gradient at the entrance.
Min et al. [69] conclude that the assumptions involved in the quasi 1D methods are
‘not accurate’ and also emphasize the importance of capturing the adverse pressure
gradients, as well as the need to capture viscosity gradient terms. With these results
in mind, it seems appropriate to focus on full 2D flow.
2.2.4 2D flow
Figure 2.3: A simplified model of Goncalves’ experiment. The active region is shaded.
The ‘activation flow’ that we investigate here is is the study of fluid entering a
region of uniform yield stress, designed as a simplification of Goncalves’ high speed
capillary slit experiment, which is shown Figure 2.3. In Goncalves, a large piston
37
pushes fluid into a rectangular duct through a smooth entrance, producing a mean
duct flow velocity v0 greater than 30 m/s. The high aspect ratio 10 mm× 1 mm
rectangular duct has a total length of 101 mm, with either an active region of
LA = 6.35 mm or 25.4 mm placed at the end of the duct, with the fluid leaving the
active region through a sharp exit into the air. These active lengths correspond
to LA/Dh = 3.49 and 13.97 respectively, and in these active regions, the field is
assumed to be uniform, a reasonable assumption as the active region is a small gap
between two large magnetic flux guides.
In order to examine the fluid dynamic effects on the yield force, we focus on
the active region, using the simplified representation shown in Figure 2.4. The active
region of length LA is treated as a spatial region where the fluid has a yield stress,
so here the fluid has zero response time. The passive region of the duct is designed
to be long enough that the flow is fully developed at the start of the active region,
which allows us to truncate the upstream portion. This allows us to simplify down
to a short passive region with LN = 2.5Dh and inlet boundary conditions as fully
developed parabolic Newtonian flow, (2.10). The short passive region is needed as
the yield stress causes fluid dynamics effects upstream of the active region. The exit
boundary condition is a uniform pressure, representing the exit into free air. The
specific LA we look at are those of Goncalves’ experiment, LA/Dh = 3.4925 and
13.97, as well as LA/Dh = 2 and 7 to investigate finite length effects and to increase
coverage of the device design space. This simplified representation has three degrees
of freedom, v0, τy and η∞, and forms the simplest possible model that we believe
than can capture the physics of high speed yield force falloff.
38
Parabolic Transition Plug 
τy = 0 τy  ≠ 0 
LN LA 
y=h 
y=0 
y 
x 
xin xout x=0 
Figure 2.4: Simulation model of the end of Goncalves duct.
By solving for the full 2D flow, we aim to determine how the flow transitions
from Newtonian parabolic flow to fully developed Bingham plastic flow, and what
losses are encountered in the transition. Typically, pressure losses of MR devices are
typically modeled by assuming fully developed flow everywhere,
∆Pslow =
(
dp
dx
)
b
LA +
(
dp
dx
)
N
LN (2.21)
neglecting any effects from the transition from Newtonian parabolic flow to biviscous
plug flow. This fully developed everywhere assumption is popular at low speeds, and
is largely successful. However, at high speeds, there is no consideration of ρv20 effects
on yield force.
An alternative model is to assume that the velocity profile is parabolic every-
where, yielding
∆Pfast =
(
dp
dx
)
p
LA +
(
dp
dx
)
N
LN . (2.22)
By construction, there are no transition losses or active region inertial effects. The
39
form is popular in high speed pressure loss models [14]. This will be valid at Re =∞,
as inertia will keep the flow in a parabolic shape in the active region. Equation 2.22
predicts lower pressure losses than the low speed model, and is typically used when
designing high speed devices.
Goncalves chooses the low speed model, but in his experiments, Re = 1400, so
we expect transition effects and inertial terms to be present, causing the low speed
analysis to incorrectly indicate that the yield stress disappears at high speeds.
In our activation flow model, we expect flow profile changes upstream of the
active region to be present but minimal, causing the velocity profile at the start of
the active region to be be nearly parabolic, and for the flow to rapidly transition to
fully developed plug flow. We expect that the length of the transition zone will be
affected by inertia and captured by Re and Cτy , but as we show below, transition
effects on wall shear stress are fairly small, and insufficient to explain high speed
force falloff. But if we do a proper control volume analysis and include momentum
flux effects, we show that basic fluid mechanics gives us a negative loss term, and a
viable explanation for the high speed force falloff effect.
2.2.4.1 Transition zone effects
We can estimate the effect of the transition from plug to parabolic by comparing
the 1D pressure loss models. If the flow changes from parabola to plug in a reasonably
gradual manner, we expect pressure losses to be bounded by the plug and parabolic
solutions. From Equation 2.16 and 2.18 fully parabolic flow has 2/3 the yield induced
40
10
0
10
2
Bi
1
1.05
1.1
1.15
1.2
β
10
-1
10
-2
10
-3
10
-4
0
Figure 2.5: Profile momentum, β vs. Bingham number for various values of η∗.
pressure gradient of fully developed flow, or Bi→ 0, or
lim
Bi→0
τwp − τwN
τwb − τwN = 2/3.
This means that transition effects, in the case of fully parabolic flow everywhere, could
cause a 33% reduction in yield force. However, since transition length should be fairly
short for most cases, the actual impact of this term will be much smaller (≈ 10%),
and thus insufficient to describe the up to 90% force falloff seen in Goncalves.
2.2.4.2 Momentum flux effects
To explain the remainder of the high speed force falloff effect, we need to look
at the effect of momentum flux. For a rectangular control volume, pressure loss is
∫ h
0
p(xin, y)−p(xout, y)dy = ρ
∫ h
0
u(xout, y)
2−u(xin, y)2 dy+2
∫ xout
xin
τw(x) dx (2.23)
41
If we assume a constant pressure at the inlet and outlet, and define profile momentum
factor,
β(x) =
∫ h
0
u(x, y)2dy
v20h
, (2.24)
pressure loss can be rearranged, using skin friction factor, Cf = τw/
1
2
ρv20 and hydraulic
diameter, Dh to account for 3-D effects, to obtain
p(xin)− p(xout)
1
2
ρv20
= 2 [β(xout)− β(xin)] + 4
Dh
∫ xout
xin
Cf (x) dx. (2.25)
For a fully developed Bingham plastic flow, β can be found in terms of Bi and δ by
substituting in (2.17) into (2.24) with η∗ = 0,
βB(Bi) =
Bi2 (δ − 1)4
16 δ
− Bi
2 (δ − 1)5
30 δ2
, (2.26)
Figure 2.5 plots β for a Bingham plastic, as well as a selection of biviscous fluids. The
biviscous solution for profile momentum factor is substantially more complicated:
βb(Bi, η
∗) =
Bi2 (15(δ − 1)4 + A)
240 δ (1− η∗)2 −
Bi2 (δ − 1)3 (8(δ − 1)2 +B)
240 δ2 (1− η∗)2 (2.27)
A = 4δ2(δ(7δ − 20) + 15)η∗2 − 20δ(2δ − 3)(δ − 1)2η∗ (2.28)
B = 20δ2 η∗2 − 25 δ2 η∗ + 25δη∗ (2.29)
For a Bingham plastic, increasing Bi reduces β, with βB(Bi = 0) = 6/5 and
βB(Bi =∞) = 1. Figure 2.5 also places the strictest limits yet on the biviscous model,
as it shows how reducing η∗ improves reproduction of Bingham plastic behavior.
42
From this, we find that the standard value of η∗ = 10−3 will accurately reconstruct
the velocity profile up to Bi ≈ 200, so that βB ≈ βb.
In the experimental context, the decrease in β with increase in Bi explains
the high speed fall off in force. Looking at pressure loss, Equation (2.25), when
Bi > 0, β(xout) − β(xin) < 0, since β(xin) occurs at Bi = 0, the maximum value
of β. Since this term is negative, we now have a steady state yield stress induced
negative velocity squared term that will reduce overall pressure loss. We believe this
momentum flux effect is the main cause of high speed yield force falloff.
2.2.4.3 Flow features of interest
To capture both the transition zone and momentum flux effects, we use CFD
to solve for the full 2D velocity profile. Since the CFD model includes inertia,
the transition zone is no longer neglected, and will extend both upstream and
downstream of x = 0, the start of the active region, with a velocity dependent length.
We decided to use fixed active lengths, instead of a velocity scaled duct, as we wanted
to investigate the effects of finite duct length.
Our activation flow model has the advantage of being simple: a 2D rectangle
with with three parameters – velocity, yield stress and active length, or in nondimen-
sional form, Re, Bi, and LA/Dh. The approximation of treating the active region
as a region of uniform yield stress is the largest assumption made here, but it is
the most useful. By assuming a uniform yield stress, we do not need to know how
yield stress varies with field, we do not need to account for body forces, and we
43
do not need to choose a basis for comparing field strength in different geometries.
This dramatically simplifies the analysis, and is also consistent with the majority
of existing 1D yield force models, which also treat the active regions as zones of
uniform yield stress. We do not need to account for the three dimensional behavior,
as [68] showed that Bingham plastics in a 10:1 aspect ratio rectangular duct can be
treated effectively as a 2D flow, and our reference experiment occurs in a 10:1 aspect
ratio rectangular duct. The assumption of fully developed flow in the inlet, combined
with sufficient passive length (LN/Dh > 1) to include effects passed upstream, make
our results independent of LN , which eliminates a model degree of freedom. We are
unaffected by the biviscous approximation, as we have shown earlier that η∗ = 10−3
is small enough that the difference between the biviscous model and the Bingham
plastic model will be negligible. While these simplifying assumptions limit our ability
to accurately model Goncalves’ results, the simplicity gained allows us to draw clear
and precise conclusions about the nature of the flow.
In this simplified 2D model, we use our CFD solution to determine:
1. That our results are sensible. We have assumed a region of uniform yield stress,
which simplifies the problem, but the tradeoff is a potentially problematic
discontinuous change in the rheological properties. We would also like to show
that the viscosity regularization is having no effect, so we must demonstrate
η∗ independence. Finally, we need to demonstrate grid independence, to show
that our results are unaffected by the CFD setup.
2. An effective nondimensional scaling of the geometry. We are using fixed length
44
active regions that include the exact active lengths of Gonvalves, but this
approach does not work for arbitrary devices. If we develop an effective
nondimensional scaling for duct length and the force correction, our numerical
results can be used with any device.
3. The change in force caused by inertial effects. For device engineering, this
is the primary concern. We would like to develop fits or tabulate this in a
way that is easily applicable to a large class of devices, and to present it in a
physically intuitive manner.
In order to validate our numerical results, we must compare to the experimental
data we have. The challenge is that in an experiment, the rheology (the yield stress in
particular) is essentially unknown. However, a CFD model requires that the rheology
of the material be specified up front, so exact matching of operating conditions is
not feasible. To get around this, we take our simplified 2D CFD model and simply
sweep all numerically feasible nondimensional operating conditions. With enough
test points, we can reasonably interpolate between them, freeing us from having to
run a new set of CFD calculations every time we modify a rheological parameter.
2.2.5 Correction Factor
Since we are investigating a force effect, characterizing the change in force in a
clear and intuitive manner is essential. Typical low speed viscoplastic entry and exit
flow analyses employ the Bagley correction [90], which effectively adds additional
contraction length to account for the viscous losses caused by the entrance/exit.
45
For controllable fluid entering an active region, we modify the Bagley correction
factor, modeling extra viscous losses as the active region getting longer, while the
passive region grows shorter by an equivalent length. This pressure correction model
can be expressed as
∆p =
(
dp
dx
)
b
(LA + npDh) +
(
dp
dx
)
N
(LN − npDh). (2.30)
where np, the pressure active length correction, is the change in active length
measured in hydraulic diameters. For an activation flow, where observed pressure
loss at high speeds is less than expected fully developed pressure loss, np < 0, a lost
active length. From an observed pressure loss, ∆p, we can rearrange to solve for np,
np =
∆P − (( dp
dx
)
N
LN +
(
dp
dx
)
b
LA
)((
dp
dx
)
b
− ( dp
dx
)
N
)
Dh
. (2.31)
We considered other forms for our correction factor. A relative change in
active region length can also be used, but for ducts of different length that are both
fully developed by the exit, the relative change would be quote different values. By
reporting a change in length in units of length, for two flows that are both fully
developed, both will have the same np, simplifying analysis. Another option is a
hydraulic ρv20 type loss coefficient. Since the β effects are scaled by a ρv
2
0 term, it is
tempting, as we would expect the loss coefficient to be dependent only on Bi, since
βB is a function of Bi only. While this is almost the case, our CFD results show
a non-negligible Reynolds number dependence, and when Re is low, hydraulic loss
46
coefficient is very sensitive to error, while np stays near zero, as the flow effect we
are characterizing is not a ρv20 effect. However, we can use βB to generate a good
analytical approximation for force loss.
We may also want to isolate the wall shear stress terms. By using Equation 2.25
to use solved quantities p + u2 instead of τw, we can get the viscous active length
correction
nv =
∆(p+ u2)− (( dp
dx
)
N
LN +
(
dp
dx
)
b
LA
)((
dp
dx
)
b
− ( dp
dx
)
N
)
Dh
(2.32)
where ∆(p+u2) is the difference of the up and downstream values of p+u2 integrated
over the duct cross section. This lost viscous length corresponds to the reduction in
viscous force on the damper walls, or the pressure loss when the effect of inlet/outlet
momentum (β) can be ignored. This term may be of use for devices where active
regions are followed by exits that can turn the momentum at the exit back into
pressure. However, in Goncalves’ experiment and typical MR energy absorbers, the
exits do not recover any pressure, and nv is not applicable.
2.2.5.1 An analytical estimate for change in active length
Since the profile momentum terms are the expected cause of the majority of
the high speed force falloff effect, we can transform those losses into np and use that
to develop some analytical insight into the problem. If we assume η∗ = 0 and that
the flow is fully developed everywhere, but include β effects,
∆P = (
dp
dx
)NLN + (
dp
dx
)bLA + (β − 6/5)ρv20
47
-8
-7
-6
-5
-4
-4
-3
-3
-2
-2
-1
-1
-0.25
-0.25
0 1000 2000 3000 4000 5000
Re
10
-4
10
-3
10
-2
10
-1
10
0
10
1
τ y
/
0
.5
ρ
v
2 0
-8
-7
-6
-5
-4
-3
-2
-1
0
np(β)
Cτy=0.4Re=120
Figure 2.6: Pressure active length correction from momentum flux effects, np(β),
Eq. (2.33). The straight pink lines are the limiting values for np = −0.25 from
Equation 2.34 and Equation 2.35.
we can then find the associated entrance correction, np,
np(β) =
βB − 1.2
2(Cτy/δ − 24/Re)
or
np(βB)
Re
=
βB − 1.2
24 (Bi/6δ − 1) (2.33)
where we have rearranged to be dimension free, and is plotted in Figure 2.6. Note that
the assumption of fully developed flow in np(β) causes np to significantly overestimate
the missing force if the flow is not fully developed.
Since the expanded form of Equation 2.33 is quite complicated, instead we
look to determine bounds on the magnitude of np(β) as a simple way of generating
insight into the problem. Our approach is take to perform a limiting analysis on
yield forces, viscous forces and inertial forces in Equation 2.33. In the limit of no
viscous effects, Re→∞,Bi→∞, sets δ → 1 and βB → 1, yielding,
lim
η→0
np(βB) = − 1
10Cτy
, (2.34)
48
corresponding to the horizontal part of the contours in Figure 2.6. This limit
produces a conservative bound on np(β), with solutions approaching the bound when
Re > 104. In the limit of no yield stress, Cτy → 0,Bi → 0, and with the help of
Matlab’s symbolics toolbox,
lim
τy→0
np(βB) = − Re
480
(2.35)
the vertical part of the contours in Figure 2.6. Since this is in the limit of zero
yield stress, this rule of thumb will only be accurate when Bi < 1, so experimentally
observing the predicted change in force in this region will be challenging. In the
limit of no inertial effects, by construction, np(βB) = 0.
The main use of these limits is to generate simple, convenient rules of thumb
for when a certain value of np will be reached, as each limit is dependent solely on a
single non-dimensional parameter. If we want to know when the momentum flux
terms will cause a 5% difference in yield force from the fully developed value, we
look for a 5% reduction in active length, or np(β) = −.05× La, and then solve for
Re and Cτy . When np is below the 5% threshold, inertial effects are negligible and
the flow is quasi-static, and will be accurately modeled by the plug flow solutions.
Substituting in, our quasi-static criteria are:
Cτy >
2
LA/Dh
and Re < 24LA/Dh
For a typical value of LA/Dh = 5, this tells us inertial effects will be noticeable when
49
Re > 120 and Cτy < 2/5. However, since most devices have multiple active regions
each with different LA/Dh, a looser criteria is more useful. Observing that many
devices have designs with LA/Dh ≈ 5, we use the specialized criteria, Re < 120 and
Cτy > 2/5 as our quasi-static criteria no matter the length.
2.2.6 Computational Fluid Dynamics
To improve our analysis beyond 2D control volumes, we use computational fluid
dynamics to obtain the full velocity profiles in our model. We chose OpenFOAM’s
finite volume simpleFoam solver, [91], as the baseline for our solver, as it is proven,
open source, cluster capable and easily modifiable CFD code. We picked this
modification of the SIMPLE method [92], as it is the classic method for solving
incompressible flow, and critically, comes with OpenFOAM by default. For this
paper, we implemented a spatially dependent rheological model, which required only
minor modifications to the viscosity calculation (replace a scalar lookup with a field
lookup), and left the solver itself unaltered.
The SIMPLE method solves the steady, incompressible Navier–Stokes equations,
ρuj
∂ui
xj
− ∂τij
∂xj
= − ∂p
∂xi
(2.36)
where for generalizable Newtonian fluids, the viscous stress tensor
τij = 2η(γ˙)eij (2.37)
50
and viscosity η = τ(γ˙)/γ˙ and τ is the 1D rheological model, and γ˙ is the shear rate.
The rate of strain tensor is
Dij =
1
2
(
∂ui
∂xj
+
∂uj
∂xi
)
, (2.38)
and shear rate is defined as
γ˙ =
√
2DijDji. (2.39)
These non-Newtonian parts add highly nonlinear couplings to the velocity equations,
and required many solver iterations to relax.
Note that by assuming a generalizable Newtonian fluid, we have assumed that
the fluid is isotropic. Experiments by Kuzhir et al. [93] have shown that yield stress
is reduced when the magnetic field is parallel to the fluid velocity. However, in the
duct flows investigated here, magnetic field and velocity are essentially perpendicular,
so anisotropic effects should be minimal.
2.3 Results
2.3.1 Numerical validation
In order to evaluate the accuracy of the solver, we consider the case of a 2D
periodic duct, and compare against the exact solution to test the accuracy of the
solver. We ended up using the default OpenFOAM schemes with a very fine grid
spacing. Here, grid spacing is characterized by N , the number of equally spaced
elements from the wall to the center line of symmetry, with the number of horizontal
51
10
-1
10
0
10
1
10
2
10
3
Bi
0
0.5
1
1.5
2
2.5
3
S
tr
es
s 
E
rr
o
r,
 %
25
50
100
200
N
(a) Wall Shear Stress
10
-1
10
0
10
1
10
2
10
3
Bi
0
0.2
0.4
0.6
0.8
1
V
el
o
ci
ty
 E
rr
o
r,
 %
25
50
100
200
N
(b) Midline Velocity
Figure 2.7: Two dimensional periodic pipe error as a function of N , the number of
elements to the midline.
elements changing equivalently. We also tried the higher order QUICK scheme, a
popular choice for finite volume viscoplastic flows [94], but this introduced small,
but problematic, oscillations in the solution, so we reverted to the default second
order schemes on a fine grid. Solver convergence was set to be quite strict, with all
residuals below 10−8, which we found was required to obtain a close match to the
steady state solution.
Figure 2.7 plots steady solution error, |u/ub − 1| and |τw/τwb − 1|, for both
midline velocity and wall shear stress, for three grids with 25, 50, and 100 uniform
elements to the center line of symmetry. Looking for sub 1% accuracy, we find that
only N=100 and N=200 achieve this criteria. However, the N=200 solutions have
a rise in error at low Bi, from running out of time due to slow convergence, with
N=200 taking twice the number of iterations than N=100. We also note that when
using over 100 elements to the center line of symmetry, the grid size can be smaller
than the particles in the fluid, as a typical particle diameter is 2–8 µm and typical
52
-2 -1 0 1 2 3
x/Dh
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
u
/
v 0
50
70
100
N
(a) Midline velocity
-2 -1 0 1 2 3
x/Dh
0
0.5
1
1.5
2
2.5
p
/
0
.5
ρ
v
2 0
50
70
100
N
-0.05 0 0.05
2
2.05
2.1
2.15
(b) Midline pressure
Figure 2.8: Activation flow solution as a function of mesh size at Re = 500 and
Bi = 10.
device gap height is 0.5 mm. For the periodic duct, we also tested for solver error as
a function of Reynolds number and found no effect, as expected, affirming the choice
to operate in Reynolds-Bingham space. From this, we chose N = 100, as it achieves
accuracy goals while improving convergence rates compared to N=200.
Next, we looked at the effect of grid size on the activation flow geometry,
considering scaled meshes with N = 50, 700 and 100 elements to the midline at a
representative test condition of Re = 500, Bi = 10. Visually, the results are nearly
identical, shown in Fig 2.8. Maximum difference in pressure loss, exit velocity, and
active length correction was 0.25%, 0.17%, and 1.2% respectively. The difference in
np from N=50 to N=100 was 0.007, a negligible amount. Since the observed changes
are small, we conclude that our results are grid independent, and that a coarser
mesh could have been used with minimal consequences.
We also checked to see if the effects of the yield stress step increase at x = 0
caused any issues. All velocities, (u, v), were continuous and oscillation free. However,
53
-0.5 0 0.5 1 1.5 2 2.5 3 3.5
x/Dh
1
1.1
1.2
1.3
1.4
1.5
u
/
v 0
1.0 × 10−2
5.0 × 10−3
2.0 × 10−3
1.3 × 10−3
1.0 × 10−3
6.7 × 10−4
5.0 × 10−4
2.0 × 10−4
η
∗
(a) Midline velocity
-0.5 0 0.5 1 1.5 2 2.5 3 3.5
x/Dh
0
5
10
15
20
25
τ
w
/
τ
w
N
1.0 × 10−2
5.0 × 10−3
2.0 × 10−3
1.3 × 10−3
1.0 × 10−3
6.7 × 10−4
5.0 × 10−4
2.0 × 10−4
η
∗
(b) Wall shear stress
Figure 2.9: Solution as a function of η∗ at Re = 2000, Bi = 100.
leading up to x = 0, there is, as expected, a large adverse pressure gradient, but
unexpectedly, a small pressure discontinuity at x = 0. However, this discontinuity
is independent of mesh size, so all results are consistent. There are also some very
small, highly damped oscillations on both sides of the discontinuity. Downstream
of x = 0, all solutions returned to the same trend regardless of grid. From these
investigations, we conclude that the simplifying benefits of a step increase in yield
stress are worth the small numerical artifacts introduced into the solution.
We also tested the effect of η∗ on the actual activation flow problem, looking
at Re = 1, 200, 1000, 2000 and at each Re, Bi = 1, 10, 100. Figure 2.9 plots midline
velocity and wall shear stress at Re = 2000, Bi = 100, the case where the largest η∗
effects were observed. The midline velocity is strongly dependent on η∗, as predicted,
but wall shear stress is essentially unaffected. The largest velocity effects are seen
as the fluid approaches steady state, with the start of the slow convergence portion
happening earlier for a system with higher η∗. Thus, when defining entrance length,
the point at which the flow has reached steady state and the transition zone ends,
54
we choose a fairly loose 90% relative convergence criteria to ensure that entrance
length results are unaffected by η∗, while still being close enough to steady state that
any further convergence can largely be neglected. With this criteria, for η∗ < 10−3,
transition length is independent of η∗.
More importantly, total change in force is nearly independent of η∗, with
a maximum difference in np of 0.038 between η
∗ = 10−3 and η∗ = 2 × 10−4 at
Re = 1,Bi = 1. This error decreased with Re, where at Re = 2000, Bi = 100,
the maximum np change was 0.009. This apparent change in active length is small
enough that effect of η∗ can be neglected. From these results, we conclude that the
biviscous model with η∗ = 10−3 is a sufficiently accurate numerical approximation of
Bingham plastic behavior for our problem.
2.3.2 Activation flow
2.3.2.1 Basic Flow visualization
In our activation flow model, we consider the two active lengths tested in
Goncalves, LA/Dh = 3.4925 and LA/Dh = 13.97, as well as LA/Dh = 2 and 7. For
all cases there is an upstream Newtonian region of length LN/Dh = 2.5. The grid had
75,000, 100,000, 125,000 and 140,000 cells for LA/Dh = 2, 3.5, 7 and 14 respectively.
On these four grids, we tested from Re = 0.1− 5000 and Bi = 0.1 = 200, for a little
over 1100 test points. For simplicity, we focus on Bi = 10, as it is a representative
test condition that shows the tested effects clearly.
We start by looking at images of velocity at a fairly short duct length. For
55
|u|/v0
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(a) Re = 0.1, Cτy = 400, He = 2
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(b) Re = 100, Cτy = 0.4, He = 2× 103
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(c) Re = 500, Cτy = 0.08, He = 10
4
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(d) Re = 2000, Cτy = 0.02, He = 4× 104
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(e) Re = 4000, Cτy = 0.01, He = 8× 104
Figure 2.10: Normalized velocity at Bi = 10. The active region starts at x = 0.
56
LA/Dh = 3.5 and Bi=10, normalized velocity, |u|/v0 is shown in Figure 2.10, where
we compare results at fixed Bingham number, so that only inertia affects the velocity
profile. In these velocity profiles, the key features to observe are the length and
starting position of the transition zone, and the region of uniform velocity, the
plug. At low Re, the transition zone starts before the active region, and reaches
the fully developed shape almost immediately after x = 0. Then, as Re increases,
we observe that the transition zone gets longer and effects upstream of x = 0 also
get smaller. Finally, at Re ≥ 2000, a close look shows that the flow does not quite
enter fully developed form. Essentially, nothing unexpected is observed: there are
no recirculation zones, no discontinuities and no oscillations. This means that there
is only one real flow feature of interest, the transition zone.
To understand the transition zone, it helps to visualize plug growth. Figure 2.11
shows images of normalized apparent viscosity, η/η∞, allowing clear observation of
the plug, or the region where η = η0. At low Re, the region of η/η∞ = 1000, an
effective proxy for the unyielded plug region, exists for the entire active region, with
a small bump at the entrance, from the flow adjusting to the presence of yield stress.
At Re=500, the low viscosity transition region can be seen, with the plug eventually
reaching steady state size. At Re=2000, the transition region extends the length of
the active region, and never reaches steady state plug form, as shown by the region
of η/η∞ = 1000 being slightly narrower at Re=2000 than at Re=500, something
difficult to see in Figure 2.10.
More rigorously, Figure 2.12 looks at normalized midline velocity, u(x, 0.5h)/v0
and wall shear stress. Looking at velocity, we confirm our previous observations,
57
η/η∞
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/
h
(a) Re = 0.1, Cτy = 400
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(b) Re = 100, Cτy = 0.4
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(c) Re = 500, Cτy = 0.08
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(d) Re = 2000, Cτy = 0.02
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(e) Re = 4000, Cτy = 0.01
Figure 2.11: Plug visualization through viscosity at Bi = 10. The active region starts
at x = 0.
58
-1 0 1 2 3 4 5 6 7
x/Dh
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
u
/
v 0
ub
Re
(a) Velocity
-1 0 1 2 3 4 5 6 7
x/Dh
1
1.5
2
2.5
3
3.5
τ
w
/
τ
w
N
τwp
τwb
1
10
100
200
500
750
1000
1250
1500
1750
2000
2200
2500
3000
3500
4000
4500
5000
ReRe
(b) Wall shear stress
Figure 2.12: Normalized velocity and wall shear stress as a function of Re for Bi = 10
and LA/Dh = 7. Dashed lines, – –, is the fully developed solution, Equation 2.17 or
2.19, and – · is parabolic wall shear stress, Equation 2.20 and · · · is the 90% entrance
length convergence criteria.
namely, significant upstream velocity changes at low Re, a lengthening transition
zone with increasing Re, and no convergence to the steady state solution at high
Re. Figure 2.12 shows the effect of finite duct length, which is the velocity ceasing
to converge to the steady state solution for the final 0.5Dh. This is caused by
wall-midline pressure difference being smoothed by the constant pressure boundary
conditions, and is a fairly small effect.
Wall shear stress tells a similar story. At low Re, shear stress rises ahead of
the active region, corresponding to the early transition. In the active region, after
a short transition length, low Re wall shear stress almost immediately approaches
τwb, while at high Re, the transition zone is longer, with shear stress reaching the
fully developed value further down the duct. Finite duct length are again visible,
where near the exit, wall shear stress stops rising, and possibly decreases slightly.
Again, no unexpected phenomenon are observed, and aside from a small oscillation
59
-2 0 2 4 6 8 10 12 14
x/Dh
1
1.1
1.2
1.3
1.4
1.5
u
/
v 0
Re=100 Bi = 0.5
Re=500 Bi = 2
Re=1500 Bi = 10
Re=4000 Bi = 100
2
3.5
7
14
LA/Dh
(a) Midline velocity
-2 0 2 4 6 8 10 12 14
x/Dh
0
5
10
15
20
25
τ
w
/
τ
w
N
Re=200 Bi = 1
Re=1000 Bi = 10
Re=1500 Bi = 50
Re=4000 Bi = 1002
3.5
7
14
LA/Dh
(b) Wall shear stress
Figure 2.13: Comparison of velocity and wall shear stress across various active
lengths.
around the onset of yield stress, wall shear rises from the parabolic value to the fully
developed value in a smooth exponential like manner.
The most critical feature of the transition zone is the distance down the duct
that the transition zone spans, which is contained in the entrance length. Previously,
we defined a 90% relative entrance length criteria, which for midline velocity, is a
90% convergence to ub from uN , and for shear stress, a 90% convergence to plug
shear stress, τwb, from parabolic shear stress, τwp. In Figure 2.12, midline velocity
and wall shear stress are plotted as a function of x and Re at Bi = 10, and shows
how the solutions grow. The key features to observe are the near linear increase in
xe with Re, and that at low Re, the entrance length can be negative, due to the flow
profile changing before x = 0.
To examine exit effects directly, Figure 2.13 compares u and τw at a represen-
tative set of low Cτy flow conditions, and demonstrates that at any x, all solutions
are extremely similar, with a small difference observed for LA/Dh = 2. The fact
60
0 1000 2000 3000 4000 5000
Re
10
-4
10
-3
10
-2
10
-1
10
0
10
1
τ
y
/
0
.5
ρ
v
2 0
0
0
.2
5
0.25
1
1
1
2
2
2
3
3 3
4
4
4
5
5
5
6
6
6
7
7
8
8
9
9
1
0
10
11
11
1
2
0
2
4
6
8
10
12
x
e
/D
h
Figure 2.14: Velocity entrance length xe,u for La/Dh = 14. Dotted black lines are
the bounds of sampled space. The pink dashed line is Cτy = 0.4.
that all active lengths produce nearly the same result at a given x, means that if we
correct for the small exit effects, we should be able to develop a model valid for any
arbitrary entrance length.
Entrance length characterizes the size of the transition zone, and is shown in
Figure 2.14 plotted as a function of Re and Cτy . Note that velocity based entrance
length can be negative, as the flow can transition upstream of the active region.As xe
increases, the transition zone is growing. When xe < 0 the flow transitions to fully
developed Bingham plastic flow before the fluid is a Bingham plastic, and Figure 2.14
places xe at Cτy = 1. To identify when the transition zone really starts growing, we
can relax the criteria slightly, choosing xe = 0.25Dh as a short entrance. This occurs
roughly at Cτy = 0.4 for xe,u, roughly where we expect β to start affecting force
output. We also note the strong resemblance in shape to the contours of np and xe,u.
Figure 2.15 displays integrated friction losses, pressure losses and both high
61
−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5
0
0.5
1
1.5
2
2.5
3
x/Dh
∆
p/
0
.5
ρ
v2 0
 
 
Friction losses
Wall pressure
Midline pressure
1−D fully developed
1−D parabolic
Figure 2.15: Pressure loss as a function of position at Re=2000.0, Bi=10.0. Fully
developed losses are from Equation 2.21, and parabolic losses from Equation 2.22.
and low speed models, and clearly shows the impact of β. The effect of the transition
zone is small, as demonstrated by the small difference between friction losses and
the fully developed model. Pressure losses, which include β effects, show a much
larger difference, dropping below the parabolic model at x/Dh = 1. Also, note that
pressure at the wall and midline diverge at the start of the active region, and do
not return to the same value until the end of the duct, where they meet due to the
exit boundary condition of a fixed, uniform pressure. The difference in pressure at
center and wall can be explained by applying Bernoulli’s equation, p+ 1
2
ρv2 = C, as
from the inviscid perspective, the fluid decelerates at the center of the flow, so there
should be a rise in pressure. In all our simulations, pmid−pwall ≤ τy, which is sensible,
as we are modeling a fluid with a yield stress, so it can support a pressure up to
τy. For most test points, this sustained pressure difference exists for a significant
duct length beyond the velocity and wall shear stress entrance length. This pressure
62
difference also explains the the acceleration of the midline velocity towards the end
of the duct seen in Figure 2.12, as the exit boundary condition specifies a uniform
pressure, so the pressure difference is resolved by accelerating the central core of the
fluid. This also means for the majority of the flow conditions, |dp/dy| > 0 . Aside
from the exit effects, this is pressure difference is largely negligible.
2.3.2.2 Graetz coordinate and fits
To further simplify these flow profiles, we note that for many situations, we
can largely eliminate Reynolds number dependence through use of a Graetz-type
coordinate [95, p. 292]. The Graetz coordinate scales length by Reynolds number,
is denoted here by a ′, where x′ = x/Dh/Re. In Figure 2.16, using the results
from LA/Dh = 14, we show midline velocity and wall shear stress in the Graetz
coordinate, and we observe that the overwhelming majority of the curves collapse
into a single curve. The curves that do not line up correspond to Cτy > 1, with
movement away from the high Re fit starting when Cτy > 0.4. In Figure 2.16 this
corresponds to Re < 100 at Bi = 10 and Re < 1000 at Bi = 100. This failure to
coalesce when Cτy > 0.4 demonstrates the effectiveness of our Cτy criteria for the
onset of inertial effects. We also note the small absolute change wall shear stress
exhibited at Bi = 100, as the difference between plug and parabolic flow is small.
Finally, for all conditions, when Re ≤ 10, scaling fails due to a lack of resolution
from the x/Dh scaled grid.
Since Graetz scaled wall shear stress and velocity curves collapse at a constant
63
-5 0 5 10 15
x/Dh/Re ×10 -3
1.4
1.42
1.44
1.46
1.48
1.5
u
/
v 0
0.1
1
10
100
200
500
750
1000
1250
1500
1750
2000
2200
2500
3000
3500
4000
4500
5000
Re
(a) Bi = 1, u/v0
-0.01 0 0.01 0.02
x/Dh/Re
1
1.05
1.1
1.15
1.2
1.25
τ
w
/
τ
w
N
0.1
1
10
100
200
500
750
1000
1250
1500
1750
2000
2200
2500
3000
3500
4000
4500
5000
Re
(b) Bi = 1, τw/τwN
-1 0 1 2 3 4 5
x/Dh/Re ×10 -3
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
u
/
v 0
0.1
1
10
100
200
500
750
1000
1250
1500
1750
2000
2200
2500
3000
3500
4000
4500
5000
Re
(c) Bi = 10, u/v0
-4 -2 0 2 4 6 8
x/Dh/Re ×10 -3
1
1.5
2
2.5
3
3.5
τ
w
/
τ
w
N
0.1
1
10
100
200
500
750
1000
1250
1500
1750
2000
2200
2500
3000
3500
4000
4500
5000
Re
(d) Bi = 10, τw/τwN
-5 0 5 10
x/Dh/Re ×10 -4
1
1.1
1.2
1.3
1.4
1.5
u
/
v 0
0.1
1
10
100
200
500
750
1000
1250
1500
1750
2000
2200
2500
3000
3500
4000
4500
5000
Re
(e) Bi = 100, u/v0
-5 0 5 10 15
x/Dh/Re ×10 -4
0
5
1
15
20
25
τ
w
/
τ
w
N
0.1
1
10
100
200
500
750
1000
1250
1500
1750
2000
2200
2500
3000
3500
4000
4500
5000
Re
(f) Bi = 100, τw/τwN
Re
Figure 2.16: Velocity and wall shear stress in terms of Graetz coordinate. The thick
dashed black lines are entrance length based fits. Dashed lines a fully developed
value, dotted line is the entrance length criteria, and dash-dot is the parabolic wall
shear stress value.
64
10
-1
10
0
10
1
10
2
Bi
0
1
2
3
4
5
6
7
8
x
e,
u
/
D
h
/
R
e
×10
-3
0.1
1
10
100
200
500
750
1000
1250
1500
1750
2000
2200
2500
3000
3500
4000
4500
5000
Re
(a) Midline velocity criteria
10
-1
10
0
10
1
10
2
Bi
0
0.002
0. 04
0.006
0.008
0.01
x
e,
τ
w
/
D
h
/
R
e
0.1
1
10
100
200
500
750
1000
1250
1500
1750
2000
2200
2500
3000
3500
4000
4500
5000
Re
(b) Wall shear stress criteria
Figure 2.17: Graetz scaled entrance length. Dashed lines are fits of xe,u and xe,τw .
Bingham number, we should be able to construct fits dependent only on Bingham
number. To do this, first we fit entrance length as a function of Bi, and then use
entrance length to define ufit(x) and τw,fit(x). Graetz scaled entrance length is shown
in Figure 2.17. Velocity based entrance length is fit as
x′e,u =
7.5× 10−3
1 + 0.2 Bi
(2.40)
which is effective for Bi > 1, and wall shear stress based entrance length is fit as,
x′e,τw =
8.7× 10−3
1 + 0.14 Bi
(2.41)
which holds up well across all Bi. We can then use the 90% convergence based
entrance length criteria in an exponential fit. This yields, for x > 0, fits for midline
65
velocity and wall shear stress
ufit(x) = uN − (uN − ub)
(
1− exp
(
log(.1)
x
xe,u
))
(2.42)
τw,fit(x) = τwp + (τwb − τwp)
(
1− exp
(
log(.1)
x
xe,τw
))
. (2.43)
These fits are shown in Figure 2.16 as the dashed black lines. While not perfect,
these very simple fits are effective enough, especially at low Cτy . We also note that
β(x) has a shape similar to u(x), so we can use xe,u to construct a fit for β(x
′),
Previously, we noted that towards the exit, both u and β stop converging in final
0.5Dh. Inserting this exit correction and fitting to minimize np error produces
βfit(x) = 6/5 + (βB − 6/5)
(
1− exp
(
log(.1)
x− 0.56Dh
xe,u
))
. (2.44)
To obtain pressure loss at any point in the duct, these fits can be placed into
our control volume analysis, Equation (2.25), giving
∆Pfit =
4
Dh
(
τwNLN +
∫ LA
0
τw,fit(x)dx
)
+ (βfit(LA)− 6/5)ρv20. (2.45)
With these fits, we have a description of the fluid behavior in ‘activation’ flow
that is dependent only on the fundamental dimensionless quantities that govern the
flow.
66
0 1000 2000 3000 4000 5000
Re
10
-4
10
-3
10
-2
10
-1
10
0
10
1
τ
y
/
0
.5
ρ
v
2 0
-1
.5
-1.25
-1
.2
5
-1
-1
-0.75
-0
.7
5
-0.5
-0
.5
-0
.2
5
0
-1.5
-1.25
-1
-0.75
-0.5
-0.25
0
np
(a) np, LA/Dh = 2, Cτy
0 1000 2000 3000 4000 5000
Re
10
-1
10
0
10
1
10
2
B
i
-2.
5
-2
.5
-2.2
5
-2
.2
5
-2
-2
-2
-1.75
-1.
75
-1
.7
5
-1.5
-1
.5
-1
.5
-1.25
-1
.25
-1
.2
5
-1
-1
-1
-0.75
-0
.7
5
-0
.7
5
-0.5
-0
.5
-0
.5
-0
.2
5
-0
.2
5
0
0
-2.5
-2.25
-2
-1.75
-1.5
-1.25
-1
-0.75
-0.5
-0.25
0
np
(b) np, LA/Dh = 3.5, Bi
-4
.5
-4
-4
-3.5
-3
.5
-3
-3
-2.5
-2
.5
-2
-2
-1.5
-1
.5
-1
-1
-0.5
-0
.5
0
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
np
0 1000 2000 3000 4000 5000
Re
10
2
10
3
10
4
10
5
10
6
H
e
(c) np, LA/Dh = 7, He
-8
-7
-6
-6
-5
-5
-4
-4
-3
-3
-2
-2
-1
-1
0
-8
-7
-6
-5
-4
-3
-2
-1
0
np
0 1000 2000 3000 4000 5000
Re
10
2
10
3
10
4
10
5
10
6
H
e
(d) np, LA/Dh = 14, He
Figure 2.18: Inertia correction in various forms. The dotted lines are the upper
and lower bounds on sampled space and the pink dashed line is Cτy = 0.4. The slid
contours are obtained directly from CFD. The light grey circles in Fig. 2.18b are the
CFD test points. Dashed contours in Fig. 2.18c are Eq. 2.45, and the brown dashed
line in Fig. 2.18d is the transition to turbulence.
67
2.3.2.3 Force correction factor
The next step is to use both the CFD and fit pressure loss data to examine the
change in force. Figure 2.18 presents contour plots of np for various active lengths, as
they reveal trends in np not otherwise visible. Figure 2.18a demonstrates that np has
limiting values in Re and Cτy as expected. We can also see that the Cτy = 0.4 rule of
thumb, the dashed pink line, very closely approaches the contour of np = 0.25 when
Re > 120 in Figure 2.18a-c. Since xe,u, np(β) = 0.25, and np coincide at Cτy = 0.4,
from here on out, we use this as our one size fits all criteria for the onset of speed
effects.
The fitted pressure model, ∆Pfit can be placed into np to obtain np,fit, which
we can then compare against CFD data. Figure 2.18c compares the solid CFD
contours with dashed line contours from the fit at LA/Dh = 7 (the case with worst
fit performance), and shows that fit is very accurate when np > −0.5LA/Dh. For
the region He = 103 − 106 and Re = 1 − 5000, across all active lengths, error
(np,fit − np,cfd)/LA is at worst 8.2% and on average, 3.5%, giving us confidence in
their accuracy.
As desired, when the flow is fully developed by the exit, np is unaffected by
LA, which can be seen in contours of np = 1 and 2 in Figure 2.18c and d. We also
note that the position of the np = 0 line is not reliable, as the precise location is very
sensitive to solution error.
Since we are considering a pipe-like flow with Re > 2000, we should consider
the role of turbulence. For a Bingham plastic in parallel plate, Hanks and Pratt
68
developed and experimentally validated a laminar-turbulent transition Reynolds
number for parallel plates [96],
He
33600
=
δc
1− δ3c
Rec =
He
12δc
(
1− 3
2
δc +
1
2
δ3c
)
where you solve for δc = τy/τwc, a critical plug thickness to obtain the transition
Reynolds number, Rec. In Figure 2.18d, we place a dashed brown line on Rec. While
the area to the right of the transition line occupies a large region in Re–He space,
at Re ≤ 5000, Bic ≤ 2. Since Bi is so low, this region will be rarely encountered in
controllable devices.
A more readily applicable way of looking at this data is shown in Figure 2.19,
which plots np as a function of Re for various He, comparing interpolated CFD data
and our curve fits. All the same sensible trends are observed - increasing Re increases
the force falloff, and raising He decreases it. In this form, we can also see mostly
clearly that the fits are accurate when np > −0.5LA/Dh. Figure 2.20 plots nv and
shows the excellent performance of τw,fit, which is accurate within 1.2%. We can
also see the very small positive nv bias at low Cτy , with typical values of nv = 0.05,
roughly the size of discretization bias error.
Finally, Figure 2.21 plots yield stress induced pressure loss, ∆Py =
(
dp
dx
)
b
(LA +
np)−
(
dp
dx
)
N
(LA + np), both with and without np. This ∆Py is normalized by the
parabolic model, and is also a normalized yield force. Here, we see a rise in ∆Py
69
0 1000 2000 3000 4000 5000
Re
-2
-1.5
-1
-0.5
0
n
p
1
×
10 6
3
×
10 5
1×10 5
3×104
1×104
1×103
1×10
6
3×10
5
1×10
5
3×10
4
1×10
4
1×10
3
He
(a) np, LA/Dh = 2
0 1000 2000 3000 4000 5000
Re
-3
-2.5
-2
-1.5
-1
-0.5
0
n
p
1×10 6
3
×
10 5
1
×
10 5
3×10 4
1×104
1×103
1×10
6
3×10
5
1×10
5
3×10
4
1×10
4
1×10
3
He
(b) np, LA/Dh = 3.5
0 1000 2000 3000 4000 5000
Re
-6
-5
-4
-3
-2
-1
0
n
p
1×106
3×10 5
1
×
10 5
3
×
10 4
1×10 4
1×10 3
1×10
6
3×10
5
1×10
5
3×10
4
1×10
4
1×10
3
He
(c) np, LA/Dh = 7
0 1000 2000 3000 4000 5000
Re
-10
-8
-6
-4
-2
0
n
p
1×106
3×10 5
1
×
10 5
3
×
10 4
1
×
10 4
1
×
10 3
1×10
6
3×10
5
1×10
5
3×10
4
1×10
4
1×10
3
He
(d) np, LA/Dh = 14
Figure 2.19: Active length correction, np from He = 10
3 − 106. Dashed lines are fits
of np.
0 1000 2000 3000 4000 5000
Re
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
n
v
1×10 6
3×10 5
1×10 5
3×10 4
1×104
1×103
1×10
6
3×10
5
1×10
5
3×10
4
1×10
4
1×10
3
He
(a) nv, LA/Dh = 2
0 1000 2000 3000 4000 5000
Re
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
n
v
1×106
3×105
1
×
10 5
3
×
10 4
1
×
10 4
1
×
10 3
1×10
6
3×10
5
1×10
5
3×10
4
1×10
4
1×10
3
He
(b) nv, LA/Dh = 14
Figure 2.20: Active length viscous correction, nv. Dashed lines are fits.
70
0 1000 2000 3000 4000 5000
Re
0
0.5
1
1.5
∆
p
y
/
(4
τ
y
L
a
/
D
h
)
1×106
3×105
1×105
3×104
1×104
1×103
He
(a) ∆Py, LA/Dh = 3.5
0 1000 2000 3000 4000 5000
Re
0
0.5
1
1.5
∆
p
y
/
(4
τ
y
L
a
/
D
h
)
1×106
3×105
1×105
3×104
1×104
1×103
He
(b) ∆Py, LA/Dh = 14
Figure 2.21: Apparent yield force, normalized by parabolic yield force using np. Solid
lines are CFD, dashed lines are fits, dash dot is the fully developed model, and
dotted is the parabolic model.
from the formation of the plug, followed by a large decrease in yield force caused
by inertia and momentum effects, one which can go below ∆Py,p. This crossing
point serves as our definition of high speed flow: high speed flows are those where
∆Py < ∆Py,p.
From this we can make a basic estimate of the yield force falloff in Goncalves.
In those experiments, He = 104 to 3 × 104, and max Re = 1400. Here, can see
that yield force is dropping 50–60% for LA/Dh = 3.5, and 10–20% for LA/Dh = 14.
While this isn’t definitive, nor enough to claim a full description of the yield force
falloff seen in Goncalves, we clearly have captured a significant portion of the trend.
In the next chapter, we examine Goncalves’ device and all of the high speed MR
devices known us. Then, by accurately predicting yield force at high speeds, we
demonstrate that we can describe high speed yield force falloff using only classic
fluid dynamics, and without a response time.
71
0.5
0.5
0.5
0.5
1
1
1
1.5
1.5
1.5
2
2
2
2.5
2.5
3
3
3.5
3.5
4
4
4.5
5 5.5
Re
H
e
 
 
0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
102
103
104
105
106
np
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
(a) np
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.6
0.8
0.8
0.8
1
1
1.2
1.2
1.4
1.4
1.6
1.8
2
Re
H
e
 
 
0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
102
103
104
105
106
nv
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
(b) nv
Figure 2.22: Deactivation correction factors for LA/Dh = 4 and LN/Dh = 13.97.
The box is the test window of Goncalves.
2.3.3 Deactivation Flow
The follow-on to looking at fluids entering the active region is to investigate the
reverse situation, a ‘deactivation’ flow where fully developed Bingham plastic suddenly
loses its yield stress and returns to Newtonian flow, or alternatively, Figure 2.4 with
the flow direction flipped left to right. The boundary conditions are the same as
before, except the inlet velocity is specified as the exact fully developed biviscous
solution, Equation (2.17).
Figure 2.22 plots correction factors for deactivation flow with a long Newtonian
region, LN/Dh = 13.97, and inlet active region LA/Dh = 4. The active region was
was lengthened, because as before, there is a wall/midline pressure difference that
needs to be resolved smoothly. The primary result is that the deactivation flow
correction factor is almost identical to the negative of the activation flow correction
factor for LA/Dh = 13.97. This indicates that in a long active region with long
72
entrances and exits, using the fully developed quasi-static pressure loss model will not
cause problems, as despite there being significant inertia effects, the activation and
deactivation corrections will cancel each other out. Then in a flow mode rheometer
where LA is long enough for the flow to be fully developed, and an exit region long
enough for the flow to return to Newtonian flow, such as the capillary rheometer in
[60], [97], we expect the activation and deactivation flow effects to cancel each other
out, and to see minimal inertia and momentum effects.
However, typical damper designs have the active region in a contraction, so
there is no exit passive region, and there will be no momentum recovery at the
exit, np will apply. But if the passive region between two active regions is smooth,
we speculate that the activation and deactivation factors will mostly cancel each
each other out, and speed effects will only be observed at the first entrance. But
in general, accurate analysis of real devices with multiple gaps will require more
detailed analysis, and we may pursue such work in the future.
2.4 Conclusions
In this study, we performed a rigorous analysis of the 2D fluid dynamics
of a controllable fluid entering or exiting an active region. To allow us to draw
generalizable conclusions, we worked with a simplified model, capable of replicating
the work of Goncalves. Using our flow model with zero yield stress response time,
we are able to produce a reduction in yield force at high speed. We conclude the
following:
73
1. The high speed yield force falloff effect in Goncalves is caused by the fluid
dynamics. The yield stress causes the velocity to develop a plug like profile,
reducing fluid momentum, causing a pressure rise.
2. This high speed force falloff effect can be viewed as a change in active length.
Fits accurately matching CFD results from He = 103 − 106 and Re = 1− 5000
are reported in Equations 2.43, 2.44 and 2.45, and plotted in Figure 2.18.
3. Momentum effects in near quasi-static flow can be estimated from Eq. 2.34
and 2.35. These can be turned into a simple criteria for when quasi-static
models can be used safely: Cτy > 0.4 and Re < 120 for a typical device with
LA/Dh = 5 wishing to stay within 5% of the fully developed solution. We also
define high speed flows as when ∆Py < ∆Py,p, or when high speed yield force
falloff becomes readily apparent.
4. Deactivation flow correction factors are almost the exact opposite of the
activation flow factors, indicating that the rare device with a long exit region
will not experience these effects.
In the next chapter, we experimentally validate these fits on a range of devices
operating from He = 104 − 106 and up to Re=4500. Going forward, we would like to
improve the efficiency of the solver, as solution convergence was slow, and there no
doubt exist more efficient methods. We would also like to examine more realistic
geometries, to capture the fluid dynamic effects present in real devices, and see how
they affect active region flow and force output. Overall, we believe that proper 2D
fluid dynamics will prove essential in accurate design of high speed energy absorbers.
74
Chapter 3: Experimental validation
3.1 Introduction
High speed yield force falloff is a phenomenon seen magnetorheological (MR)
fluid flow mode devices, where the yield force, defined as Fy , Fon − Foff , drops
as active gap velocity increases. This can be clearly seen in Goncalves’ high speed
capillary rheometer tests [11], [12], the impact absorber tests of Mao [13] and Mao
et al. [14], and the seismic dampers built by the Sanwa Tekki corporation [15], [56].
This is phenomenon is not predicted by existing yield force models, where existing
models predict either an increasing or constant yield force with increasing speed.
Goncalves hypothesized that the yield force falloff is caused by a fundamental
response time, where below a certain fluid active region residence time, there is no
yield stress. Mao [13] and Mao et al. [14] used an alternate active region force model,
where the amplification of the viscous force by the yield stress induced plug does not
occur. The Sanwa Tekki group tried to resolve the missing yield force by removing
upstream Newtonian valve losses [98], but we believe the yield stress altering the
fluid dynamics far upstream is unlikely.
We believe that the reduction in yield force is caused by active region fluid
dynamics, but note that existing methods for predicting Fy and Fon are not grounded
75
in rigorous fluid dynamics. These methods obtain Fy by assuming a 1-D velocity
profile, and subtracting off the fully developed Newtonian losses. To obtain Fon, first
a model of the offstate is constructed, and then Fy is added to the off-state forces
to obtain on-state forces, or Fon,model = Foff,model + Fy,model. Aside from the fact
that it works at low speeds, this procedure is questionable, as it involves adding the
difference between the solutions to two different nonlinear fluid dynamics problems.
At high speed, where on-state force predictions are inaccurate, we need to
question the three components of predicting on state forces: off-state forces, adding
and yield forces. At high speeds, Min Mao’s dissertation [13] demonstrated that
off-state force modeling can be quite accurate, so this is unlikely to be the source of
error. There are then two candidates for the source of inaccuracy: the assumption
that yield force can be superimposed and the accuracy of the yield force model. The
ability to add yield force to off-state losses is a powerful assumption for the device
designer, as predicting off-state forces is fairly straightforward, and existing yield
force models are simple analytical expressions. If we can’t add a yield force, it means
that we need to perform full active Bingham plastic CFD of the entire device, which
in our experience is cumbersome and impractically slow. Then from a perspective
where we seek simplicity and productivity, it is much preferable to assume that we
can superimpose a yield force, and then seek to create a more accurate yield force
model.
The obvious flaw in existing yield force models is that they are based on simple
1D solutions. Our approach is to develop a correction based on Goncalves’ experiment
and apply it to practical devices, using the last chapter’s exploration of the fluid
76
dynamics in Goncalves’ device. To analyze Goncalves’ experiment, we created
‘activation flow’, a simplified 2D CFD model where fully developed Newtonian flow
enters a region of uniform yield stress with a constant pressure exit. This simplified
model is the first rigorous examination of high speed variable yield stress fluid flow,
and we claim it captures the fluid dynamics common to flow-mode devices. In our
analysis of activation flow, the main cause of high speed force falloff is a change in
control volume momentum flux from the velocity profile developing a ‘plug’. We
then used the CFD results to develop a fit for yield force based on control volume
analysis of activation flow, as analytical criteria for the onset of fluid dynamic effects.
In this paper, we show that the yield force obtained from the analysis of activation
flow accurately predicts yield force at high speeds in practical devices.
To validate this model we need to identify devices which operate in the high
speed regime where the fluid dynamics affect yield forces. Table 3.1 collects piston
velocity vp and yield force Fy for a series of devices that caught our eye. But since
our explanation for this behavior is a fluid dynamic one, we include the scaled
active length, LA/Dh, and the two governing nondimensional numbers: Reynolds
and Hedstrom number, Re and He. These fluid dynamic quantities are defined using
the properties of the active gap, with Re = ρvgapDh/η∞, the dimensionless velocity
and He = ρτyD
2
h/η
2
∞, the dimensionless yield stress. Excluded from this list are
papers that do not list enough geometric and fluid dynamic properties to compute
the required properties. The advantage of nondimensional analysis is that it allows
comparison of physically dissimilar devices, expanding the range of devices we can
validate against. For example, it may not be readily apparent that a high-force
77
(a) Physical units
(b) Dimensionless
Figure 3.1: Fluid dynamic properties of MR dampers in both raw units and nondi-
mensional form. See Table 3.1 for details.
78
Author Fy, kN vp, m/s Re He LA/Dh
S.B. Choi et al. [99] 0.3 0.2 240 4× 104 9.2
S.B. Choi et al. [100] 2 0.4 170 3.4× 104 5
Fujitani et al. [98] 300 0.25 570 2.1× 105 2.7
Goncalves et al. [12]* 35 0.036 1400 2.7× 104 3.9 - 15.7
Gordaninejad et al. [101] 100 0.1 83 1.5× 105 3.6
Li and Wang [102] 3 3.6 1200 1.1× 105 7.5
Li and Xu [103] 20 0.13 36 6.3× 103 13
Parlak and Engin [74] 0.8 0.2 96 2.6× 104 5
Rodriguez et al. [104] 16 0.07 860 9.7× 104 5.5
Sodeyama et al. [56] 100 0.2 1100 1.2× 105 2.5
Sodeyama et al. [15]* 160 0.4 1700 7× 104 6
Tu et al. [105] 400 0.2 93 2.8× 103 10
Yang [106] 170 0.08 20 1.7× 103 5.5
UMD
Y.T. Choi et al. [107]* 7 1 1600 4.6× 104 6.25
Mao et al. [14], 1* 6 5.1 4500 1.4× 105 6.2
Mao et al. [14], 2* 5 4.9 4300 2.4× 105 5.1
Mao et al. [108], 3 2 6 1000 2.1× 104 2.5
Singh et al. [109]* 12 4.4 3223 3.3× 105 4.7
Table 3.1: Fluid dynamic properties of potential high speed magnetorheological
dampers. We compare our model to the devices marked with an asterisk.
79
low-speed seismic damper [15] and a high speed impact absorber [107] have similar
fluid dynamics, but Table 3.1 shows that they do, and later we demonstrate that
both devices see similar amounts of high speed force falloff.
Figure 3.1 displays Table 1 visually, comparing the traditional presentation
of device size and speed, Fy vs vp, and our dimensionless form, Re vs He. In the
dimensionless presentation, we leverage our analysis to separate the space into five
regions. Using a typical geometry of LA/Dh = 5, we split space into the three areas:
the quasi-static regime where Cτy > 0.4 and Re < 120, and Fy ≈ Fy,B; the low speed
regime where yield force will larger than parabolic model prediction Fy, act > Fy,p; and
the high speed regime Fy, act < Fy, p where yield force falls below the high parabolic
model, and thus where high speed yield force falloff should be large. We also indicate
the region where controllable force is low, Fy, p < Foff , or Bi < 6, as well as the region
of turbulent flow. This also motivates the extensive nondimensional analysis in the
previous chapter, as all four nondimensional numbers—Re, He, Bi and Cτy—are
needed to describe various elements of the fluid dynamics.
This paper compares our activation flow yield force model to all the devices in
the high speed region. We do not look at devices in the low speed and quasi-static
regions as in this region, as existing models perform well for those devices, and by
construction, our model is largely indistinguishable. First, we look at Goncalves’
capillary slit rheometer with a single active region [12], the inspiration for the
activation flow model, and show reasonably accurate predictions. We follow this with
the two double rod dampers of Mao et al. [14], which operate at the largest Reynolds
numbers ever, and demonstrate that activation flow yield force predictions are highly
80
accurate. We then examine Singh’s 12 kN damper [109], which has the largest
Hedstrom number of any known device, and show excellent model performance.
Next, we look at the 2 kN the landing gear damper in Choi with gas reservoir and
two active regions, and show reasonably effective yield force predictions. We exclude
Li and Wang [102], who do not present any force–velocity data, and Mao’s bi-fold
damper [108], which has a large bend in the active region and a significant spring
term. We conclude by looking at a damper from the Sanwa Tekki Corporation [15],
and discuss several of their other seismic devices and what they tell us about high
speed force falloff.
3.2 Background
We claim that activation flow is a more accurate yield force model, and allows
use of yield force superimposition at high speeds. To do so, we need to account
for off-state forces, yield forces, how we apply this to a typical contraction device,
turbulence effects, and concerns about time domain effects.
3.2.1 Off-state forces
To obtain experimental yield forces, we need to obtain the off-state force at ar-
bitrary values, as many experiments do not have on-state and off-state measurements
at the exact same piston velocities. Off-state forces are composed of viscous (∼ v)
and dynamic terms (∼ v2), with dynamic losses becoming substantial at high speed.
At even larger speeds, turbulent losses also become significant. Since we are working
81
with models that treat yield force as totally independent of the off-state forces, we
don’t need to know any details of the off-state fluid dynamics. For most of the
devices here, this allows us to ‘predict’ off-state forces by curve fitting experimental
off-state forces. However, when looking at Goncalves’ slit rheometer, we perform a
full model of the offstate forces.
In general, we fit the force–velocity data to a quadratic or higher polynomial.
The quadratic term force term F = ... + ApK
1
2
ρv2gap + ... is of importance, as
conservation of momentum requires that K > 1 for a typical device with a lossy exit.
This is of critical importance when examining Goncalves’ device. For several devices
higher order polynomials are needed to capture the increase in force caused by the
transition to turbulence.
3.2.2 Yield Force
In our previous paper, we developed a set of fits designed to describe activation
flow, where fully developed Newtonian flow enters an active region, instantly develops
a yield stress and then converges to fully developed Bingham plastic flow. Our model
82
for pressure loss in activation flow is below:
Re =
ρv0Dh
η∞
Bi =
τyh
η∞v0
He =
ρτyD
2
h
η2∞
Cτy =
τy
1
2
ρv20
(3.1)
τwN = 12vgapη∞/Dh τwB = τy/δ τwp = τy + τwN (3.2)
δ = 2a cos(1/3 cos−1(a−3)− 2pi/3) a = √1 + 4/Bi (3.3)
β =
∫ h
0
u(x, y)2dy
v20h
βB =
Bi2 (δ − 1)4
16 δ
− Bi
2 (δ − 1)5
30 δ2
(3.4)
xe,u =
7.5× 10−3
1 + 0.2 Bi
ReDh xe,τw =
8.7× 10−3
1 + 0.14 Bi
ReDh (3.5)
τw, act(x) = τwp + (τwB − τwp)
(
1− exp
(
log(.1)
x
xe,τw
))
(3.6)
βact(x) = 6/5 + (βB − 6/5)
(
1− exp
(
log(.1)
x− 0.56Dh
xe,u
))
(3.7)
∆Py, act =
4
Dh
∫ LA
0
(τw,act(x)− τwN) dx+ (βact(LA)− 6/5)ρv2gap (3.8)
This fit should be an effective description of the CFD data from 103 < He < 106,
0 < Re < 5000, and 2 < LA/Dh < 14, a range that covers nearly every ER and MR
flow mode device built. We also have the existing yield force models,
∆Py,B =
4(τwB − τwN)LA
Dh
(3.9)
∆Py,p =
4(τwp − τwN)LA
Dh
=
4τyLA
Dh
(3.10)
the fully developed and parabolic flow yield force models respectively. Yield force in
a flow mode device is typically obtained from pressure loss, when neglecting wall
shear stresses, from F = Ap∆P , where Ap is the area of the damper piston head.
83
For improved accuracy, we include the piston wall shear stress forces, so
F = Ap∆P +
∫ xout
xin
τw 2pirdx,
where ∆P contains dynamic losses, and the second term contains the shear stresses
on the piston. This can be rewritten, separating wall shear stress losses and dynamic
loss terms K,
F = Ap
[
4
∫ xout
xin
τw dx
Dh
(
1 +
pirpDh
2Ap
)
+K
1
2
ρv2gap
]
(3.11)
with β effects as part of K. This also demonstrates the utility of constructing a
fit based on control volume analysis, as it allows it to be easily modified to suit
alternate geometries. In all the devices here, the extra wall shear stress contribution
pirpDh/2Ap is small, increasing wall shear stresses by less than 5%, but provides an
improvement.
Our goal to is demonstrate that the activation flow pressure losses contained in
∆Py, act can accurately predict yield force at high speeds. We do so by comparing yield
force prediction from it and from the two existing flow models: fully developed flow
and parabolic flow. Fully developed flow, ‘plug flow’, is characterized by a growing
yield force with increasing speed, eventually limiting out at 1.5× the yield force at
zero speed, but assumes that there are no inertial effects, or Re = 0. Parabolic flow
assumes that the velocity profile is unchanged from the fully developed Newtonian
case (Re = ∞), and is just the addition of the yield stress to the Newtonian wall
84
1	
0	 1	 2	 3	
Ap	
Ares	
Figure 3.2: Schematic of a step contraction.
shear stress. Parabolic flow is accurate at large Bi, where plug amplification is small,
as well as at moderate velocities where yield force falloff is present, but a small
effect. The activation flow model exhibits an increase in Fy at low velocities, like
plug flow, followed by its unique feature, a decrease in Fy as velocity increases, where
it eventually predicts forces that are less than the parabolic model, which simply
adds a yield stress. We define the onset of high speed flow as when the activation
flow model predicts a yield force below the parabolic model, ∆Py, act/∆Py,p < 1. In
Fig. 3.1 this is used to identify the set of high speed devices which we will validate
the activation flow yield force model against.
3.2.3 Contraction flow
Activation flow is an idealized and simplified representation of Goncalves’
device, but here we claim that you can apply this model to a typical MR damper.
These devices are very different in construction, where Goncalves’ device is an active
region following a long passive region, while most ER and MR dampers place the
85
active region in a short step-like contraction. An example of this is shown in Fig.
3.2, where the contraction is the region between points 1 and 2.
In order to justify applying activation flow to yield force predictions in step
contractions, it first makes sense to examine Newtonian flow in a step contraction.
For a typical Newtonian contraction with length LA, after a careful reading of White
[95, p. 290-293] and Merritt [110, p. 39-48], we identify these useful observations:
1. There are minimal viscous losses between 0 and 1 in Fig. 3.2, so from conserva-
tion of momentum, P0 − P1 = 12ρv2gap.
2. Losses inside the contraction are modeled using fully developed viscous losses
and a velocity squared correction term, or P1 − P2 = 4τwNLA/Dh + 12Kρv2gap,
with K capturing the extra losses encountered in the contraction.
3. At the exit, P2 = P3, as there is no pressure recovery due to turbulent losses.
4. Combining, total pressure loss is P0 − P3 = 4τwNLA/Dh + 12(1 +K)ρv2gap
5. Fully developed viscous losses imply that we have assumed a fully developed
velocity profile between 1 and 2.
6. The correction factor, K, includes the change in momentum from changing
velocity profile. For a Newtonian fluid in a rectangle, K ≈ 0.5− 0.67 of which
0.4 is from momentum (β) effects, with the rest coming from an increased τw
in the entrance of the flow.
7. The classic simplified model of a contraction flow, entrance flow, is a straight
pipe/duct with a uniform velocity inlet.
86
These assumptions tell us that all the interesting fluid dynamics occur in the
contraction between 1 and 2 in Fig. 3.2. Then to obtain the yield force, we should
solve a contraction/entrance flow type problem and subtract off Newtonian losses.
For Bingham plastic entrance flow, we should get a decreasing yield force as speed
increases, as Chen et al. [88] shows that in quasi-1D models, K decreases with
increasing Bi, due to the reduced change in momentum and shorter entrance length
reducing extra viscous losses. However, 1D methods, while capturing broad trends,
are known to be inaccurate for Bingham plastic entrance flow problems [69], [70].
Second, the inlet boundary condition of uniform flow causes a large non-physical
wall shear stress spike, as in a controllable yield stress fluid, large changes happen
upstream of the region of yield stress. Fixing this requires abandoning entrance flow
and using the more complex contraction flow geometry. However, fully solving for
flow past a contraction has various subtle challenges [111], and accurately adding a
variable yield stress will only add more complications.
The state of the art in active region MR fluid modeling at high speeds [13],
[109] uses an unchanged K factor and the parabolic model for τw. If yield stress
is independent of velocity, then the unchanged K and assumed wall shear stresses
must be the source of error. Using the insights gained from activation flow and
our knowledge of Newtonian contraction flow, we can construct an estimate for the
true losses in a contraction with yield stress with the following three requirements.
First, we expect a reduced K compared to the Newtonian case, as we know that β
contribution to K will be smaller. Also, the β effects should be reduced if the flow is
not fully developed, x < xe,u. Second, we expect a reduction in the τw contribution
87
to K, as both the extra wall shear stresses are less (the steady state profile is flatter),
and the entrance length is shorter. Third, to be grounded in solid physics, the yield
force model should reduce to plug flow at low speeds. Activation flow satisfies all
three criteria, allowing us to use our model of activation flow to predict yield force in
contraction flow. While the entrance lengths and wall shear stresses from activation
flow are unlikely to be a close match to the real contraction flow values, we show that
the activation flow entrance lengths provide an accurate prediction of yield force.
An alternate reasoning for this choice comes from invoking linear superposition.
In a Newtonian flow, viscous losses in the contraction are found from fully developed
parabolic flow, with an additional K term to capture any losses. Then to estimate
yield force from the current solution of parabolic flow everywhere, we superimpose
the transition from parabolic to plug (activation flow) to obtain the yield forces.
While this is a conceptually simple explanation, superposition of solutions to different
equations is unlikely to be strictly valid.
3.2.4 Turbulence
In a paper on experimentally verifying a laminar flow duct model when Re >
2000, turbulence is a major concern. Fortunately, the yield stress adds viscosity,
delaying the onset of turbulence. Hanks [96] develops a theory for the critical
Reynolds number as a function of Hedstrom number for traditional Bingham plastics
88
in a duct:
He
33600
=
δc
1− δ3c
(3.12)
Rec =
He
12δc
(
1− 3
2
δc +
1
2
δ3c
)
(3.13)
and finds excellent agreement with experiments conducted on 0.8 µm to 4.8 µm sized
aluminum silicate pigment particles, which is roughly equivalent to an MR fluid with
no field. Figure 3.1 plots Rec and show that none of the devices we validate against
operate in the turbulent regime, with only a few of Mao’s test points approaching
the turbulent transition.
While it appears easy to rule out turbulence in active flow, turbulence is actually
a potential explanation for high speed yield force falloff. In particular, Govier and
Aziz [82, p. 229-233] note the consistent observation that in a turbulent pipe flow,
at equivalent Reynolds numbers, Bingham plastic friction factors are expected be
less than or equal to the Newtonian friction factor when He < 106. However, this
is only true for fully turbulent flow. According to Hanks’ turbulence criteria, the
experiments discussed in this paper should never be fully turbulent.
Even then, the presence of turbulence is challenging to rule out, as we do
encounter scenarios where Re > Rec in the passive region but Re < Rec in the active
region. Then if the flow is turbulent in the passive region, but in the active region,
Re < Rec, but only slightly, it seems plausible that the active region would still be
turbulent due to the upstream turbulence. Second, if the passive region is turbulent
and the active region flow instantly becomes laminar, then the assumed parabolic
89
inlet profile is no longer valid, as turbulent flows have a much flatter inlet profile.
This would cause us to overestimate β effects and thus high speed yield force falloff.
For devices with a straight slot/pipe type design, the effect of upstream turbulence
cannot be ruled out. This is a potential issue in testing Singh’s large damper, but
one that doesn’t seem to arise. In contraction type devices, upstream turbulence
should be less critical, as the flow profile will dominated by the effects of contraction.
Experimentally, evidence of turbulence in traditional Bingham plastic flows
is expected to be sharp and clear [112]. However, a brief examination of the MR
literature shows no sharp transitions or onstate measurements dropping below the
off-state values. Mao et al. [14] comes close to turbulence and sees a small reduction
in yield stress, but we show below this can be explained by the laminar activation
flow model. Browne et al. [9] conducts a series of impacts tests, which are likely in
the turbulent regime, but show a near constant yield force at high speeds. The lack
of evidence for turbulence is perhaps unsurprising, as the Bingham plastic literature
is based on suspensions with τy < 100 Pa. For MR fluids, yield stresses are orders
of magnitude larger and caused by an external applied field, so the assumption of
Bingham plastic behavior may not hold. However, Gu¨th and Maas [58] show that in a
shear cell with large gap, Taylor-Couette instabilities are suppressed with increasing
magnetic field, a result consistent with the fluid behaving as a Bingham plastic.
Govier and Aziz [82] review turbulent Bingham plastic and Herschel-Bulkley
friction factor equations, and show substantial variation in form and value of the
estimates. They conclude that the turbulent Bingham plastic friction factors should
be estimated using the Newtonian friction factor. However, we never conclusively
90
10 -1 10 0 10 1 10 2 10 3
Bi
0
0.05
0.1
0.15
0.2
0.25
T
r
=
t r
η
∞
/
ρ
h
2
Dapra and Scarpi
Fit
Figure 3.3: Response time for Bingham plastic channel start up flow.
reach the turbulent regime, so choosing a model is not necessary. For CFD, Malin
[113] argue that existing turbulence models, k −  and k − ω, are sufficient to
describe the flow of Bingham plastics, but such claims are based on comparison to
measurements with τy < 10 Pa. Overall, we conclude that it is unlikely that any of the
devices here are turbulent in the active region, but that any strong conclusions about
the nature of turbulence in MR fluids will require careful experimental measurement.
3.2.5 Time domain effects
We also need to rule out time domain effects, since many of the high speed
tests we look at are impact tests taking 20 ms or less. The experiments here are not
looking at force rise time to an applied current, but rather the force rise time to an
applied velocity. This flow response time, tr, can be approximated by classic startup
flow, where a constant pressure gradient is suddenly applied to a stationary fluid in
a duct. For Newtonian fluids, Schlichting [114, p. 85] shows that fluid response time
is independent of velocity, with a dimensionless response time, Tr = trη/ρh
2.
91
The same scaling holds for Bingham plastics, with an analytical solution for
startup flow of a Bingham plastic between infinite parallel plates available in Dapra`
and Scarpi [115]. The solutions are presented here in our notation, in terms of
applied pressure gradient ∇P , the time dependent plug thickness δt, and steady
state plug thickness, δ. By solving for plug thickness,
δT (T ) = δ
{
1− 4
pi
∞∑
k=0
(−1)k
2k + 1
exp
[
−pi
2
4
(
2k + 1
1− δ
)2
4T
]}−1
(3.14)
and velocity, v∗ = 4uµ/h∇P ,
v∗(T ) =
1
2
(1− δT )2 − 16(1− δT )
2
pi3
∞∑
k=0
(−1)k
(2k + 1)3
exp
[
−pi
2
4
(
2k + 1
1− δT
)2
4T
]
(3.15)
for the time at which velocity reaches 90% of the steady state value, we can obtain a
response time.
The response time is shown in Figure 3.3. Since solving these equations is
cumbersome, we construct fit Tr = 0.235/1 + 0.2Bi, which is also shown. For a
typical experiment here, h = 1 mm, ρ = 3 g/cm3, and η∞ = 0.1 Pa s and Bi > 10,
Tr < 0.1, so tr < 3 ms, which is close to, but below, the time scales seen in the
experiments investigated here. We then expect no difficulties in using our steady
state activation flow model on any of the drop stand experiments.
We also note that the response time to an applied field has been analyzed
elsewhere, but we believe incorrectly. Choi and Wereley [116] solves the transient 1D
flow profile using Newtonian fluid basis functions, which can introduce an unwanted
92
stress discontinuity. Phillips [86] attempts a simple estimate for response time, but
flips a negative sign in the pressure gradient, yielding an incorrect result.
93
Parameter Mao 1 [14] Mao 2 [14] Singh [109] Goncalves [12] Sodeyama [15] Choi [107]
Dh, mm 1.83 2.42 5.18 1.62 3.0 1.67
LA, mm 15.2 16.3 30.5 6.35 or 25.4 10 14
rp, mm 19.7 19.43 28.1 - - 15.9
Ap, mm
2 963 927 1968 8107 2.9× 104 2553
Ares/Agap 9.8 7.5 5.1 1026 67 30.4
η∞, Pa s 0.072 0.072 0.112 0.13 – 0.17 0.13 0.092
ρ, g/cm3 3.52 3.52 3.08 3.06 3.06 3.08
Table 3.2: Key parameters for the high speed devices in Table 3.1.
94
3.3 Devices
With these observations in mind, the most direct course of action is to simply
see if the model contained in Eq. 3.8 works. We will compare it against experimental
measurements from all of the high speed devices in Fig. 3.1, with details in Table
3.2. Experimental forces and velocities are taken from the authors’ filtered results,
with no additional post-processing.
We will show both total modeled force and yield force for each device using
the three yield force models, Fy,p, Fy,B and Fy,act. For the yield force diagrams, the
right axis maps yield force into the parabolic apparent yield stress,
τy,app =
FyDh
4ApLA(1 +
pirpDh
2Ap
)
(3.16)
This allows us to see what the apparent yield stress falloff is.
3.3.1 Goncalves’ slit rheometer
Goncalves tests a capillary slit rheometer at large gap speeds. In this device,
a large piston pushes Lord MR132LD into a slit with an active region placed at
the end of the duct. The fundamental observation is that as speed rises, there is
a reduction in yield force at high speeds. There is also an increase in yield force
at low speeds over the zero speed value, as expected from fully developed Bingham
plastic flow. This experiment is the only steady-state high speed test of MR fluid,
and forms an essential test case for our activation flow analysis.
95
In general, the constitutive properties of Lord MR132LD are not clear, and
it appears to have some level of plasticity in the off-state. Goncalves’ dissertation
contains off-state rheometer measurements from the fluid datasheet, which are
performed on a very low speed cup and bob rheometer (0 < γ˙ < 160 s−1), and indicate
that at 40 ◦C the fluid is a Bingham plastic with τy = 12 Pa and η∞ = 0.17 Pa s. As
we discuss later, Sodeyama et al. [15] also shows datasheet measurements of viscosity
of Lord MR132LD, here at larger shear rates, (0 < γ˙ < 1000 s−1), and also observes
substantial shear thinning. These off-state measurements also indicate Bingham
plastic behavior in the off-state, with τy = 13 Pa and η∞ = 0.11 Pa s. This indicates
that there is substantial uncertainty about the actual viscosity of the fluid, but
in both cases the yield stress is small enough to be negligible. For consistency we
assume the fluid is Newtonian and fit η∞ and K, a hydraulic loss coefficient, to the
off-state measurements.
To validate our analysis, we extract the pressure loss data from Goncalves’
thesis using a visual data extraction tool, and compare our activation flow model to
the experimental results. We also make the following three observations about the
experiment: there is a large uncorrected temperature rise in the slit, there are no v2
effects from the contraction and there is uncertainty in the geometric properties.
3.3.1.1 Temperature effects
In the experiment, the measured temperature rise across the slit is 11 ◦C,
sufficient to cause substantial changes to the viscosity of the fluid. There is also a
96
gradual 2 ◦C temperature rise during testing. A back of the envelope calculation
indicates this should change viscosity by 20-30%, which would produce a change in
force as large as the observed yield stress falloff. This means that to make precise
conclusions, we need to carefully develop a temperature correction.
However, we do not have comprehensive Lord MR132LD temperature viscosity
data, so we need to estimate the viscosity temperature effects. We construct an
estimate using the principle that suspension viscosity over carrier viscosity is inde-
pendent of viscosity [117], an assumption that we verified on other Lord fluids [40].
This allows us to use measurements of the carrier viscosity temperature dependence
to correct the suspension data. For Lord MR132LD, the carrier fluid is a Chevron
Phillips Synfluid 4 cSt polyalphaolephin oil [11], and the data sheet tells us that
kinematic viscosity ν = 3.9 mm2/s at 100 ◦C, 16.8 mm2/s at 40 ◦C and 2498 mm2/s
at −40 ◦C. From this, we fit to the ASTM D341 temperature correction standard
for hydrocarbon oils [118], which gives kinematic viscosity, ν, as a function of T:
Z = exp[exp(A−B log T )]
ν = (Z− 0.7)− exp[−0.7847− 3.295(Z− 0.7) + 0.6119(Z− 0.7)2− 0.3193(Z− 0.7)3]
where here, exp and log are in base 10, and A = 8.7452 and B = 3.4684 are fluid
properties fit using the listed data.
97
We estimate the temperature rise in the duct due to viscous heating:
∆T = ∆Tvisc
v2
v2ref
(
LN
Ltot
+
(
1 +
Bi
6
)
LA
Ltot
)
+
∆Tres
vref
.
This is based off the measured off-state temperature rise from entrance to to exit
of ∆Tvisc = 11
◦C, at the maximum velocity, vref . The additional and unmeasured
heating from yield stress forces is estimated through the parabolic model in the
active region, appearing as the 1 + Bi/6 terms. The term on the end comes from
the measured reservoir temperature rise ∆T = ∆Tres/vref , where ∆Tres = 2
◦C. This
equation is split up to estimate temperature at the entrance, start of active region,
and exit. Then, viscosity in the active and passive region is calculated using the
mean temperature of each region. This predicts a max temperature rise of 20 ◦C and
an up to 40% change in viscosity from the entrance to the exit for LA = 25.4 mm at
H = 200 kA/m. We do not include any change in yield stress due to temperature
due to a lack of data.
3.3.1.2 Velocity squared effects
Second, control volume analysis tells us that there’s a missing ρv2gap term.
Goncalves’ device works by pushing a large piston which forces the fluid through
a nozzle into a small slit, and roughly matches the step contraction type device in
Fig. 3.2.
If we assume that there are no viscous losses in the nozzle, and that Ares/Agap 
1, then via Bernoulli: P0 + 0.5ρv
2
res ≈ P0 = P1 + 0.5ρv2gap. If the viscous pressure
98
loss is P1 − P2 = ∆PN , then the measured pressure difference between the reservoir
and exit is P0 − P2 = ∆PN + 0.5ρv2gap. For vgap = 30 m/s, 0.5ρv2gap = 1 MPa, and
should introduce a clearly visible v2 pressure loss in the measurements, but this is
not observed, a rather surprising result. However, we will show that by including our
previously developed temperature correction model, we can obtain realistic values of
K.
3.3.1.3 Geometry
Third, in a personal communication with Fernando Goncalves, he noted that the
slit cross section used in the analysis was 8.889 mm×0.889 mm, not the 1 mm×10 mm
listed. This resolves a discrepancy in the area ratio calculation, as with the listed
properties the area ratio is 811:1, not the 1000:1 listed. For all results, we use
h = 0.889 mm. This makes LA/Dh = 15.7 and 3.925 instead of LA/Dh = 13.97 and
3.4925. While the CFD results that underlie the fit in Eq. 3.8 have a maximum
LA/Dh = 13.97, we still use the activation flow fits, as we are confidant that Graetz
coordinate scaling will allow accurate extrapolation.
As an aside, when Goncalves is determining the fluid properties, off-state
measurements are conducted on both active lengths, and report a 15% difference
in viscosity with a 1.8 ◦C temperature difference. A 15% change is excessive, and
indicates that there may be other uncertainties in the device geometry.
99
0 500 1000 1500
Reynolds number
0 10 20 30 40
Gap velocity, m/s
0
2
4
6
8
10
12
14
∆
P
,
M
P
a
200
100
0
H, kA/m
0 500 1000 1500
Reynolds number
0
10
20
30
40
50
60
A
p
p
ar
en
t 
Y
ie
ld
 S
tr
es
s,
 k
P
a
0 10 20 30 40
Gap velocity, m/s
0
1
2
3
4
∆
P
M
R
,
M
P
a
2.7×104
1.3×104
He
(a) LA = 25.4 mm
0 200 400 600 800 1000 1200 1400
Reynolds number
0 10 20 30 40
Gap velocity, m/s
0
2
4
6
8
10
12
∆
P
,
M
P
a
200
100
150
0
H, kA/m
0 200 400 600 800 1000 1200 1400
Reynolds number
0
20
40
60
80
A
p
p
ar
en
t 
Y
ie
ld
 S
tr
es
s,
 k
P
a
0 10 20 30 40
Gap velocity, m/s
0
0.2
0.4
0.6
0.8
1
1.2
1.4
∆
P
M
R
,
M
P
a
2.5×104
1.9×104
1.1×104
He
(b) LA = 6.35 mm
Figure 3.4: As reported. For LA = 25.4 mm, η∞ = 0.116 Pa s, K = 0.01, and for
LA = 6.35 mm, η∞ = 0.137 Pa s, K = 0.02. Solid lines are activation flow based fits,
dotted lines the parabolic model, dash-dot plug flow. Pink dashed line is Cτy = 0.4.
100
3.3.1.4 Results
In Fig. 3.4, we compare the fully developed Bingham plastic model, the
parabolic model and the activation flow model with no temperature correction to
the experimental pressure loss measurements. We can also identify the three flow
regimes, a quasi-static, low speed and high speed regimes. The quasi-static regime
should have yield forces that are accurately predicted by the plug flow model, and is
estimated to occur when Cτy > 0.4. Fig. 3.4 shows that when Cτy > 0.4 experimental
forces are quite close to plug flow, and very accurately predicted by activation flow,
confirming our basic claims about the applicability of activation flow. However, the
criteria fairs poorly for the long LA/Dh = 15.7 duct, as it is based on a LA/Dh = 5
duct. In the low speed regime, where Cτy < 0.4 and Fy,act > Fy,p, activation flow is
reasonably accurate in the long duct, but underestimates the yield force decrease in
the short duct. High speed is defined as Fy,act < Fy,p, and in this region, activation
flow, while predicting a reduced yield stress, fails to correctly estimate the magnitude
of yield force falloff. Overall, activation flow clearly improves accuracy, but struggles
at high speeds.
We can also assess accuracy by comparing our values to those obtained in
Goncalves. There, yield stress is calculated using the Phillips polynomial, which fits
τy assuming fully developed flow, and η∞ coming from the off-state measurement at
the equivalent speed. Here, we assume that η∞ is a constant, and that there is a
velocity independent K term, and determine these quantities by first fitting to Foff .
We then using the off-state fit values of η∞ and K for a fit of τy on low speed Fon
101
data. The main differences between our approach and Goncalves’ is our inclusion of
a K term and assumption of constant η∞ and τy.
As expected, both methods produce highly similar results. Here, K ≈ 0 as
expected, and η∞ = 0.11 Pa s and 0.14 Pa s, close to the reported η∞ = 0.14 Pa s in
Goncalves. For the two methods, yield stresses at low speeds are similar, where at
H = 200 kA/m, for LA = 25.4 mm, we fit τy = 45 kPa vs. 48 kPa in Goncalves, and
for LA = 6.35 mm we fit 58 kPa vs. 55 kPa in Goncalves. Part of the difference is
caused by the activation flow model assuming constant viscosity (no shear thinning),
which increases apparent yield stress at low speed by 5-10%. However, when compared
to the datasheet value of τy = 42 kPa at H = 200 kA/m, both models predict yield
stresses that are too large. Overall, for the long duct, our simplified model captures
60% of the apparent yield force reduction, but captures only 30% in the short duct.
While this constitutes an improvement, it remains plausible that the yield force
falloff is caused by some other effect.
We attempt to resolve these uncertainties in Fig. 3.5 by applying the tempera-
ture corrected viscosity model, and otherwise using the same procedure. Including
the temperature changes reduces the reference viscosity by about 5%, and enables a
realistic K > 1 factor for both ducts. Temperature induced shear thinning affects
all three models, as the yield stress induced temperature rise causes a reduction in
viscosity, reducing Fy,p and Fy,B slightly due to the lower η∞ in the active region. In
the activation flow model, temperature has a much larger effect, as the reduction
in η∞ raises Re and Bi at a given velocity, increasing yield force falloff. Figure 3.5
shows that including temperature effects makes the activation flow prediction highly
102
(a) LA = 25.4 mm
(b) LA = 6.35 mm
Figure 3.5: Temperature increase modeled by ∆T ∝ v∆P , and the ASTM D341
viscosity-temperature correction for the carrier fluid. For LA = 25.4 mm, η∞ =
0.110 Pa s, K = 1.07, and for LA = 6.35 mm, η∞ = 0.131 Pa s, K = 1.22. Pink
dashed line is Cτy = 0.4.
103
accurate for the long duct, and in the short duct captures 65% of the force falloff
for the short duct. Note that τy is unaffected by the temperature changes, as we fit
τy at low speeds. Our main conclusion is that the inclusion of temperature effects
allows for a realistic K > 1 and improves the the accuracy of force predictions to
the point where activation flow seems to be accurate.
In this experiment, the activation flow model is accurate in quasi-static and low
speed flows, but at high speeds in short ducts, yield force falloff is underestimated.
This prevents us claiming confirmation that yield stress falloff is caused by fluid
dynamic effects, but the activation flow force model does capture the essential trends
of a reduced yield force at high speeds, and allows estimation of when these effects
will occur. In particular, the one-size-fits-all quasi-static criteria, τy/0.5ρv
2
gap > 0.4,
successfully identifies the onset of fluid dynamic effects for both the short and long
duct. However, we noted many uncertainties in the experiment, and while including
a temperature–viscosity correction improves accuracy, the poor performance in the
short duct means questions still remain.
3.3.2 Min Mao’s double ended rod damper
Next we look at the two practical high speed impact absorbers designed by
Mao et al. [14]. These dampers operate at Re > 4000 in the active region, the
highest active region Reynolds numbers that we are aware of. These devices are
of a double piston rod damper design and have minimal spring terms as there is
no accumulator. Both devices have 4 contraction type active regions, two with the
104
listed LA in Table 3.2 and two active regions with 0.5LA. The dampers were tested
with the same custom fluid, with properties listed in Table 3.2, and we assume all
active regions have the same yield stress. These devices also serve as the first test of
our claim that the activation flow analysis can be applied to practical devices with
contraction-type active regions.
The focus of this paper is on the yield force, so we avoid the substantial
challenges of accurate off-state modeling, the focus of Mao’s thesis [13], by curve
fitting the off-state experimental data to a cubic polynomial, and then adding Fy,model
to Foff,fit obtain Fon,model. Similarly, to obtain an experimental Fy for fitting purposes,
we use Fy = Fon−Foff,fit, since on and off state measurements don’t occur at the same
speeds. We note that since predictions of yield force model are set by fluid properties
and the geometry of the gap, yield force predictions are completely unaffected by
the specific details of the off-state model.
The devices here of a typical damper design, where the active region is an
annulus and one of the walls is moving. This poses a challenge as the activation flow
model is based on a parallel plate solution with stationary walls. However, the gap
width to piston diameter ratio is quite low, so we are firmly within the rectangular
limit, as a concentric annulus with b < r < a can be treated as a rectangle for
b/a > 0.7, White [95, p. 120-124], a criteria achieved for all the devices in this paper.
We neglect the moving walls, as it has previously been shown to have a negligible
effect for devices with Ap/Agap > 3 [119]. We also operate in the piston frame, where
the flow rate through gap is Q = Aresvp, so vgap = Ares/Agapvp. This is a slightly
different choice than the typical wall frame Q = Apvp, but we choose piston frame
105
0 1000 2000 3000 4000
Active gap Reynolds number
0 1 2 3 4 5
Peak piston velocity, m/s
0
5
10
15
20
25
30
P
ea
k
 f
o
rc
e,
 k
N
3.25A
2A
1A
0.5A
0A
0 1000 2000 3000 4000
Active gap Reynolds number
0
10
20
30
40
50
60
70
A
p
p
a
re
n
t
y
ie
ld
st
re
ss
,
k
P
a
0 1 2 3 4 5
Peak piston velocity, m/s
0
1
2
3
4
5
6
7
Y
ie
ld
 f
o
rc
e,
 k
N
1.2×105
9.7×104
5.2×104
2.5×104
He
Figure 3.6: Min Mao’s SSMREA1 [14]. Solid lines are activation flow based fits,
dotted lines the parabolic model, dash-dot plug flow. The x symbols correspond to
sinusoidal testing data, circles are drop stand data. In the yield force diagrams, pink
dash dot is τy,app/0.5ρv
2
gap = 0.4, and dashed brown is the onset of turbulence for a
Bingham plastic.
106
0 1000 2000 3000 4000
Active gap Reynolds number
0 1 2 3 4 5
Peak piston velocity, m/s
0
5
10
15
P
ea
k
 f
o
rc
e,
 k
N
4A
2A
0A
Figure 3.7: Mao’s SSMREA2 [14]. Solid lines are activation flow based fits, dotted
lines the parabolic model, dash-dot plug flow. The x symbols correspond to sinusoidal
testing data, circles are drop stand data. In the yield force diagrams, pink dash dot
is τy,app/0.5ρv
2
gap = 0.4, and dashed brown is the onset of turbulence for a Bingham
plastic.
107
as it is easier to model in CFD and should be equally accurate. The choice of the
piston frame flow rate for defining gap velocity also increases flow rate, reducing
some of the error from assuming stationary walls.
For the practical devices, our procedure for experimental validation is simpler
than in Goncalves - we just use the given geometric and fluid properties, which fixes
Re and LA/Dh, and then fit τy(He) to minimize error in Fy. We note that since
∆np ∝ log(∆He), the force curve shape is fairly insensitive to the precise value of
τy. We also include both sinusoidal and drop stand testing data on the same plot, a
rarity, as they are often separated since don’t show the same trends, but include it
here, as it demonstrates the flexibility of our model, which is accurate for both low
and high speed cases.
Figure 3.6 and 3.7 shows that the activation flow model accurately describes
yield force at any speed, while the parabolic and plug flow models capture none of
the observed trends at low and high speeds. The off-state force fit has an RMS force
error of 220 N and 320 N in device 1 and 2 respectively, which serve as a reasonable
estimates of experimental uncertainty. Activation flow yield force has RMS error of
220 and 340 N, indicating our fits are within the experimental uncertainty. The fit
τy are also physically realistic, where at maximum applied current τy = 55 kPa for
SSMREA1 and τy = 62 kPa for SSMREA2, close to the design goal of τy = 60 kPa
for both devices.
For SSMREA1’s low current cases, I = 0.5 A and 1 A, model performance is
worse once Re > 2800 and turbulence starts to be a concern. We also see that the
quasi-static Cτy > 0.4, and high speed criteria are effective, as expected from the
108
excellent performance of the fit. The excellent model performance observed here give
us a high level of confidence that the activation flow force model can be applied to
this and other contraction type devices.
3.3.3 Harinder Singh’s large damper
Singh et al. [109] designs a 12 kN MR damper for impact mitigation. This
damper is single rod design with pressurized diaphragm, but here we’ve made no
attempt to correct for spring terms, as for the data available they are expected to be
relatively small. The device has six active regions, four with the LA in Table 3.2, two
with 0.5LA, and we assume all regions have the same τy. The device is nominally of
the contraction-type design, but the passive regions were later smoothed into a flat
shape so as to reduce turbulent exit losses. This means that our modeled boundary
conditions may not hold up, as at high speeds, the flow exiting one active region may
not return to parabolic shape by the time it reaches the next active region. Similarly,
if the flow goes turbulent, our active region model will break down. However, for
this device, these issues seem to be a concern for only one point.
Offstate forces were fit to a cubic, in order to capture the rising forces near the
exit. The device was tested with Lord MR132DG, with maximum τy ≈ 50 kPa and
we use the manufacturer specified properties in our model, shown in Table 3.2. Due
to drop stand height limitations and load cell ringing, there is limited high-speed
high-force drop stand data, with sinusoidal MTS data used for I = 5.5 A and I = 4 A.
Figure 3.8 compares the three active flow models to all available experimental
109
Figure 3.8: Singh’s large damper [109]. Solid lines are activation flow based fits,
dotted lines the parabolic model, dash-dot plug flow. In the yield force diagram,
pink dash dot is τy,app/0.5ρv
2
gap = 0.4, and dashed brown is the onset of turbulence
for a Bingham plastic.
110
measurements. Again, the activation flow model matches the experimental data
extremely well, accurately capturing the I = 1 A force-velocity data, and performing
well for the low speed I = 2.25 A tests. At the higher currents, where the flow is
quasi-static, the activation flow model captures the yield stress amplification from
the plug. For the test point at vp = 3.5 m/s and I = 2.25 A, model performance is
poor, possibly due to either the turbulence or exit profile concerns outlined above.
The fit yield stress estimates are accurate where at I = 5.5 A, we fit τy = 43 kPa,
in excellent agreement with the FEA estimate of 45 kPa. We also clearly see the
effectiveness of the quasi-static criteria, Cτy > 0.4, with all tests points in that region
conforming to the plug flow model.
3.3.4 Choi and Robinson
Choi et al. [107] build an 8 kN MR oleo-pneumatic damper for helicopter landing
gear. Oleo-pneumatic dampers have gas and fluid mixed in a single chamber, with
device expansion and contraction causing the fluid to flow through the annular active
region. Spring effects are substantial in this design, making analysis challenging.
Specific properties are listed in Table 3.2, and note both the small gap and very
large area ratio.
In Figure 3.9, we compare our model to a series of MTS ramp experiments
where the damper was pressurized at 50 PSI, and we have attempted to remove spring
forces. Yield stress is fit to the high speed Fy measurements using our previous
procedure. The fit at high speed means that high speed yield force predictions
111
Figure 3.9: Choi’s landing gear damper [107]. Solid lines are activation flow based
fits, dotted lines the parabolic model, dash-dot plug flow. In the yield force diagram,
pink dash dot is τy,app/0.5ρv
2
gap = 0.4, and dashed brown is the onset of turbulence
for a Bingham plastic.
112
are accurate, but low speed model performance is as poor as existing models. Fit
yield stress at 4A is 36 kPa, which is below the design value of 45 kPa. Overall, the
activation flow model is an improvement over existing models, capturing the high
speed decrease in yield force. However, at low speeds performance is poor, possibly
due to the high compliance inherent to the oleopneumatic design.
3.3.5 Sanwa Tekki seismic dampers
Surprisingly, the same fluid dynamics seen in the previous high speed impact
devices can also be seen in building scale seismic dampers. Due to the high forces
required for a building scale device, Ares/Agap can be very large, sending vgap into
the high speed region. While a variety of seismic dampers have been built [106], only
the Sanwa Tekki designs enter into the high speed regime [15], [56], [98], as other
groups building seismic MR dampers choose fluids with very high viscosity, so as to
prevent settling [105], [106], [120]. The Sanwa-Tekki group typically use a double
rod piston design with an external bypass for the active region, indicating that the
device should to conform to the activation flow model.
While the authors do not specifically call out a high speed force falloff, we note
in [56], that the 200 kN device they built has has LA/Dh = 2.5 and sees a reduced
yield force when vp = 20 cm/s or Re ≈ 1100, He ≈ 1.2× 105. However, there is no
point validating our model against this data, as the force-velocity plots in [56] are
unclear, and a conference proceeding version [121] has a 50 kN spread in force data.
However, a basic activation flow analysis indicates that there should be a small but
113
noticeable reduction in yield force. It is worth noting that these tests were conducted
with Lord MR132LD in a short gap (LA/Dh < 4), the same scenario our model
struggled with when examining Goncalves duct.
Even more interestingly, in the same paper, the authors also test the same
device with a custom, highly viscous MR fluid, and no disappearing yield force is
observed. This is consistent with our fluid dynamic hypothesis, but contradicts the
trends expected from a chain/rheological response time. Previously we demonstrated
that the particle dynamics response time tr ∝ ηc/µ0M2. So when the fluid is made
more viscous, the rheological response time should increase, which should make the
yield stress falloff more visible, not less. We also note that earlier in [56], they test a
2 kN and 20 kN devices and observe no force falloff. In these devices, Re and He are
such that no falloff is expected, as seen in Fig. 3.1.
Later, when designing a 400 kN device [98], they take a new approach to
predicting device force, and remove upstream orifice losses when the current is on.
This approach reduces the apparent yield force, but it seems unlikely that yield
stress would affect behavior so far upstream. However, this effort appears unneeded,
as no force falloff is seen in their experimental data, consistent with Re = 560 and
He = 2× 105.
Finally, in [15], they publish a close look at a 300 kN seismic damper, a double
rod design with annular contraction style active region with LA/Dh = 6 and ten active
regions, using Lord MR132LD, like Goncalves, with other details in Table 3.1. They
present clear F vs vp results from a sinusoidal excitation test, which we extract for
use with our model. Note that vgap = 27 m/s at vp = 0.4 m/s, which means that this
114
Figure 3.10: Sodeyama’s 300 kN seismic damper [15]. Solid lines are activation flow
based fits, dotted lines the parabolic model, dash-dot plug flow. In the yield force
diagram, pink dash dot is τy,app/0.5ρv
2
gap = 0.4
115
device is firmly in the high speed regime. They also show a rheometer measurement of
η at zero field, which exhibits strong shear thinning, and η∞ = 0.11 Pa s at 1000 s−1.
We compare our activation flow model to their experimental data in Fig. 3.10, where
τy = 43 kPa at 3 A, a reasonable value. While no major force falloff is observed, our
model produces a small rise in yield force followed by a small reduction, a trend
seen in the data. With just four measurements at each input current, the amount
of measurement uncertainty is difficult to estimate. However, the model developed
here clearly captures the observed decrease in yield force at high speeds that the
other models cannot. Overall, this is a successful validation of the activation flow
analysis, as it is clearly more accurate than existing models.
3.4 Conclusions
In this paper, we validated our claim that the activation flow force model will
predict device yield force against all of the known high speed MR fluid devices. We
designed our activation flow model to capture the fluid dynamics seen in the high
speed capillary rheometer of Goncalves, and despite some experimental uncertainties,
showed we could reproduce a significant portion of the measured high speed force
falloff effect seen in his experiment. We then looked at four impact absorbers designed
by our group, and showed outstanding agreement between the activation flow model
and experimental yield force data. Finally, we looked at one of the very large dampers
built by the Sanwa-Tekki corporation and showed that they were also experiencing
high speed force falloff, and that our model could accurately predict the observed
116
yield forces. While we didn’t look at low speed or quasi-static experiments, we
emphasize that our model should be as or more accurate as existing models in those
regimes. From this we conclude that high speed force falloff is a consequence of fluid
dynamics, not a response time.
However, in constructing our model we omitted many details so we could get
a clear understanding. For future work, we would like start adding in the details
we omitted to see how they refine the solution. In particular, experimental data
seems to indicate worse model performance in short active regions, LA/Dh < 4,
possibly due to the assumption of uniform yield stress no longer being valid in a low
aspect-ratio duct. Future work in this area should look at higher fidelity models,
ones that include the effect of the moving wall, non-uniform yield stresses and better
models of the upstream flow.
Validation of these high fidelity models will require high-quality experimental
data, which necessitates an improved version of Goncalves rheometer. In particular,
highly accurate viscosity data is needed, and that requires a device that is temperature
controlled, or at least one that measures wall temperature, so fluid film temperature
can be estimated. The device would also need to correct for entrance effects, ideally
by obtaining pressure loss through pressure taps in the passive region. To verify that
this succeeded, significant effort would need to be spent calibrating the rheometer
against fluids with known properties across the operating temperatures and pressures.
After a similar set of measurements on the carrier fluid, only then could active
field measurements begin with enough accuracy to make definitive claims on the
effectiveness of high fidelity CFD models and secondary concerns like turbulence.
117
In conclusion, we believe that the fluid dynamic model outlined in the previous
two chapters provides an accurate and effective description of the fluid dynamics
in flow mode ER/MR devices. We demonstrated this by accurately modeling yield
force for a comprehensive series of devices. This should enable accurate design of
ER and MR devices in an expanded design space, enabling a next generation of
controllable seismic and shock absorbers.
118
Chapter 4: Conclusions
This dissertation is an investigation into the fundamentals of controllable
yield stress fluids, focusing on the disappearance of yield force at high speeds in
magnetorheological fluids devices. First, we briefly discuss the dynamics of the
particles in MR fluids, and show that current models are limited by their lack of
friction. Then, we showed that high-speed yield force falloff has all the symptoms of
a fluid dynamics problem, turning this dissertation into an exploration of the fluid
dynamics of controllable yield stress fluids.
Chapter 2 rigorously investigates the fluid dynamics of controllable yield stress
fluids entering an active region. The key observation that is at high speeds, the
formation of the plug reduces the momentum of the fluid, causing a pressure rise.
This pressure rise explains much of the observed high speed yield force falloff. To
obtain generalizable results, we created a simplified model of a controllable yield
stress fluid entering an active region, the ‘activation flow’, designed to capture
the fluid dynamics of any ER/MR flow mode device. We then modified and and
validated an OpenFOAM CFD solver, and used it to solve for the full 2-D profile over
a nondimensional space large enough to include nearly every MR ever built. Using
the CFD data, we developed a set of fits for wall shear stress and profile momentum,
119
which allows the prediction of high speed force falloff in arbitrary flow mode ducts.
We also showed that Reynolds and Hedstrom of the active gap are the most
convenient nondimensional numbers for describing the fluid dynamics of the system,
as they act as the dimensionless velocity and yield stress respectively. By reporting
these two quantities and the device geometry, a complete description of the flow
mode fluid dynamics can be obtained. However, the Reynolds and Bingham number
are the most fundamental description of the fluid dynamics, and should be used in
rigorous analyses. Here, Reynolds and Bingham number are then used in a control
volume analysis to estimate the magnitude and onset of high speed effects, resulting
in Equation 2.34 and 2.35. These simple equations allow prediction of the onset and
magnitude of inertial effects in an extremely compact form. Notably, Equation 2.34
depends solely yield stress coefficient, Cτy = τy/0.5ρv
2
0, allowing speed effects to be
predicted with a single nondimensional number. We also argued that changes in
force are best viewed as changes in apparent active length, as it is easy to analyze
and gives geometry independent results that are intuitive at low speeds.
Chapter 3 uses the nondimensional analysis tools in the previous chapter to
identify which experimental devices will exhibit high speed yield force falloff, and then
compares the yield force predictions of Chapter 2 to experimental force measurements
from those devices. After addressing concerns about time-domain fluid dynamics
and turbulence, we demonstrate that activation flow accurately describes high-speed
yield force falloff in devices that operate at high speeds, and is identical to existing
yield force models at low speeds.
These results also expand our understanding of the behavior of magnetorhe-
120
ological fluids outside of the realm of high-speed devices. By showing that fluid
dynamics explains a complex experimental phenomenon, we demonstrate that CFD
and classic control volume analysis should be used to investigate other phenomenon
and other geometries. There are many devices that do not conform to the flow mode
step-contraction geometry we discuss here, and applying the methods used here
should enable both accurate force prediction and even more creative designs.
4.1 Future work
There are many possible directions for future work. One basic step would be
to validate the model against a set of ER devices, as all of the physical models used
here are valid for ER fluids. Another option is to refine the particle model through
an investigation of the tribological interactions of the particles. This will require
extensive characterization of the fluid, additives and particles, and then taking that
data and using it to develop particle force models. With a comprehensive set of
experiments, such a test campaign should enable accurate magnetorheological fluid
design.
Improving the bulk modeling is likely to be the most productive choice. More
efficient numerical methods and schemes are the most important next step to take,
as our approach while accurate, was highly inefficient. The biviscous regularization
model is not ideal, and solver performance can be improved substantially by choosing
a smooth regularization [78], [79], as there is no real need to validate against exact
1-D solutions. Grids could also be much coarser, as the sub 1% error threshold used
121
here was unnecessarily strict. Second, the solver we chose, the SIMPLE method, is
not particularly sophisticated, and performance was poor. Performance could be
increased by resolving the non-Newtonian elements through an explicit inclusion in
the solver equations [122]. When combined with a smooth viscosity model, major
performance improvements should be possible.
With a better numerical method, the next step would be to increase the fidelity
of the physical model. There are lots of basic improvements to make, like making the
duct an annular channel and including the moving damper wall. A more challenging
improvement would be to include a spatially varying magnetic field and thus spatially
varying rheology. To handle the spatially varying rheology, a Mason number master
viscosity curve would provide a smooth and consistent viscosity model, providing
accuracy while maintaining nondimensionality. Adding the varying field would
improve model fidelity as well as removing discontinuities, but specializes results
to the fluid with the given master curve, and most significantly, makes the analysis
much harder. Alternate rheological models may provide additional accuracy, but
should give results substantially similar to those using the Bingham plastic model.
Beyond refinements to activation flow type models, different flow geometries
can be investigated, such as bifold and radial flow designs [101], [123]. Another option
is to investigate how deactivation flow type behavior impacts solutions, so as to
enable the analysis of flow mode devices that do not conform to the step contraction
model. Similarly, actual contraction geometries could be examined, though this risks
needing to model and verify the turbulent behaviors of MR fluid.
The other direction is to build an improved version of Goncalves’ experiment.
122
It should be capable of reaching even larger gap velocities vgap > 50 m/s, but more
importantly, would include more rigorous controls for fluid and thermal properties.
In particular the device should be extensively calibrated using the known properties
of an existing fluid, so as to ensure device analysis accuracy. Similarly, the tested
fluid should be extensively measured on different rheometers, to provide a strong
baseline of comparison. It’s worth noting that that the operating pressure of any
new device would be large, ∆P > 10 MPa, and that while compressibility is not a
concern, oils do change viscosity at large applied pressures. Second, to rigorously
avoid entrance effect terms, pressure taps need to be used to obtain pressure loss.
Third, the device needs to be actively cooled, or at least have entrance, wall and
exit temperature measured to enable accurate correction for thermal effects. Such a
device would provide definitive measurements of high speed force yield force falloff,
and could begin to address active region turbulence effects.
But the best option is to take the experimentally validated force models from
this thesis and use them to build efficient high speed devices. With analytical
expressions for force, the activation flow model is amenable to use in any design
and analysis code, and should dramatically expand the design space of flow mode
magnetorheological devices. This expanded design space will allow a next generation
of impact and seismic energy absorbers capable of saving lives in ever more dangerous
situations.
123
Appendix A: Flow Images
This appendix contains images of normalized velocity and viscosity from Bi = 1
to Bi = 300 from Re = 0.1 to Re = 4000 at LA/Dh = 3.4925. They are presented at
constant Bi, which helps reveal the Cτy related trends in the data. The active length,
LA/Dh = 3.4925 was chosen solely for clarity of presentation, and all of the trends
seen here are independent of LA/Dh.
124
|u|/v0
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(a) Re = 0.1, Cτy = 40, He = 0.2
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(b) Re = 100, Cτy = 0.04, He = 2× 102
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(c) Re = 500, Cτy = 0.008, He = 10
3
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(d) Re = 2000, Cτy = 0.002, He = 4× 103
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(e) Re = 4000, Cτy = 0.001, He = 8× 103
Figure A.1: Normalized velocity at Bi = 1. The active region starts at x = 0.
125
|u|/v0
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(a) Re = 0.1, Cτy = 120, He = 0.6
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(b) Re = 100, Cτy = 0.12, He = 6× 102
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(c) Re = 500, Cτy = 0.024, He = 3× 103
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(d) Re = 2000, Cτy = 0.006, He = 1.2× 104
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(e) Re = 4000, Cτy = 0.003, He = 2.4× 104
Figure A.2: Normalized velocity at Bi = 3. The active region starts at x = 0.
126
|u|/v0
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(a) Re = 0.1, Cτy = 400, He = 2
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(b) Re = 100, Cτy = 0.4, He = 2× 103
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(c) Re = 500, Cτy = 0.08, He = 10
4
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(d) Re = 2000, Cτy = 0.02, He = 4× 104
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(e) Re = 4000, Cτy = 0.01, He = 8× 104
Figure A.3: Normalized velocity at Bi = 10. The active region starts at x = 0.
127
|u|/v0
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(a) Re = 0.1, Cτy = 1200, He = 6
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(b) Re = 100, Cτy = 1.2, He = 6× 103
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(c) Re = 500, Cτy = 0.24, He = 3× 104
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(d) Re = 2000, Cτy = 0.06, He = 1.2× 105
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(e) Re = 4000, Cτy = 0.03, He = 2.4× 105
Figure A.4: Normalized velocity at Bi = 30. The active region starts at x = 0.
128
|u|/v0
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(a) Re = 0.1, Cτy = 4000, He = 20
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(b) Re = 100, Cτy = 4, He = 2× 104
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(c) Re = 500, Cτy = 0.8, He = 10
5
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(d) Re = 2000, Cτy = 0.2, He = 4× 105
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(e) Re = 4000, Cτy = 0.1, He = 8× 105
Figure A.5: Normalized velocity at Bi = 100. The active region starts at x = 0.
129
η/η∞
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/
h
(a) Re = 0.1, Cτy = 40, He = 0.2
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/
h
(b) Re = 100, Cτy = 0.04, He = 2× 102
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(c) Re = 500, Cτy = 0.008, He = 10
3
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(d) Re = 2000, Cτy = 0.002, He = 4× 103
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/
h
(e) Re = 4000, Cτy = 0.001, He = 8× 103
Figure A.6: Normalized viscosity at Bi = 1. The active region starts at x = 0.
130
η/η∞
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/
h
(a) Re = 0.1, Cτy = 120, He = 0.6
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/
h
(b) Re = 100, Cτy = 0.12, He = 6× 102
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(c) Re = 500, Cτy = 0.024, He = 3× 103
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(d) Re = 2000, Cτy = 0.006, He = 1.2× 104
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/
h
(e) Re = 4000, Cτy = 0.003, He = 2.4× 104
Figure A.7: Normalized viscosity at Bi = 3. The active region starts at x = 0.
131
η/η∞
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/
h
(a) Re = 0.1, Cτy = 400, He = 2
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/
h
(b) Re = 100, Cτy = 0.4, He = 2× 103
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(c) Re = 500, Cτy = 0.08, He = 10
4
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(d) Re = 2000, Cτy = 0.02, He = 4× 104
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/
h
(e) Re = 4000, Cτy = 0.01, He = 8× 104
Figure A.8: Normalized viscosity at Bi = 10. The active region starts at x = 0.
132
η/η∞
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/
h
(a) Re = 0.1, Cτy = 1200, He = 6
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/
h
(b) Re = 100, Cτy = 1.2, He = 6× 103
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(c) Re = 500, Cτy = 0.24, He = 3× 104
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(d) Re = 2000, Cτy = 0.06, He = 1.2× 105
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/
h
(e) Re = 4000, Cτy = 0.03, He = 2.4× 105
Figure A.9: Normalized viscosity at Bi = 30. The active region starts at x = 0.
133
η/η∞
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/
h
(a) Re = 0.1, Cτy = 4000, He = 20
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/
h
(b) Re = 100, Cτy = 4, He = 2× 104
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(c) Re = 500, Cτy = 0.8, He = 10
5
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/h
(d) Re = 2000, Cτy = 0.2, He = 4× 105
−0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
x/Dh
y
/
h
(e) Re = 4000, Cτy = 0.1, He = 8× 105
Figure A.10: Normalized viscosity at Bi = 100. The active region starts at x = 0.
134
Appendix B: Correction factors
135
Table B.1: np for LA/Dh = 13.97
Bi
0.1 0.2 0.5 1 2 3 5 7.5 10 15 20 30 50 100 200
Re
0.1 0.20 0.12 0.11 0.11 0.11 0.09 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.08 0.10
1 0.19 0.12 0.11 0.10 0.11 0.09 0.07 0.06 0.06 0.05 0.05 0.06 0.06 0.08 0.10
10 0.16 0.09 0.08 0.07 0.07 0.07 0.06 0.05 0.05 0.05 0.05 0.05 0.06 0.08 0.10
100 −0.19 −0.24 −0.24 −0.22 −0.19 −0.16 −0.13 −0.10 −0.08 −0.05 −0.03 0.00 0.03 0.06 0.09
200 −0.59 −0.61 −0.58 −0.55 −0.47 −0.41 −0.32 −0.26 −0.21 −0.15 −0.11 −0.06 −0.01 0.04 0.08
500 −1.68 −1.65 −1.58 −1.49 −1.31 −1.16 −0.93 −0.74 −0.62 −0.46 −0.36 −0.25 −0.13 −0.03 0.05
750 −2.52 −2.48 −2.38 −2.24 −1.98 −1.76 −1.43 −1.15 −0.96 −0.72 −0.57 −0.40 −0.23 −0.08 0.02
1000 −3.32 −3.26 −3.14 −2.96 −2.63 −2.34 −1.91 −1.55 −1.30 −0.98 −0.78 −0.55 −0.33 −0.13 −0.01
1250 −4.05 −3.98 −3.85 −3.64 −3.25 −2.91 −2.39 −1.95 −1.64 −1.23 −0.99 −0.71 −0.43 −0.19 −0.04
1500 −4.70 −4.63 −4.49 −4.26 −3.83 −3.45 −2.86 −2.34 −1.97 −1.49 −1.20 −0.86 −0.53 −0.24 −0.07
1750 −5.27 −5.20 −5.05 −4.82 −4.37 −3.96 −3.31 −2.72 −2.30 −1.75 −1.41 −1.01 −0.64 −0.30 −0.10
2000 −5.78 −5.70 −5.55 −5.32 −4.86 −4.44 −3.74 −3.10 −2.63 −2.01 −1.62 −1.17 −0.74 −0.35 −0.13
2200 −6.13 −6.06 −5.91 −5.67 −5.21 −4.79 −4.07 −3.39 −2.89 −2.22 −1.79 −1.29 −0.82 −0.40 −0.15
2500 −6.60 −6.52 −6.38 −6.15 −5.69 −5.27 −4.53 −3.81 −3.27 −2.52 −2.04 −1.47 −0.94 −0.46 −0.18
3000 −7.23 −7.15 −7.01 −6.79 −6.37 −5.96 −5.22 −4.47 −3.87 −3.02 −2.46 −1.78 −1.14 −0.58 −0.24
3500 −7.72 −7.64 −7.50 −7.30 −6.91 −6.52 −5.81 −5.06 −4.43 −3.50 −2.87 −2.09 −1.34 −0.69 −0.30
4000 −8.10 −8.02 −7.90 −7.71 −7.34 −6.99 −6.32 −5.57 −4.94 −3.97 −3.27 −2.40 −1.54 −0.79 −0.36
4500 −8.41 −8.34 −8.21 −8.04 −7.70 −7.37 −6.74 −6.03 −5.41 −4.40 −3.66 −2.70 −1.75 −0.90 −0.42
5000 −8.67 −8.59 −8.48 −8.31 −8.00 −7.69 −7.10 −6.43 −5.82 −4.81 −4.04 −3.00 −1.95 −1.01 −0.48
136
Table B.2: nv for LA/Dh = 13.97
Bi
0.1 0.2 0.5 1 2 3 5 7.5 10 15 20 30 50 100 200
Re
0.1 0.20 0.12 0.11 0.11 0.11 0.09 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.08 0.10
1 0.20 0.12 0.11 0.10 0.11 0.09 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.08 0.10
10 0.18 0.11 0.10 0.09 0.09 0.08 0.07 0.06 0.05 0.05 0.05 0.06 0.06 0.08 0.10
100 0.02 −0.03 −0.04 −0.03 −0.02 −0.01 0.00 0.01 0.01 0.03 0.03 0.04 0.06 0.08 0.10
200 −0.16 −0.19 −0.18 −0.17 −0.14 −0.11 −0.07 −0.05 −0.03 −0.01 0.01 0.03 0.05 0.07 0.10
500 −0.62 −0.61 −0.58 −0.55 −0.47 −0.40 −0.30 −0.22 −0.17 −0.11 −0.07 −0.02 0.02 0.06 0.09
750 −0.96 −0.94 −0.90 −0.84 −0.73 −0.63 −0.49 −0.37 −0.28 −0.19 −0.13 −0.07 0.00 0.05 0.09
1000 −1.27 −1.24 −1.19 −1.12 −0.98 −0.86 −0.67 −0.51 −0.40 −0.27 −0.19 −0.11 −0.03 0.04 0.09
1250 −1.55 −1.51 −1.46 −1.38 −1.22 −1.07 −0.84 −0.65 −0.52 −0.35 −0.26 −0.15 −0.05 0.03 0.08
1500 −1.80 −1.76 −1.70 −1.62 −1.44 −1.27 −1.01 −0.79 −0.63 −0.43 −0.32 −0.19 −0.08 0.02 0.08
1750 −2.02 −1.98 −1.92 −1.83 −1.64 −1.46 −1.18 −0.92 −0.74 −0.52 −0.38 −0.24 −0.10 0.01 0.07
2000 −2.21 −2.17 −2.11 −2.02 −1.83 −1.64 −1.33 −1.05 −0.85 −0.60 −0.45 −0.28 −0.13 0.00 0.07
2200 −2.35 −2.31 −2.25 −2.16 −1.96 −1.77 −1.45 −1.15 −0.94 −0.67 −0.50 −0.31 −0.15 −0.01 0.06
2500 −2.53 −2.49 −2.43 −2.34 −2.14 −1.95 −1.62 −1.30 −1.07 −0.76 −0.57 −0.36 −0.18 −0.03 0.06
3000 −2.77 −2.74 −2.68 −2.59 −2.40 −2.21 −1.86 −1.53 −1.27 −0.92 −0.70 −0.45 −0.23 −0.05 0.05
3500 −2.97 −2.93 −2.88 −2.79 −2.61 −2.42 −2.08 −1.73 −1.46 −1.07 −0.82 −0.53 −0.28 −0.07 0.04
4000 −3.12 −3.09 −3.04 −2.96 −2.77 −2.59 −2.26 −1.91 −1.63 −1.22 −0.94 −0.62 −0.33 −0.10 0.03
4500 −3.25 −3.21 −3.16 −3.09 −2.91 −2.74 −2.41 −2.06 −1.78 −1.35 −1.06 −0.70 −0.38 −0.12 0.02
5000 −3.35 −3.32 −3.27 −3.20 −3.03 −2.86 −2.54 −2.20 −1.91 −1.48 −1.17 −0.79 −0.43 −0.14 0.01
137
Table B.3: np for LA/Dh = 7
Bi
0.1 0.2 0.5 1 2 5 7.5 10 15 20 30 50 100 200
Re
1 0.02 0.02 0.04 0.05 0.06 0.04 0.03 0.03 0.03 0.03 0.03 0.03 0.04 0.05
10 −0.01 −0.01 0.01 0.02 0.03 0.02 0.02 0.02 0.02 0.02 0.03 0.03 0.04 0.05
100 −0.34 −0.32 −0.30 −0.27 −0.23 −0.16 −0.13 −0.10 −0.07 −0.05 −0.03 0.00 0.02 0.04
200 −0.70 −0.67 −0.63 −0.58 −0.51 −0.36 −0.29 −0.24 −0.17 −0.13 −0.09 −0.04 0.00 0.03
500 −1.68 −1.64 −1.57 −1.47 −1.30 −0.94 −0.76 −0.64 −0.48 −0.39 −0.27 −0.16 −0.06 0.00
750 −2.37 −2.32 −2.24 −2.12 −1.89 −1.41 −1.15 −0.97 −0.74 −0.60 −0.42 −0.26 −0.12 −0.03
1000 −2.91 −2.86 −2.77 −2.65 −2.41 −1.84 −1.53 −1.30 −1.00 −0.80 −0.58 −0.36 −0.17 −0.06
1250 −3.32 −3.27 −3.18 −3.06 −2.83 −2.23 −1.88 −1.61 −1.25 −1.01 −0.73 −0.46 −0.23 −0.09
1500 −3.63 −3.58 −3.50 −3.39 −3.16 −2.58 −2.20 −1.90 −1.49 −1.22 −0.88 −0.56 −0.28 −0.12
1750 −3.88 −3.83 −3.75 −3.64 −3.44 −2.88 −2.49 −2.18 −1.73 −1.42 −1.04 −0.67 −0.34 −0.15
2000 −4.07 −4.02 −3.95 −3.85 −3.66 −3.13 −2.75 −2.43 −1.95 −1.62 −1.19 −0.77 −0.39 −0.18
2200 −4.20 −4.15 −4.08 −3.99 −3.81 −3.30 −2.94 −2.62 −2.12 −1.77 −1.31 −0.85 −0.44 −0.20
2500 −4.35 −4.31 −4.24 −4.15 −3.99 −3.53 −3.18 −2.87 −2.37 −1.99 −1.49 −0.97 −0.50 −0.23
3000 −4.56 −4.51 −4.45 −4.37 −4.23 −3.82 −3.51 −3.22 −2.73 −2.33 −1.77 −1.17 −0.61 −0.29
3500 −4.71 −4.66 −4.60 −4.53 −4.40 −4.04 −3.76 −3.50 −3.03 −2.64 −2.05 −1.37 −0.72 −0.35
4000 −4.83 −4.78 −4.72 −4.66 −4.55 −4.22 −3.97 −3.73 −3.29 −2.90 −2.30 −1.56 −0.83 −0.41
4500 −4.93 −4.88 −4.82 −4.77 −4.66 −4.36 −4.13 −3.91 −3.50 −3.14 −2.54 −1.76 −0.94 −0.47
5000 −5.01 −4.96 −4.91 −4.85 −4.76 −4.48 −4.27 −4.07 −3.69 −3.34 −2.75 −1.94 −1.05 −0.52
138
Table B.4: nv for LA/Dh = 7
Bi
0.1 0.2 0.5 1 2 5 7.5 10 15 20 30 50 100 200
1 0.02 0.03 0.04 0.06 0.06 0.04 0.03 0.03 0.03 0.03 0.03 0.03 0.04 0.05
10 0.01 0.01 0.03 0.04 0.05 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.04 0.05
100 −0.13 −0.12 −0.10 −0.08 −0.06 −0.03 −0.02 −0.01 0.00 0.01 0.02 0.03 0.04 0.05
200 −0.28 −0.26 −0.23 −0.21 −0.17 −0.11 −0.08 −0.06 −0.03 −0.02 0.00 0.02 0.03 0.05
500 −0.66 −0.63 −0.60 −0.56 −0.48 −0.33 −0.25 −0.20 −0.13 −0.09 −0.05 −0.01 0.02 0.04
750 −0.92 −0.89 −0.85 −0.80 −0.71 −0.50 −0.39 −0.31 −0.21 −0.16 −0.09 −0.03 0.01 0.04
1000 −1.12 −1.09 −1.06 −1.01 −0.91 −0.66 −0.52 −0.42 −0.29 −0.22 −0.13 −0.06 0.00 0.04
1250 −1.28 −1.25 −1.22 −1.17 −1.06 −0.80 −0.64 −0.52 −0.37 −0.28 −0.18 −0.08 −0.01 0.03
1500 −1.40 −1.38 −1.34 −1.29 −1.19 −0.92 −0.75 −0.62 −0.45 −0.34 −0.22 −0.11 −0.02 0.03
Re 1750 −1.50 −1.47 −1.44 −1.40 −1.30 −1.03 −0.85 −0.71 −0.53 −0.40 −0.26 −0.13 −0.03 0.02
2000 −1.58 −1.55 −1.52 −1.48 −1.38 −1.12 −0.94 −0.80 −0.60 −0.46 −0.30 −0.16 −0.04 0.02
2200 −1.63 −1.60 −1.57 −1.53 −1.44 −1.18 −1.00 −0.86 −0.65 −0.51 −0.34 −0.18 −0.05 0.01
2500 −1.69 −1.67 −1.64 −1.60 −1.51 −1.26 −1.09 −0.94 −0.72 −0.57 −0.39 −0.21 −0.07 0.01
3000 −1.77 −1.75 −1.72 −1.68 −1.60 −1.36 −1.20 −1.06 −0.84 −0.67 −0.46 −0.26 −0.09 0.00
3500 −1.83 −1.81 −1.78 −1.75 −1.67 −1.44 −1.28 −1.15 −0.93 −0.76 −0.54 −0.31 −0.11 −0.01
4000 −1.88 −1.85 −1.83 −1.80 −1.72 −1.50 −1.35 −1.22 −1.00 −0.84 −0.61 −0.35 −0.13 −0.02
4500 −1.91 −1.89 −1.87 −1.84 −1.76 −1.55 −1.40 −1.28 −1.07 −0.91 −0.67 −0.40 −0.16 −0.03
5000 −1.95 −1.92 −1.90 −1.87 −1.80 −1.59 −1.45 −1.32 −1.12 −0.96 −0.73 −0.45 −0.18 −0.04
139
Table B.5: np for LA/Dh = 3.5
Bi
0.1 0.2 0.5 1 1.5 2 3 5 7.5 10 15 20 30 50 100 200
Re
0.1 −0.02 −0.00 0.02 0.03 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.03
1 −0.03 −0.01 0.01 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.03
10 −0.06 −0.04 −0.02 −0.00 −0.00 −0.00 0.00 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.02 0.02
20 −0.09 −0.07 −0.05 −0.03 −0.03 −0.03 −0.02 −0.02 −0.01 −0.01 −0.00 0.00 0.01 0.01 0.02 0.02
50 −0.20 −0.17 −0.15 −0.13 −0.12 −0.11 −0.10 −0.08 −0.06 −0.05 −0.03 −0.02 −0.01 −0.00 0.01 0.02
100 −0.37 −0.34 −0.31 −0.29 −0.27 −0.25 −0.22 −0.17 −0.14 −0.11 −0.08 −0.06 −0.04 −0.02 0.00 0.01
200 −0.70 −0.67 −0.63 −0.59 −0.55 −0.51 −0.45 −0.37 −0.29 −0.25 −0.18 −0.15 −0.10 −0.06 −0.02 0.00
500 −1.47 −1.43 −1.38 −1.31 −1.24 −1.18 −1.07 −0.90 −0.74 −0.63 −0.49 −0.39 −0.28 −0.18 −0.08 −0.03
750 −1.83 −1.79 −1.74 −1.68 −1.62 −1.56 −1.45 −1.26 −1.07 −0.93 −0.73 −0.59 −0.43 −0.28 −0.14 −0.06
1000 −2.05 −2.01 −1.97 −1.91 −1.86 −1.81 −1.71 −1.53 −1.34 −1.19 −0.95 −0.79 −0.58 −0.38 −0.19 −0.09
1250 −2.19 −2.15 −2.11 −2.07 −2.02 −1.98 −1.89 −1.74 −1.56 −1.40 −1.15 −0.97 −0.73 −0.48 −0.25 −0.11
1500 −2.29 −2.25 −2.22 −2.17 −2.13 −2.10 −2.02 −1.89 −1.73 −1.58 −1.33 −1.13 −0.87 −0.58 −0.30 −0.14
1750 −2.36 −2.33 −2.29 −2.26 −2.22 −2.19 −2.12 −2.00 −1.85 −1.72 −1.48 −1.28 −1.00 −0.67 −0.36 −0.17
2000 −2.42 −2.39 −2.36 −2.32 −2.29 −2.26 −2.20 −2.09 −1.96 −1.84 −1.61 −1.42 −1.12 −0.77 −0.41 −0.20
2200 −2.46 −2.43 −2.40 −2.36 −2.33 −2.31 −2.25 −2.15 −2.03 −1.91 −1.70 −1.51 −1.21 −0.84 −0.46 −0.22
2500 −2.51 −2.48 −2.45 −2.42 −2.39 −2.37 −2.32 −2.22 −2.11 −2.01 −1.81 −1.64 −1.34 −0.95 −0.52 −0.26
3000 −2.58 −2.55 −2.52 −2.49 −2.47 −2.44 −2.40 −2.32 −2.22 −2.13 −1.96 −1.80 −1.52 −1.12 −0.63 −0.32
3500 −2.64 −2.60 −2.57 −2.54 −2.52 −2.50 −2.46 −2.39 −2.31 −2.23 −2.08 −1.93 −1.67 −1.27 −0.74 −0.37
4000 −2.68 −2.65 −2.61 −2.59 −2.57 −2.55 −2.52 −2.45 −2.38 −2.31 −2.17 −2.04 −1.80 −1.41 −0.84 −0.43
4500 −2.72 −2.68 −2.65 −2.63 −2.61 −2.59 −2.56 −2.50 −2.43 −2.37 −2.24 −2.13 −1.90 −1.53 −0.94 −0.49
5000 −2.75 −2.71 −2.68 −2.66 −2.64 −2.63 −2.60 −2.54 −2.48 −2.42 −2.31 −2.20 −1.99 −1.63 −1.04 −0.55
140
Table B.6: nv for LA/Dh = 3.5
Bi
0.1 0.2 0.5 1 1.5 2 3 5 7.5 10 15 20 30 50 100 200
Re
0.1 −0.02 −0.00 0.02 0.03 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.03
1 −0.02 −0.00 0.02 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.03
10 −0.04 −0.02 0.00 0.02 0.02 0.02 0.02 0.02 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.03
20 −0.05 −0.03 −0.01 0.00 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.02
50 −0.09 −0.07 −0.05 −0.04 −0.03 −0.03 −0.02 −0.01 −0.01 −0.00 0.00 0.01 0.01 0.01 0.02 0.02
100 −0.16 −0.14 −0.12 −0.10 −0.09 −0.08 −0.07 −0.05 −0.04 −0.02 −0.01 −0.00 0.00 0.01 0.02 0.02
200 −0.29 −0.26 −0.24 −0.22 −0.20 −0.19 −0.16 −0.12 −0.09 −0.07 −0.04 −0.03 −0.01 0.00 0.01 0.02
500 −0.58 −0.55 −0.53 −0.50 −0.47 −0.44 −0.40 −0.32 −0.25 −0.20 −0.14 −0.10 −0.06 −0.02 0.00 0.02
750 −0.72 −0.69 −0.67 −0.64 −0.62 −0.59 −0.54 −0.45 −0.36 −0.30 −0.22 −0.16 −0.10 −0.05 −0.01 0.02
1000 −0.80 −0.78 −0.76 −0.73 −0.71 −0.68 −0.64 −0.55 −0.46 −0.39 −0.29 −0.22 −0.14 −0.07 −0.02 0.01
1250 −0.86 −0.84 −0.82 −0.79 −0.77 −0.75 −0.70 −0.62 −0.53 −0.46 −0.35 −0.28 −0.18 −0.10 −0.03 0.01
1500 −0.90 −0.88 −0.86 −0.84 −0.82 −0.79 −0.75 −0.67 −0.59 −0.52 −0.40 −0.32 −0.22 −0.12 −0.04 0.00
1750 −0.93 −0.91 −0.89 −0.87 −0.85 −0.83 −0.79 −0.71 −0.63 −0.56 −0.45 −0.37 −0.26 −0.15 −0.05 −0.00
2000 −0.95 −0.93 −0.91 −0.89 −0.88 −0.86 −0.82 −0.74 −0.67 −0.60 −0.49 −0.41 −0.29 −0.17 −0.06 −0.01
2200 −0.96 −0.95 −0.93 −0.91 −0.89 −0.87 −0.84 −0.76 −0.69 −0.62 −0.52 −0.43 −0.32 −0.19 −0.07 −0.01
2500 −0.98 −0.97 −0.95 −0.93 −0.91 −0.89 −0.86 −0.79 −0.72 −0.65 −0.55 −0.47 −0.35 −0.21 −0.09 −0.02
3000 −1.01 −0.99 −0.97 −0.96 −0.94 −0.92 −0.89 −0.82 −0.75 −0.69 −0.59 −0.51 −0.40 −0.25 −0.11 −0.03
3500 −1.02 −1.01 −0.99 −0.98 −0.96 −0.94 −0.91 −0.84 −0.78 −0.72 −0.62 −0.55 −0.43 −0.29 −0.13 −0.04
4000 −1.04 −1.02 −1.01 −0.99 −0.97 −0.96 −0.92 −0.86 −0.80 −0.74 −0.65 −0.58 −0.46 −0.32 −0.15 −0.05
4500 −1.05 −1.03 −1.02 −1.00 −0.99 −0.97 −0.94 −0.88 −0.81 −0.76 −0.67 −0.60 −0.49 −0.35 −0.17 −0.06
5000 −1.06 −1.04 −1.03 −1.01 −1.00 −0.98 −0.95 −0.89 −0.82 −0.77 −0.68 −0.61 −0.51 −0.37 −0.19 −0.07
141
Table B.7: np for LA/Dh = 2
Bi
0.1 0.2 0.5 1 2 5 7.5 10 15 20 50 100 200
Re
1 −0.05 −0.02 −0.01 −0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
10 −0.08 −0.05 −0.04 −0.03 −0.02 −0.01 −0.01 −0.00 0.00 0.00 0.01 0.01 0.01
100 −0.37 −0.35 −0.32 −0.29 −0.25 −0.18 −0.14 −0.12 −0.09 −0.07 −0.03 −0.01 0.00
200 −0.67 −0.63 −0.60 −0.55 −0.49 −0.36 −0.29 −0.25 −0.19 −0.15 −0.06 −0.03 −0.01
500 −1.13 −1.10 −1.06 −1.02 −0.96 −0.78 −0.67 −0.59 −0.46 −0.38 −0.18 −0.09 −0.04
750 −1.28 −1.25 −1.22 −1.19 −1.14 −1.00 −0.90 −0.81 −0.66 −0.56 −0.28 −0.14 −0.07
1000 −1.36 −1.34 −1.31 −1.29 −1.24 −1.13 −1.04 −0.96 −0.82 −0.71 −0.37 −0.20 −0.10
1250 −1.42 −1.39 −1.37 −1.35 −1.31 −1.21 −1.14 −1.07 −0.95 −0.84 −0.47 −0.25 −0.12
1500 −1.46 −1.43 −1.41 −1.39 −1.36 −1.28 −1.21 −1.15 −1.04 −0.94 −0.55 −0.31 −0.15
1750 −1.49 −1.47 −1.44 −1.43 −1.40 −1.32 −1.27 −1.21 −1.11 −1.02 −0.63 −0.36 −0.18
2000 −1.52 −1.49 −1.47 −1.45 −1.43 −1.36 −1.31 −1.26 −1.17 −1.09 −0.71 −0.41 −0.21
2200 −1.54 −1.51 −1.49 −1.47 −1.45 −1.39 −1.34 −1.29 −1.21 −1.13 −0.76 −0.45 −0.23
2500 −1.56 −1.53 −1.51 −1.50 −1.47 −1.42 −1.37 −1.33 −1.26 −1.19 −0.84 −0.51 −0.27
3000 −1.60 −1.57 −1.54 −1.53 −1.51 −1.46 −1.42 −1.39 −1.32 −1.26 −0.94 −0.61 −0.33
3500 −1.62 −1.59 −1.57 −1.55 −1.54 −1.49 −1.46 −1.43 −1.37 −1.32 −1.03 −0.70 −0.38
4000 −1.64 −1.61 −1.59 −1.57 −1.56 −1.52 −1.49 −1.46 −1.41 −1.36 −1.10 −0.77 −0.44
4500 −1.66 −1.63 −1.61 −1.59 −1.58 −1.54 −1.51 −1.49 −1.44 −1.40 −1.16 −0.84 −0.50
5000 −1.68 −1.65 −1.62 −1.61 −1.59 −1.56 −1.53 −1.51 −1.47 −1.43 −1.21 −0.91 −0.55
142
Table B.8: nv for LA/Dh = 2
Bi
0.1 0.2 0.5 1 2 5 7.5 10 15 20 50 100 200
Re
1 −0.05 −0.02 −0.01 −0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
10 −0.06 −0.03 −0.02 −0.01 −0.01 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.01
100 −0.17 −0.14 −0.13 −0.11 −0.10 −0.06 −0.04 −0.03 −0.02 −0.01 0.00 0.01 0.01
200 −0.28 −0.25 −0.23 −0.21 −0.18 −0.13 −0.10 −0.08 −0.05 −0.03 −0.00 0.01 0.01
500 −0.45 −0.43 −0.41 −0.39 −0.36 −0.28 −0.23 −0.19 −0.14 −0.10 −0.03 −0.00 0.01
750 −0.51 −0.49 −0.48 −0.46 −0.43 −0.36 −0.30 −0.26 −0.20 −0.16 −0.05 −0.01 0.00
1000 −0.54 −0.53 −0.51 −0.50 −0.47 −0.40 −0.35 −0.31 −0.25 −0.20 −0.08 −0.03 0.00
1250 −0.56 −0.55 −0.53 −0.52 −0.50 −0.43 −0.39 −0.35 −0.29 −0.24 −0.10 −0.04 −0.00
1500 −0.58 −0.56 −0.55 −0.54 −0.51 −0.45 −0.41 −0.37 −0.31 −0.27 −0.12 −0.05 −0.01
1750 −0.59 −0.57 −0.56 −0.55 −0.53 −0.47 −0.43 −0.39 −0.33 −0.29 −0.14 −0.06 −0.01
2000 −0.60 −0.58 −0.57 −0.56 −0.54 −0.48 −0.44 −0.41 −0.35 −0.31 −0.16 −0.07 −0.02
2200 −0.60 −0.59 −0.58 −0.56 −0.54 −0.49 −0.45 −0.42 −0.36 −0.32 −0.17 −0.08 −0.02
2500 −0.61 −0.59 −0.58 −0.57 −0.55 −0.50 −0.46 −0.43 −0.37 −0.33 −0.19 −0.09 −0.03
3000 −0.62 −0.60 −0.59 −0.58 −0.56 −0.51 −0.47 −0.44 −0.39 −0.35 −0.21 −0.11 −0.04
3500 −0.63 −0.61 −0.60 −0.59 −0.57 −0.52 −0.48 −0.45 −0.40 −0.36 −0.23 −0.12 −0.05
4000 −0.63 −0.62 −0.61 −0.60 −0.58 −0.53 −0.49 −0.46 −0.41 −0.37 −0.24 −0.14 −0.06
4500 −0.64 −0.62 −0.61 −0.60 −0.58 −0.53 −0.50 −0.47 −0.42 −0.38 −0.25 −0.15 −0.07
5000 −0.64 −0.62 −0.61 −0.60 −0.59 −0.54 −0.50 −0.47 −0.42 −0.39 −0.26 −0.16 −0.07
143
Appendix C: Temperature dependence of MR fluids
This is a reproduction of ”Scaling temperature dependent rheology of magne-
torheological fluids” by Stephen G Sherman, Louise A Powell, Andrew C Becnel and
Norman M Wereley, published in Journal of Applied Physics [40].
C.1 Introduction
Magnetorheological (MR) fluids are suspensions of magnetic particles in oil,
that upon the application of magnetic field, develop a field controllable yield stress,
giving a suspension with a controllable apparent viscosity. This controllable viscosity
enables a damper or other energy absorbing device to produce a rapidly controllable
force output. These devices, due to their energy absorbing nature, operate over
a wide range of temperatures (e.g. -40 to 150 ◦C), ideally with minimal change in
performance with respect to temperature. However, in practice, MR devices see a
significant reduction in force output with increase in temperature. Wilson et al. [124]
characterized the temperature dependent performance of a flow mode damper using
Lord MRF132DG fluid and finds substantial variation in device performance in the
temperature range of 0 - 100 ◦C, with yield stress varying by up to 30% and post yield
damping varying by up to 80%. Gordaninejad and Breese [125] tested a series of MR
144
dampers, and observed a reduction in peak force of up to 50% over a temperature
range of 20 - 70 ◦C. This dramatic drop-off in force is typically attributed to a
reduction in yield stress and viscosity due to an increase in temperature.
Rheometer experiments by Weiss and Duclos [126] reported a 10% reduction
in yield stress and 95% reduction in viscosity over the range of -40 to 150 ◦C, which
in damper tests corresponded to a predicted 16% force loss vs. 18% measured.
Later, Weiss et al. [127] patented a series of MR fluids with relatively independent
temperature properties through the identification of carrier fluids that exhibit minimal
temperature changes, an approach which we validate. Sahin et al. [128] characterized
the yield stress of a custom grease-based MR fluid and shows that the temperature
dependence can be expressed through an exponential relationship, and that the
temperature change in viscosity is independent of shear rate. Finally, Lemaire et al.
[129] examined the effects of Brownian motion by varying particle size and found
that even when Brownian forces are relatively small, such forces can still affect the
yield stress.
In this study, we investigate the effects of temperature on MR fluid off-state
viscosity, as viscosity and its device-scale cousin, post yield damping, undergo the
largest relative temperature changes. First, we analyze the field free state using the
three fundamental forces acting on the particles (Brownian, shear, and magnetic) ,
through the relative role that they play in affecting fluid properties, using three non-
dimensional numbers: the Mason number, Mn, the Peclet number, Pe and the ratio
of magnetic to thermal energy, λ. Then, we validate our analysis through rheometer
measurements on a simple fluid designed to conform to the assumptions present
145
in the analysis, followed by measuring the temperature effects on a commercially
available fluid. We also investigate the effects of remanent magnetization on fluid
performance, whereas typical off-state experiments use previously unmagnetized
fluid.
C.2 Background
MR fluids are typically modeled as Bingham plastics,
τ = τy + ηpγ˙, (C.1)
where under shear they resist movement up until a field dependent yield stress τy,
after which they flow with a field independent viscosity ηp. When no field is applied,
τy is small and apparent viscosity η = τ/γ˙ = ηp for large shear rates. Apparent
viscosity of the MR fluid suspension can be broken down into three elements,
η = ηM + ηB + ηH (C.2)
ηM is the component of viscosity from interparticle (magnetic) forces, ηB is the
component of viscosity caused by Brownian motion, and ηH is the component of
viscosity from the hydrodynamic drag of the particles.
The first factor affecting viscosity is the magnetic interactions of the particles.
Typically, with no applied magnetic field, ηM is assumed to be negligible. However,
even at zero field, remanent magnetization in the particles will cause the particles to
146
cluster, potentially inducing a residual yield stress or increase in viscosity, giving
some nonzero value to ηM . Since the particles for MR fluids are chosen to have
minimal hysteretic effects, the magnetic interactions in the particles are extremely
weak, so particle cluster formation will be opposed by Brownian motion. This can
be characterized through the ratio of magnetic energy to thermal energy [39], known
as the λ parameter, and is defined as
λ =
pia3µ0M
2
18kBT
. (C.3)
Typically λ is defined using H instead of M , but in this case using M is sensible as
H = 0. This magnetic-thermal ratio characterizes the ability of the particle magnetic
forces to overcome diffusive Brownian motion, and can be used to characterize
temperature effects on particle structure. Typically, Brownian effects on the magnetic
interactions are assumed to be negligible for λ 0, however, from experiments on
adjacent particle chains, values of λ > 100 can increase particle aggregation [130]
through long-ranged fluctuations. From experiments on MR fluid particle size, effects
on field-on stress attributed to Brownian motion have been reported up to λ = 2000
[129]. For typical field-on cases, with low representative values of M = 100 kA/m,
T=400 K, and a = 1 µm, λ > 106, indicating that temperature, through Brownian
motion, plays no role in the behavior of the particles. However, in the field-off case,
where the Mrem ≈ 5 kA/m, λ ≈ 103, thermal effects are strong enough to disrupt
particle structure. It is then possible that temperature could affect ηM , and thus
overall suspension viscosity.
147
The magnitude of the contribution of ηm makes to the overall viscosity can be
expressed by the Mason number,
Mn = 144
ηcγ˙
µ0M2
, (C.4)
the ratio of hydrodynamic forces to magnetic forces. Using ηc = 0.01 Pa s and
γ˙ = 100 s−1, Mn > 1, indicatesthat hydrodynamic forces will be larger than the
magnetic forces. Thus, it is reasonable to expect that ηM will be relatively small, so
the expected temperature effects from the λ analysis should have minimal impact on
the overall viscosity.
The relative contribution to fluid stress by Brownian motion, ηB, is characterized
by the Peclet number [131],
Pe =
6piηcγ˙a
3
kBT
, (C.5)
the ratio of advection by the flow to the rate of diffusion by Brownian motion. In this
definition, Pe = 3/4Mnλ. From the theory of hard sphere suspensions, Brownian
effects on viscosity are known to be negligible for Pe > 102. Using the previous values
for typical fluids, Pe > 103 indicates that most MR fluids will see no contribution to
viscosity from Brownian motion.
Finally, there is the hydrodynamic contribution to viscosity from the addition of
the particles. In non-Brownian hard sphere theory, for a random suspension of spheres,
the only parameter affecting viscosity is volume fraction, yielding predictions of the
form η = ηcf(φ), typically given in the form of relative viscosity ηr = η/ηc = f(φ).
148
Relative viscosity can be described for these suspensions through the semi-empirical
Quemada equation [131],
ηr =
(
1− φ
φmax
)−2
. (C.6)
Since temperature will have a negligible effect on volume fraction, relative viscosity
will remain unchanged with temperature, and since the magnetic and Brownian
components are also so small, relative viscosity of an MR fluid will remain fixed.
Thus, we hypothesize that viscosity-temperature effects in an MR fluid are caused
only by the changes in the viscosity of the carrier fluid. However, hard sphere theories
operate under the assumption of a Newtonian carrier fluid, whereas practical MR
fluids introduce additives or use carriers with non-Newtonian behavior to reduce
particle settling[128], so extending these theories to practical fluids must be done
with care.
In order to model temperature effects of the fluid, we choose the simple, well
known Reynolds equation [132],
η(T ) = ηref exp [−k (T − Tref)] . (C.7)
Then if we characterize the viscosity of multiple MR fluids and their carrier fluid,
across a range of temperatures, and they all share a common value of k, then our
hypothesis that carrier fluid effects are the only source of post yield viscosity changes
will be confirmed.
149
C.3 Experiments
In order to characterize the temperature effects of viscosity in MR fluids and
to demonstrate common scaling, we measure the off state viscosity of two MR fluids
and their carrier fluid to show that all the fluids have the same scaling coefficient.
Off-state viscosity was measured on an Anton Paar MCR300 rheometer, with an
MRD180 measuring cell using parallel plate geometry with a 0.5mm gap and 150 µL
samples. Temperature was controlled using a Julabo F25 refrigerating/heating
circulator, with the sample left to rest for 3 minutes to achieve thermal equilibrium.
To ensure uniformity, samples were mixed using a high shear mixer for 30 minutes
before each test. Each test measured shear stress at shear rates from 10-1000 s−1 in
an ascending log profile.
C.3.1 Simple MR Fluids
Two samples of simple MR fluid were prepared such that they closely matched
non-Brownian particle theory (containing no stabilizers or additives) in order to
avoid non-Newtonian behavior of the carrier fluid and to prevent any stabilizers
from forming a coating of unknown thickness. To be representative of typical
commercial fluids, we chose BASF CM grade spherical carbonyl iron powder (> 99.5%
Fe, a = 3.5 − 4.75 µm, ρ = 7.78 g/cm3), having the standard large spherical
particles and high saturation magnetism. The carrier fluid was Exxon Mobil SHF-41
polyalphaolephin hydrocarbon oil (ρ = 0.82 g/cm3). Two 15ml samples were mixed by
weight, at 20 and 30 vol% (70.3 and 80.3 wt%), known as PlainFE20 and PlainFE30
150
−1500 −1000 −500 0 500 1000 1500−1500
−1000
−500
0
500
1000
1500
H (kA/m)
M
 (k
A/
m)
−4 −2 0 2 4
−20
−10
0
10
20
H (kA/m)
M
 (k
A/
m)
Figure C.1: Particle magnetization in the PlainFE30 fluid as a function of applied
field.
respectively. Remanent particle magnetization was obtained through measurements
of PlainFE30 fluid on a LakeShore 7400 series VSM. The results were scaled to
individual particle magnetization, shown in Fig. C.1, yielding a measured remanence
of M = 8.4 kA/m. Thus, at 300K, the measured value of ηc and γ˙ = 100 s
−1,
Pe = 4.1× 105, λ = 2.4× 105 and Mn = 2.2. Therefore, the particles are expected
to have minimal thermal effects on viscosity of the fluid for these model fluids.
Then, unmagnetized samples of the simple MR fluids and their carrier fluid
were measured on the rheometer at temperatures from 5 to 55 ◦C, corresponding to
the minimum chiller temperature and maximum rheometer temperature, in steps
of 10 ◦C, with each fluid measured at least three times at each temperature. For
each measurement, the flow curve was fit to (F.5) using the high shear rate values,
γ˙ > 150 s−1, with viscosities plotted against temperature in Fig. C.2. Then to
confirm our hypothesis of common temperature scaling, we fit (C.7) to viscosity
vs. temperature for PlainFE30, PlainFE20, and the carrier fluid SHF41, with the fit
151
0 10 20 30 40 50 60
10−3
10−2
10−1
100
Temperature (C)
V
isc
os
ity
 (P
a s
)
 
 
SHF41
PlainFE20
PlainFE30
PlainFE30−Magnetized
Figure C.2: Viscosity vs. temperature for the simple fluids, including the magnetized
sample.
Table C.1: Custom Fluid Properties
Fluid Name ηT=30C (mPa s) k ηr
SHF41 14.3 0.0361 1
PlainFE20 36.6 0.0368 2.57
PlainFE30 94.1 0.0392 6.60
Magnetized FE30 100.5 0.0418 7.03
shown in Fig. C.2. A table of coefficients listed in Table C.1 shows that all three
fluids have a value of k within 5% of the mean, which confirms our hypothesis that
all the fluids scale similarly. We can also validate our results through the Quemada
equation, which for a plausible value φmax = 0.50, gives results close to the values in
Table C.1.
To examine the effects of magnetic hysteresis, two sets of measurements were
conducted at T=25 ◦C. In the first, samples were magnetized by a 1 T neodymium
152
0 10 20 30 40 50 60
0
5
10
15
20
Temperature (C) 
O
ff−
sta
te
 Y
ie
ld
 S
tre
ss
 (P
a)
 
 
SHF41
PlainFE20
PlainFE30
FE30−MCR
FE30−Nd
Figure C.3: Offstate yield stress for the simple fluids. FE30-MCR corresponds to
samples magnetized by the rheometer and FE30-Nd for samples magnetized by a
neodymium magnet.
magnet and then measured on the rheometer as before. In the second, the samples
were placed in the rheometer with field applied at a nominal B = 0.35 T. Then
the flux return of the rheometer was demagnetized with alternating fields with a
period of two seconds, with the intention that the particles should reorient before
they demagnetize significantly. When tested at room temperature, both methods
produced viscosities indistinguishable from the unmagnetized samples, as shown
in Fig. C.2. However, when looking at the off-state ‘yield stress’ from the high
shear rate fit, shown in Fig. C.3, the magnetized samples clearly have a higher yield
stress, with the more strongly magnetized sample having a higher yield stress, as
expected. We also note that uniquely, the PlainFE20 sample shows a decrease in τy
with temperature. However, the measured yield stresses are extremely small and can
be effectively ignored for γ˙ > 10 s−1.
153
C.3.2 Commercial MR Fluids
The previous fluids were designed to work with non-Brownian suspension
theory, but practical MR fluids are synthesized using carrier fluids with multiple
fluid additives, surfactants, and stabilizers, which can make the carrier fluid non-
Newtonian, violating the assumptions of theory. To investigate temperature scaling in
practical fluids, we tested commercially available MR fluids sold by Lord Corporation,
which use a proprietary hydrocarbon oil and additive mixture, and large particles
a = 3− 5 µm. To measure the rheology of the carrier fluid, the particles in a sample
of fluid were allowed to settle, and the clear supernatant layer was decanted, allowing
measurement of its viscosity in a rheometer. However, this procedure does not
guarantee that the extracted fluid will be precisely representative of the carrier
fluid, as the additives may not uniformly separate, introducing a possible source of
error. Using the same experimental procedure as before, three commercially available
fluids, Lord MRF132DG, LordMRF126CD and LordMRF122EG, and the carrier
fluid from MRF132 and MRF126 are characterized. Particle remanent magnetization
for MRF132 was found to be 5 kA/m. Assuming common composition for these
fluids, Pe = 2.0× 105, Mn = 3.2, and λ = 8.5× 104, indicates that there should be
no particle contribution to viscosity. Fig. C.4, and Table C.2 show that all three MR
fluids scale nearlys identically, having values of k within 4% of the mean. However,
the carrier fluids do not scale with a similar exponent, which we attribute to fluid
additives that did not separate. This argument is bolstered by the fact that the
relative viscosities in Table C.2 are much larger than those predicted by Eq. F.19,
154
0 10 20 30 40 50 60
10−3
10−2
10−1
100
Temperature (C)
V
isc
os
ity
 (P
a s
)
 
 
MRF132DG
MRF126CD
MRF122EG
Carrier 132
Carrier 126
MR132−MCR
Figure C.4: Viscosity for temperature for a series of commercially available fluids
and their extracted carrier fluid.
Table C.2: Lord Fluid Properties
Fluid Name ηT=30C (mPa s) k ηr
MRF132DG 135.0 0.0396 16.5
MRF126CD 72.2 0.0398 10.9
MRF122EG 43.1 0.0416 -
132C 8.2 0.0313 1
126C 6.6 0.0355 1
indicating the particle hydrodynamics are not the ones expected by hard sphere
theory. Similarly, measurements of magnetized Lord fluids found, as previously
mentioned, a minimal change of viscosity due to remanent magnetization.
C.4 Conclusion
Magnetorheological (MR) fluid viscosity is known to vary by over an order of
magnitude throughout temperature operating range of MR fluids. In this paper,
155
we perform a thorough analysis of the effect, and attribute it solely to changes
in the viscosity of the carrier fluid. This means that if the carrier fluid viscosity-
temperature relation is known, then the same relative change in viscosity will be
present in an MR fluid made with that carrier liquid, an approach that has been
used in patented temperature-stable MR fluids [127]. Theoretically, this claim is
validated through a nondimensional analysis, finding that even in in the offstate,
for typical MR fluids, Pe  1, λ  1 and Mn > 1, indicating that the magnetic
and Brownian contributions to fluid viscosity will be zero. In order to obtain these
numbers, the magnetic remanence of the particles in the fluid was measured, and
this result was confirmed by performing measurements of viscosity before and after
cycling the field, with no resulting change in viscosity. The effects of magnetic
hysteresis on fluid properties were investigated, and we found a negligible increase in
yield stress and no change in viscosity. Finally, we experimentally validated this claim
by performing measurements of viscosity across temperature for a pair of custom
MR fluids and corresponding carrier liquid, and we found that all of these fluids had
an indistinguishable change in viscosity, a result we also found in measurements of
related commercially available MR fluids. This development of a common scaling
law allows for the temperature dependence of MR fluids to be characterized in fewer
experiments, or possibly using only known carrier fluid properties.
156
Appendix D: Sheet formation at large scales
This work [19] originally appeared as “Parallel Simulation of Transient Magne-
torheological Direct Shear Flows Using Millions of Particles” by Stephen G. Sherman,
Derek A. Paley and Norman M. Wereley, published in IEEE Transactions on Mag-
netics .
D.1 Introduction
Magnetorheological (MR) fluids and electrorheological fluids are non-Newtonian
fluids consisting of polarized particles suspended in a carrier fluid. Upon the ap-
plication of magnetic/electric field, the particles form chains, causing the fluid to
develop a field-dependent stress. The adjustable nature of these fluids makes them
useful in a variety of applications, such as dampers and clutches, so developing an
understanding of the particle structure dynamics is essential to understanding the
behavior of these fluids.
The static equilibrium structure for ER/MR fluids is a thick column, but
under shear these fluids develop into a lamellar sheet structure. These sheets have
significant transient effects on the fluid [133], and their effects are not always agreed
upon [134]. Similarly, the formation of these sheets has shown to be a complex
157
Figure D.1: Structural pattern of ER fluid under 3 conditions: (a) no electric field or
shear, showing a random distribution of particles, (b) electric field is applied, causing
chain formation and (c) with both electric field and shear flow, showing lamellar
sheet formation.
function of time, shear rate, and field strength [135]. Figure D.1 reproduces a
diagram from Cao et al. [136], showing the formation of chains and sheets in a
SrCO3-based ER fluid.
The primary method for gaining insight on the particle dynamics of these
fluids is through simulations. However, boundary conditions are known to have
significant effects on the particle behavior and rheological properties of the fluid,
both experimentally [48] and numerically [137]. Therefore the simulation gap height
could have a substantial effect on the results. MR fluid devices have gap heights on
the order of 1 mm, which for a typical fluid would require simulating over 1 000 000
particles. Unfortunately, existing simulations [26] use at maximum 10 000 particles.
In order to simulate at this scale, we use Nvidia’s CUDA environment to run code
on a commercial graphics card, allowing us to reach these length scales and simulate
158
over one million particles [138].
We use our code to investigate the lamellar sheet formation at length scales
on the order of 1 mm. At these large volume scales, direct visualization of sheet
formation becomes difficult, so several nondimensional metrics are used to measure
both chain and sheet formation independent of length scale. We then characterize
the effects of simulation scale, shear rate, and magnetic field strength on lamellar
sheet formation.
D.2 Model
We model the behavior of a magnetorheological fluid by simulating the trajec-
tories of the particles in the fluid. Our model follows the work done by Klingenberg,
[28], [34]. In these simulations, the particles are treated as induced point dipoles,
with magnetization M , and the particle boundaries are represented by an exponential
repulsion force. Since the carrier fluid is a viscous oil, particle accelerations can
be neglected, simplifying the simulation to a kinematic model, offering a dramatic
speedup. Brownian motion is also neglected without loss of generality. The fluid is
treated as a continuum under uniform shear, and particle disturbances on the flow
are neglected. Other simulation methods include finite element models and two-fluid
models [139].
The simulations take place in the cubic volume shown in Fig D.2, with the
external field H0 applied in the y direction. Fluid is sheared in the positive x
direction. Lamellar sheet formation is expected to occur in the z direction. We
159
Figure D.2: Schematic of simulation volume.
enforce a no slip condition, which is required to generate a rheological response [137],
at the upper and lower walls, pinning the particles to the surface of the wall. Periodic
boundary conditions are used in the x and z directions.
In order to simulate at volume scales approaching 1mm, we use Nvidia’s CUDA
programming environment. CUDA runs highly parallel code on a commercial graphics
cards, offering a speed up of 10–100x over single processor code [31]. The structure
of the code is algorithmically quite similar to many molecular dynamics codes [140],
allowing for straightforward implementation.
D.3 Results
Two sets of simulations were done, examining the effects of field strength and
volume scale respectively. All the simulations were performed at shear rates of
γ˙ =250 s−1, 500 s−1, 1000 s−1 and 2000 s−1, and run until a shear of γ = 20. The
first set were done at M =300 kA/m and 600 kA/m at a simulation volume size of
h = 0.5 mm, corresponding to N = 208 803 particles and a volume fraction of φ = 0.3.
160
To investigate the effects of volume size, simulations volumes of h =0.175 mm, 0.5 mm
and 0.875 mm were used, corresponding to N =8952, 208 803 and 1 119 058 particles,
at a field strength of M = 300 kA/m. The particles had radius a = 3.5 µm, and the
carrier fluid had viscosity ηc = 0.1 Pa s. Figure D.3 shows the graphical output of
the simulation code. We nondimensionalize time by reporting it in units of shear.
Figure D.3: Graphical output of the simulation code, at shear γ = 8 with M =
300 kA/m and γ˙ = 500 s−1, looking in the direction of shear.
The simulations were able to produce lamellar sheet formation, as shown in
Figure D.3. At simulation start, the particles quickly form chains spanning top
to bottom, and by γ = 2–3 the chains begin to form into wavy, interlinked sheet
structures. By γ = 6–8, the structure has straightened out, and aggregated into
sheets two particles thick, with strands of particles linking sheets. As time passes,
these strands disappear and the two-particle-thick sheet mode dominates. Due to
the random initial particle positions, occasionally a thicker structure will form.
The two-particle-thick sheets are thinner than those seen experimentally [136].
161
However our simulations cover at most 80 ms of time, and use uniformly sized
particles in a uniform field. Experimental measurements are typically taken after
several minutes of shearing, and time is known to have a significant effect on sheet
formation [135]. Gradients in the magnetic field could cause large movements of
the sheets, causing them to merge. Similarly, a distribution of particle sizes would
promote a more irregular structure, increasing the adhesion between sheets. However,
we note that the force cutoff of 8a means that this periodicity may be a numerical
artifact.
0 0.05 0.1 0.15 0.2 0.25 0.3−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
z (mm)
n
/
n
0
−
1
 
 
γ = 20 γ = 8 γ = 4 γ = 0
Figure D.4: Incremental number density of particles at H0 = 100 kA/m and γ˙ =
250 s−1 in a portion of a h = 0.5 mm volume as γ increases.
At high particle counts, directly observing particle positions is difficult. Noting
that there is a large variation in the particle distribution in the direction normal to
both field and shear, Fig. D.4 shows particle density in the z direction for a portion
of the volume when divided into bins one particle diameter thick. The number
density n is nondimensionalized by the expected value for a uniform distribution of
162
particles n0. This plot shows that from a random initial condition at γ = 0, regions
of high and low are accentuated until the trend of two-particle-thick sheets becomes
clear at γ = 8, and achieves a steady state by γ = 20.
To allow for better analysis of the time history of sheet formation, we use the
standard deviation of the number density in the z direction as a metric of sheet
formation. A higher value indicates a larger variance of particle densities, and thus
implies increased sheet formation. Figure D.5a plots the time histories for both
M =300 kA/m and 600 kA/m. Increasing magnetization causes a decrease in the
sheet formation time and an increase in magnitude of the sheet metric. Increasing
shear rate increases the response and decreases the steady state value of the sheet
metric. The results for M = 600 kA/m, γ˙ = 1000 s−1 and M = 300 kA/m, γ˙ =
250 s−1 overlap, as do M = 600 kA/m, γ˙ = 2000 s−1 and M = 300 kA/m, γ˙ =
1000 s−1 due to identical Mason numbers [35]. This metric of density variation
effectively shows the sheet formation trend, although it is quite sensitive to initial
conditions.
We can also use two metrics of chain formation: the connectivity, c, and
chain length, L, to help measure the sheet formation. Connectivity is the number
particle contact points, where two particles are in contact if the separation distance
is small, or rij−2a < 0.1a. Connectivity and chain length are nondimensionalized by
comparing their values to the case of a single-width particle chain spanning the top
and bottom plates. This yields an idealized value for connectivity, c0 = N(1− 2a/h)
and for chain length, L0 = h/2a. Time histories for both c/c0 and L/L0 are plotted
in Fig. D.5b and D.5c. Figure D.5b shows that, at high fields and low shear rates, the
163
extended shearing associated with sheet formation causes a decrease in chain length,
due to the separation of sheets, but connectivity increases monotonically (Fig. D.5c),
indicating that the dense sheet structure allows for more particle interactions.
In Fig. D.6 examines the effects of volume size on the chain and sheet metrics.
Figure D.6c shows that increasing volume size causes a slight increase in both the
steady state connectivity and the time required to reach steady state, indicating
increased aggregation in the larger volumes. Nondimensional chain length is plotted
in Fig. D.6b, and shows a small drop in value with an increase in volume. All three
metrics show a reduction in noise with an increase in volume size.
Figure D.6a shows that larger volume sizes decrease the both the rate of
formation and the magnitude of the density variation in the fluid. Since connectivity,
which measures particle packing, increases with volume, this result means that the
straight, narrow sheets present at smaller volumes are instead being replaced by a
dense but structure; and visual inspection of the simulation output agrees with this.
These irregular structure are likely the early stages of thick structure formation.
This sheet formation effect can be explained by the larger volume size reducing
the effects of boundary conditions. At small volume sizes, Pappas and Klingenberg
[137] showed that the elimination of the pinned boundary condition causes rapid
aggregation into thick structures, while a no-slip condition caused the formation of
one-particle-thick sheets. It has also been shown that chain aggregation into columnar
structures is dependent on chain length, with increasing chain length increasing the
potential for aggregation [141]. Thus operating at the large length scales enabled by
our high performance CUDA code is necessary to develop an accurate understanding
164
of the chain and sheet formation dynamics of magnetorheological fluid.
D.4 Conclusion
We successfully simulated over a million particles, and demonstrated sheet
formation at experimental volume scales. At high particle counts, direct visualization
of the behavior of the fluid is difficult, so we computed the standard deviation of
the particle density as a way to measure lamellar sheet formation of the fluid. We
also used two metrics of chain formation, the chain length and connectivity, to aid
in measuring and understanding the fluid structure.
Our simulations showed that simulation volume has a significant effect on the
chain formation of the fluid, with larger volume sizes promoting the formation of
thicker lamellar sheet structures, due to the decreased effects of boundary conditions.
We also demonstrated that lamellar sheet formation is inhibited by increasing shear
rates, and aided by increasing magnetic field strength.
For future work, we propose the use of a more direct metric of sheet formation,
such as a directional connectivity [141], as well as examining the effects of particle
volume fraction on the resulting structure.
165
0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Shear, γ
st
d
(n
/
n
0
)
 
 
γ˙ = 250
γ˙ = 500
γ˙ = 1000
γ˙ = 2000
(a) Standard deviation of n/n0
0 2 4 6 8 10 12 14 16 18 202
3
4
5
6
7
8
9
10
11
Shear, γ
L
/
L
0
 
 
γ˙ = 250
γ˙ = 500
γ˙ = 1000
γ˙ = 2000
(b) Nondimensional chain length
0 2 4 6 8 10 12 14 16 18 201
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
Shear, γ
c
/
c
0
 
 
γ˙ = 250
γ˙ = 500
γ˙ = 1000
γ˙ = 2000
(c) Nondimensional connectivity
Figure D.5: Nondimensional chain parameters versus shear at h = 0.5 mm. Blue and
red lines correspond to M = 300 and 600 kA/m respectively.
166
0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Shear, γ
st
d
(n
/
n
0
)
 
 
h = .175mm
h = .500mm
h = .875mm
(a) Standard deviation of n/n0.
0 2 4 6 8 10 12 14 16 18 200
1
2
3
4
5
6
7
8
9
Shear, γ
L
/
L
0
 
 
h = .175mm
h = .500mm
h = .875mm
(b) Nondimensional chain length
0 2 4 6 8 10 12 14 16 18 201
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
Shear, γ
c
/
c
0
 
 
h = .175mm
h = .500mm
h = .875mm
(c) Nondimensional connectivity
Figure D.6: Nondimensional chain parameters versus shear at M = 300 kA/m. Blue
and red lines correspond to γ˙ = 500 and 2000 s−1 respectively.
167
Appendix E: The effect of particle size distribution on chain struc-
tures in MR fluids
E.1 Summary
This work originally appeared as “Effect of particle size distribution on chain
structures in magnetorheological fluid” by Stephen G. Sherman and Norman M.
Wereley and published in IEEE Transactions on Magnetics [41].
E.2 Introduction
Magnetorheological fluids (MRFs) consist of micron scale carbonyl iron (CI)
particles suspended in a carrier fluid. When external field is applied, interparticle
magnetic forces cause the CI particles to align and form chain-like structures. These
chain structures cause the MRF to develop a field controllable resistance to stress, in
that these chains must break before the fluid will flow. Thus, if these chain structures
are modified because of changes in particle composition, scale or morphology, then
it follows that the field controllable resistance to stress will also change. Existing
simulation and theoretical work has focused on monodisperse MRFs, or fluids where all
the spherical particles have a uniform size. However, practical fluids are polydisperse,
168
and such a distribution of particle sizes alters the chain structure that form in the
presence of magnetic field. We simulate the effect of particle size distribution on fluid
stress and on formation of particles into chain structures in shear flows of MRFs with
a volume fraction of φ = 0.3. Such a volume fraction is representative of practical
MRFs used in vibration and shock isolation applications.
Particle size distribution plays a key role in bidisperse fluids, which have a
bimodal particle distribution and have been shown to exhibit a synergistic increase in
the yield stress of an MRF versus each particle suspended in a fluid alone [142]. This
bidisperse effect is caused by the interactions of different size particles. Because the
bidisperse effect is believed to be caused by the interactions of different particle sizes,
modeling the size distribution is necessary for an accurate model of chain formation
in MRFs.
For MRFs consisting of spherical particles suspended in a carrier fluid, experi-
mental and theoretical work by Lemaire et al. [129] indicates that at small particle
sizes and low volume fractions, particle size and size distribution strongly affects fluid
properties, which the authors attribute to Brownian motion. To measure the role of
Brownian motion in the fluid, Lemaire uses the ratio of magnetic forces to Brownian
forces, λ ∝ a3M2/T , where a is the particle radius, M is the particle magnetic
moment, and T is the temperature. At λ = 103 (a < 0.5 µm, M < 20 kA/m) and
a low volume fraction φ = 0.05, and a large particle size distribution, the smaller
particles in the distribution are influenced by Brownian motion, reducing the fluid
stress. However, for ferrofluids with silica particles, Brownian motion ceases to play
a role at λ = 109 and large volume fractions φ = 0.2.
169
There has been extensive research on simulating MRFs because simulations
can offer unique insights into how particles form chain structures, and, in turn, how
these chain structures affect MRF performance. While some simulation research
has looked at bidisperse fluids, using a mixture of two uniform particle sizes [29],
[143]. However, to our knowledge, no numerical investigations on the effects of
the particle size distribution have been performed. Current simulations are also
limited to extremely small numbers of particles, N < 104, so that there may be an
in insufficient number of particles to accurately and effectively represent particle
size distribution. However, we have developed a high performance CUDA-based
simulation code capable of modeling up to 106 particles, allowing us to simulate
chain formation in MRFs using a near experimental scale volume [19], [138]. In this
study, we modify this simulation code to investigate the effects of log-normal particle
size distribution on the performance of a typical MRF.
E.3 Background
To represent particle size range in MRF simulations, a log normal distribution is
selected because it is the typical distribution observed in carbonyl iron (CI) powders,
and is suitable for particle simulations [117]. The particle distribution function (see
Fig. E.1) is given by
f(a) =
1
aσ0
√
2pi
exp
[
−1
2
(
ln a− ln a0
σ0
)2]
, (E.1)
170
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Partic le Radius (µm)
P
ro
b
.
D
e
n
s.
(1
/
µ
m
)
 
 
σ0 = 0.0
σ0 = 0.05
σ0 = 0.1
σ0 = 0.2
σ0 = 0.3
Figure E.1: Particle size distribution.
where a0 is the median particle radius, and σ0 is the distribution parameter. For
σ0 ≤ 0.1, σ0 ≈ std(a/a0). Typically, distributions with σ0 ≤ 0.1 are considered to be
effectively monodisperse, and fluids with σ0 ≥ 0.1 are considered to be polydisperse.
In a typical MRF, commercial CI powder are used, such as those made by BASF,
where the particle size ranges is nominally 0.2 < σ0 < 0.3, and the smallest size
distributions in Lemaire et al.[129] have σ0 ≈ 0.1. Note that this distribution is
skewed, so for fixed a0, the mean (and mass weighted mean) radius will increase
with increasing σ0. Thus, for a fluid with constant volume fraction, increasing σ0
will reduce the number of total particles.
Prior work simulating MRFs has focused on monodisperse fluids. The simula-
tion developed here follows the uniform particle size simulations of Klingenberg [28],
[34], but particles of different size can interact. In this model, particles are magne-
tized solely by the external field, H0 = H0ey, giving them a uniform magnetization
171
M = Mey, and gives particle i with radius ai the magnetic moment, mi = 4/3pia
3
iM.
From that, the force on dipole i from dipole j is given by:
Fmij =
3µ0
4pir4ij
[(mi · rˆij)mj + (mj · rˆij)mi + (mi ·mj)rˆij
− 5(mi · rˆij)(mj · rˆij)rˆij], (E.2)
where rij is the position of particle i with respect to j. To avoid solving an n-body
problem, we truncate forces at 4(ai + aj), which corresponds to a cut off when forces
have fallen below 0.5% of their in-contact maximum. Observing that the attraction
force has magnitude
F0(ai, aj) =
3µ0mimj
4pi(ai + aj)4
, (E.3)
with peak attraction force of 2F0 and a maximum repulsion force of −F0.
Particle collisions are modeled using the repulsion force,
Fcij = 2F0(ai, aj) exp
[
−k
(
rij
ai + aj
− 1
)]
rˆij, (E.4)
where the stiffness parameter, k = 50. This model has the disadvantage that force
is applied outside of the particle sphere, but spring-based methods cause simulated
particles to overlap. Note that this stiffness parameter is twice as large as used in
[34]. This larger stiffness parameter is used to prevent overlap in the two-particle
attractive case, although overlap may still occur due to long range forces. Minimizing
overlap using the larger stiffness parameter enables more accurate measurements of
172
particle structure.
Stress on the particles in volume V is computed via
τxy = − 1
V
N∑
i=1
(Fi · ex)(yi − h/2) (E.5)
where ex is a unit vector in the x direction, yi is distance from the bottom surface,
and h is height of the simulation volume. A reference stress is computed
τ ∗ ==
1
16
piµ0M
2, (E.6)
which is independent of the particle radius, which implies that MRF performance
should theoretically be unaffected by changes in particle size distribution.
Here, Couette flow of an MRF is simulated in the fluid volume in Fig. E.2.
Despite the high particle volume fraction (φ = 0.3) [117], the simplified model in
[34] is used, where particles travel in an implicit carrier fluid where particles do not
couple to the fluid. In steady state shear flow, the carrier fluid moves with velocity,
v∞ = γ˙y ex. (E.7)
Periodic boundary conditions are placed in the x and z directions, and the particles
are given a slip boundary condition on the y border. We choose a slip condition for
the sake of simplicity and ease of integration, though particle boundary conditions
can have an effect on fluid stress and particle chain formation [48].
The drag force on a spherical particle moving with velocity vi is modeled by
173
Figure E.2: Simulation volume.
Stokes drag,
Fdi = 6piai(vi − v∞) (E.8)
Stokes drag is valid here because of the Reynolds number, Re < 10−2, due to the
small particle size and viscous carrier fluid.
Drag force is used to compute the Mason number, Mn, that is, the ratio of
hydrodynamic shear to magnetic shear forces:
Mn =
6piηca
2
0γ˙
F0
=
9ηcγ˙
2µ0M2
, (E.9)
where ηc is carrier fluid viscosity. Thus, Mn is independent of particle size. The
viscous nature of the fluid can also be used to neglect particle acceleration, as the
velocity response time is less than the duration of the forces [117], with response
time tv = 2ρpa
2/9ηc = 40 ns, and ρp as particle density. Thus, if the particle is
174
always at terminal velocity, then the particle dynamics can be represented by a first
order model. Buoyancy forces are neglected because simulation time is much less
(milliseconds) than settling times (days). Because simulated magnetizations are fairly
large, M = 300 kA/m, λ ≈ 108, Brownian forces are also neglected.
To measure the chain structures formed in the MRF, we use metrics quantifying
chain formation from graph theory[19]. The particles are the vertices of the graph,
and two particles are considered adjacent if rij < 1.05(ai + aj). A value of 1.05 was
chosen based on visual criteria. The first metric, total particle connectivity, c, is
based on the total number of graph edges. By considering a fluid consisting solely of
single width chains that span from top to bottom, the idealized total connectivity
is c0 = N/(1− 2a0/h). A second metric is the mean particle cluster size, S, or the
mean number of vertices in each connected component. Connected components
are identified through a depth-first search [144]. A similar ideal value for S can be
deduced based on the single-particle-width strand assumption, giving S0 = h/2a0.
E.4 Results
To analyze the effects of particle size distribution on fluid stress and chain
formation, we investigated the effects of distribution parameter, σ0, in a fluid with
constant particle volume fraction, φ = 0.3 and constant particle magnetization,
M = 300 kA/m, at various shear rates. A range of discrete values of σ0 were
considered: 0 (uniformly sized particles), 1× 10−3, 2.5× 10−3, 5× 10−3, 0.01, 0.025,
0.05, 0.1, 0.2 and 0.3. The properties were measured under shear at γ˙ =30 s−1, 60 s−1,
175
0 1 2 3 4 5 6 7 8
x 10−3
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Mason Number
τ
/
τ
∗
 
 
σ0=0
σ0=0.1
σ0=0.2
σ0=0.3
Figure E.3: Stress versus Mason number.
125 s−1, 250 s−1, 500 s−1, 1000 s−1, 1500 s−1 and 2000 s−1 until a nondimensional shear
of γ = 8, corresponding to simulation times of 4 ms to 267 ms, and Mason numbers
from 8× 10−3 to 1.2× 10−4 . Steady state quantities are computed from time
averages during γ = 4 to γ = 8, and then results were averaged over 3 runs. All
simulation runs occurred in a cubic volume with a constant median particle radius
of a0 = 4 µm, a gap height of 0.5 mm, and with a carrier fluid viscosity of 0.1 Pa s.
The integration method was a classic fourth-order Runge-Kutta scheme with a time
step of 450 ns, with a limiter that recomputes the iteration at half the time step if
the maximum particle displacement relative to the fluid is greater than 0.035a0.
Fig. E.3 depicts the flow curve, which shows that stress increases with Mn,
and is broadly consistent for all fluids at high Mn. However at low Mason numbers,
Mn < 10−3, the uniform particle distribution has a significantly higher stress, and at
176
(a) σ0 = 0.0 (b) σ0 = 0.05
Figure E.4: Particle structure at γ = 4 and Mn = 2.4× 10−4 with a volume height
of 0.35 mm. Colors correspond to force on the particles.
high Mn, Mn > 5× 10−3, fluids with large particle size distribution exhibit a small
increase in fluid stress, which indicates that particle size distribution effects MRF
performance.
In order to identify over what range of distributions size parameter this occurs,
Fig. E.5 τ/τ ∗ versus σ0 for a variety of Mason numbers. At low Mason numbers, Mn
< 10−3, fluids with with a distribution of σ0 > 0.025, exhibit a 25% decrease in stress
over fluids with σ0 ≤ 0.01. At high Mason numbers, fluids with σ0 > 0.05, increasing
σ0 causes a slight increase in stress. From these results, particles size distribution
ceases to play a role when σ0 ≤ 0.025, which is significantly less than the value of
σ0 = 0.1 that is typically used in simulations. To explain why these changes occur,
we hypothesize that the particle size distribution is altering the particle structure,
and that the resulting new structure reduces the apparent viscosity.
One way to identify the change in particle structure caused by the size distri-
177
bution is visually; Fig. E.4 compares a representative polydisperse fluid, σ0 = 0.05,
with a monodisperse MRF, σ0 = 0, at Mn= 2.4 × 10−4. These images show that
the chain structure in the monodisperse MRF forms a lattice-like sheet structure,
while the polydisperse fluid develops a more irregular structure. As values of σ0 and
Mn increase, the chain structure irregularity also increases. These trends can be
quantified using the following chain structure metrics.
The first metric of chain formation, mean cluster size, S/S0, is plotted against
distribution size parameter, σ0, in Fig. E.6. Generally, increasing σ0 and Mn causes
a decrease in S. Increasing Mason number causes cluster size to decrease, because
increasing shear force causes the particle clusters to break apart. The decrease
in S/S0 for σ0 > 0.025 is due to the increasing distribution size parameter, which
prevents particles from forming the lattice-like sheet structures of monodisperse MRFs.
However, at low Mn, and for values of 0.025 ≤ σ ≤ 0.2, the cluster size increases, so
that the increase occurs over a similar range of distribution size parameter as the
drop in stress observed in Fig. E.5. Here, the distribution size prevents particles
from forming the regular lattice sheets observed in a monodisperse MRF, but the
shear rate is low enough that the particles can form irregular structures with tenuous
connections between different clusters, causing S/S0 to rise significantly. The values
of S/S0 are so large that less than 10 clusters are in the simulated fluid volume. This
further motivates the need for high particle count simulations.
The second metric is particle connectivity, c/c0, which measures the number of
particles with which each particle is in contact, or localized density. Figure E.7 plots
particle connectivity, c/c0, against σ0. As Mn increases, connectivity drops. which
178
10−3 10−2 10−1
0.01
0.015
0.02
0.025
0.03
0.035
σ0
τ
/
τ
∗
 
 
Mn
8.0×10−3
6.0×10−3
4.0×10−3
2.0×10−3
1.0×10−3
5.0×10−4
2.4×10−4
1.2×10−4
 Increasing
Mason Number
Figure E.5: Stress versus distribution size for various Mason numbers.
10−3 10−2 10−1
100
101
102
103
104
σ0
S
/
S
0
 
 
Mn
8.0×10−3
6.0×10−3
4.0×10−3
2.0×10−3
1.0×10−3
5.0×10−4
2.4×10−4
1.2×10−4
 Increasing
Mason Number
Figure E.6: Cluster size versus distribution size for various Mason numbers.
10−3 10−2 10−1
1.5
2
2.5
3
σ0
c/
c
0
 
 
Mn
8.0×10−3
6.0×10−3
4.0×10−3
2.0×10−3
1.0×10−3
5.0×10−4
2.4×10−4
1.2×10−4
 Increasing
Mason Number
Figure E.7: Connectivity versus distribution size for various Mason numbers.
179
indicates that increased shearing is actually separating the majority of the particles.
At low Mason numbers, for σ > 0.025, increasing distribution size parameter causes
a drop in connectivity, because the distribution particle size promotes irregular
structures over lattice-like structures. However, at high Mn and for σ0 ≥ 0.2, there
is a slight rise in connectivity caused by shear forces pushing smaller particles into
contact with larger ones.
However, all these results are over the range of γ = 4 to 8, the changes in
particle structure may be coming from changes to the behaviors of the transients,
rather than the steady-state properties.
E.5 Conclusions
Effects of log-normal particle size distribution on MRF behavior was inves-
tigated via high particle count (on the order of 106 particles) simulation based
on CUDA. While simple theoretical models indicate that particle size distribution
should have no effect on MRF performance, these large particle count simulations
indicate that MRF behavior changes substantially as the size parameter, σ0, increases.
Simulations were done for median particle radius of a0 = 4 µm, carrier fluid viscosity
of 0.1 Pa s, and particle volume fraction of φ = 0.30. The significant findings are as
follows.
1. MRFs behave as monodisperse fluids for narrow particle distributions having a
size parameter, σ0 < 0.025. This is well below practical size parameters in CI
powders, typically σ0 ≥ 0.1, therefore, monodisperse MRFs would be difficult
180
to realize in practice.
2. Simulations show that particle sheets in monodisperse MRFs form lattice-like
structures and, as the particle distribution size parameter increases, particles
tend to form more irregular structures.
3. These increasingly irregular structures manifest as a 25% reduction in MRF
shear stress at low Mn, and a slight increase in stress at high Mn.
4. Wide particle size distributions correspond to a reduction in particle cluster
size, and a small increase in connectivity. The changes in cluster size also
correspond well to the simulated stress values.
Based on these simulation results, we conclude that particle size distribution
has a substantial effect on the structure and performance of magnetorheological
fluids, so that accuracy of simulation codes may improve when realistic particle size
distributions are utilized.
181
Appendix F: The Bingham and Mason numbers
F.1 Summary
This is a reproduction of the paper, ”Relating Mason number to Bingham
number in magnetorheological fluids” published in the Journal of Magnetic Materials
and Magnetism [53], and presented at the 10th Inertnaional Conference on the
Scientific and Clinical Applications of Magnetic Carriers in Dresden, Germany.
F.2 Introduction
Magnetorheological (MR) fluids are a fluid composed of micron scale magnetiz-
able particles suspended in a carrier fluid. Upon the application of field, the particles
in the field align to form chain like structures, and these chains cause the fluid to
develop a field dependent yield stress. The primary application of MR fluid has
been in MR dampers and MR energy absorbers, where the controllable apparent
viscosity allows for a controllable damping force or stroking load, which enables high
performance vibration isolation [145] or shock mitigation that can adapt to payload
weight and impact severity [146], [147].
Models of magnetorheological fluids have typically taken two perspectives:
182
either modeling the MR fluid as a collection of microscopic particles floating in a
carrier fluid, or as a bulk fluid continuum. Microscopic modeling of MR fluids focuses
on the behavior of the particles [26], [34], [141] by examining the formation and
destruction of chain structures in the fluid with the goal of predicting yield stress.
The primary forces on the particles that govern chain formation are viscous drag
of the carrier fluid on the particle and the interparticle magnetic forces. The ratio
of particle magnetic forces to viscous forces is known as the Mason number, Mn,
[35], [148], [149], named after the work of S. G. Mason et. al. on the behavior of
fluid droplets in the presence of electric field [150]. In the equations of motion, the
Mason number is the governing parameter of the shear response of a particle in an
MR fluid, and is an essential part of research on dynamic models of chain formation.
The Mason number, Mn also has value in the analysis of experimental data, such as
when apparent viscosity is plotted against Mason number, the apparent viscosity
curves collapse to a single curve, thereby reducing the dimensionality of a dataset
[35], [148].
At the bulk scale, one of the idealized descriptions of MR fluids is as a Bingham
plastic [151], in which the applied magnetic field additively induces a field controllable
yield stress to a Newtonian fluid. The Bingham number, Bi, which is the ratio of yield
stress to viscous stress, describes the extent to which the controllable yield stress can
exceed the viscous stress (typically Bi 1), and is an essential descriptor of Bingham
plastic behavior. The Bingham number can be used to calculate flow rates, flow
profiles, and pressure losses in devices using Bingham plastic fluids [80]. In particular,
for shear mode MR devices, the Bingham number represents the controllable force
183
ratio [152], and since MR fluids are used for the purpose of generating controllable
forces, the Bingham number is an essential and fundamental parameter for the
understanding and analysis of MR fluids at the bulk scale.
We seek to relate the Bingham number to the Mason number, two nondimen-
sional numbers that represent fundamental descriptions of the behavior of MR fluid
at macroscopic and microscopic scales, respectively. In particular, we focus on MR
fluids typically used in energy absorbing devices. These MR fluids are typically
suspensions of 1–10µm diameter carbonyl iron particles with solids loading ranging
from 20 to 50 volume percent, and well described by the Bingham plastic model. By
mathematically relating Bingham number to Mason number, we enable microscopic
Mason number based analyses to be directly extended to macroscopic or device scale
Bingham number based problems. Alternatively, experimental Bingham number
based results can be scaled down for comparison to Mason number based particle
level analyses.
In this study, the Bingham number and the Mason number are developed, and
it is shown that if microscopic forces map linearly to macroscopic forces, then the
Bingham number and the Mason number are inversely related, or that the product
of Bingham number and Mason number is a constant. This notion is confirmed
through measurements of apparent viscosity. We experimentally validate the claim
that microscopic and macroscopic forces are linearly related, and that this is akin to
assuming that MR fluids are well described by the Bingham plastic model. Finally,
the relationship between Mason number and Bingham number is used to examine
the experimental relevance of a Mason number based result, as well as how such a
184
Figure F.1: Idealized rheogram or shear stress vs. shear rate diagram for an MR
fluid
relationship would inform the MR fluid and/or device design process.
F.3 Background
To motivate the usage of these nondimensional numbers, both numbers are
derived in the analytical context in which they arise.
F.3.1 The Bingham Number
For device scale analyses, the fluid is treated as a continuum with nonlinear
rheological properties. A typical MR fluid shear stress vs. shear rate graph is shown
in Fig. F.1. These shear stresses for each field strength are typically modeled by the
Bingham plastic model,
τ = τy + ηplγ˙, (F.1)
185
which has a plastic viscosity, ηpl, and a yield stress, τy. The yield stress is magnetic
field dependent, and it is typical to assume that ηpl is independent of field strength,
and equivalent to the off-state viscosity, ηoff . In MR fluids, ηpl is chosen to be the
slope of the high shear rate asymptote of the shear stress curve, and τy corresponds
to the intersection of the high shear rate asymptote with the stress axis at γ˙ = 0.
A typical way in which MR fluid is used in damper design is the shear mode
damper [152], where an upper plate moving with velocity, v, and area, A, moves
over a stationary lower plate with a gap of d between the two plates. Here, the fluid
velocity profile is linear, and the force on the upper plate is
Fd = (τy + ηpl
v
d
)A. (F.2)
The force in conventional viscous dampers can be written in the form Fd = c0v,
where c0 is the damping, and for a Newtonian fluid in shear mode c0 = ηA/d. For
the shear mode MR damper, rearranging into this form yields,
Fd = (
τyd
ηplv
+ 1)ηpl
A
d
v = ceqv, (F.3)
where ceq is the equivalent damping for a fluid with a yield stress. The ratio of
equivalent damping to Newtonian damping yields the damping coefficient
ceq
c0
= 1 +
τyd
ηplv
= 1 + Bi (F.4)
which describes the effect that the addition of a yield stress has on damping force. For
186
an MR fluid, where the yield stress is field controllable, this ratio is the controllable
force ratio. The term that governs controllability is the Bingham number,
Bi =
τy
ηplγ˙c
, (F.5)
the ratio of magnetic forces (τy) to viscous forces (ηplγ˙c) in the fluid, where γ˙c is the
characteristic shear rate of the system, which for a shear mode damper is γ˙c = v/d.
Since the purpose of MR fluids is to generate a field controllable force, and the
Bingham number represents the controllable force ratio of an MR device, it is clear
that Bingham number is a fundamental representation of the behavior of MR fluids.
In more complicated geometries, such as in pipe flow, the Bingham number becomes
an essential intermediate quantity in the determination of the flow rate, flow profile,
and controllable force output of an MR fluid device [80]. But at the fluid level, the
Bingham number is a descriptive, empirical quantity, and doesn’t tell us anything
about what causes the MR effect, or how a fluid can be modified to improve its
performance.
F.3.2 Mason Number
Modeling MR fluid at the particle level allows us to develop predictive models of
fluid behavior, providing insight into the chain formation that underlies the MR effect.
At the microscopic scale, MR fluids consist of magnetizable particles suspended
in a carrier fluid under the influence of an applied magnetic field, H0. Figure F.2
contains a diagram of two interacting particles under shear and applied magnetic field.
187
Figure F.2: Diagram of two particles in a shearing fluid.
Typically, these are spherical carbonyl iron particles with diameter σ = 1− 10 µm.
The particles are usually modeled as perfect spheres with fixed point dipole moments
mi aligned with the applied field, mi = piσ
3Mp/6, where magnetization, Mp = χpH0
is also aligned with the applied field. The force on particle i from particle j is
Fij =
F0
(rij/σ)4
[
(3 cos2 θij − 1)er + sin 2θij eθ
]
(F.6)
with force magnitude
F0 =
3µ0m
2
4piσ4
=
pi
48
µ0M
2
pσ
2. (F.7)
This force magnitude can be turned into a reference stress, τ ∗ = F0/σ2, which will
be used to normalize shear stress.
The fluid interacts with particles via viscous drag, Fi = −Cd(x˙i − vc), where
ηc is the carrier fluid viscosity, x˙i and vc are the particle and carrier fluid velocity
respectively, and Cd is the coefficient of drag, where for Stokes drag, Cd = 3piσηc.
188
In most microscopic models, the carrier fluid moves independently of the particles,
allowing us to assume a carrier fluid velocity profile. In shear, the bulk fluid, and
thus the carrier fluid moves with velocity vc = γ˙yex, where y is the distance from
the stationary surface.
Solving for the trajectory of the particle, the equations of motions for the
particle are
mix¨i + Cdx˙i −
N∑
j=1
j 6=i
Fij(xi,xj) = Cdγ˙yi ex, (F.8)
where xi is the position of the ith particle, and for this equation only, mi is the mass.
The particle mass is small, so that the force contributions from particle acceleration
are also small and occur at such a short time scale that mix¨i can be set to zero [34].
This allows the full equations of motion to be placed into a kinematic form, which
when placed in a dimensionless form yields
Cd
F0
x˙i =
N∑
j=1
j 6=i
Fij(xi,xj)
F0
+ Mn
yi
σ
ex, (F.9)
where Mn is the Mason number, and is the sole term governing the shear response
of the particles in the fluid. The Mason number is the ratio of microscopic shear
forces to microscopic magnetic forces, defined as
Mn =
Cdσγ˙
F0
=
3piηcγ˙
τ ∗
= 144
ηcγ˙
µ0M2p
. (F.10)
Because Mason number governs shear behavior of the particle structures, it will
189
govern the breaking and reforming of the chains in the fluid, and can be used to
predict the shear response of the fluid.
When apparent viscosity, τ/γ˙, is plotted against Mason number for experimental
data, the apparent viscosity curves collapse to a single master curve, so that apparent
viscosity is solely a function of Mason number. Thus, the Mason number acts as
a nondimensionalized input condition that yields one output condition, and it is
through this result that we intend to relate the Bingham and Mason numbers. This
nondimensionalization of experimental data allows low shear rate experimental data
to be extrapolated to high shear rates, and will be used in our relationship between
Bingham number and Mason number. In particular, this is useful as shear rates
in practical MR devices can exceed γ˙ > 10 000 s−1, while rheometer experiments
typically operate at γ˙ < 1000 s−1.
The Mason number is often defined using the H field, where M = 3βH with
β = (µp−µc)/(µp+2µc) from the textbook problem of an isolated, linearly susceptible
sphere in a uniform applied field, yielding
Mn(H) =
16 ηcγ˙
µ0µcβ2H2
, (F.11)
where µp and µc are the relative permeability of the particle and carrier fluid
respectively. The H-based form has several benefits, among which are its amenability
to theoretical analysis, applicability to inverse ferrofluids, and its ease of relation to
the electric analogue used in electrorheological fluid analysis. While M is a derived
quantity, experimentally, the magnetization curves of the fluid must always be known
190
to determine the magnetic field within the testing device. The H-based analysis
also requires measuring µp(H) for β, a process which involves measuring M . Using
M or µp from a magnetometer invokes the assumption that the particle structure,
and thus magnetization, is the same in both magnetometer and device, however in
practice, this assumption appears successful [153]. Most importantly, Klingenberg
showed that approaching saturation, the H based Mason number fails to deliver
the desired coalescing of data onto a master curve [35]. In the context of device
engineering, nonlinear magnetization must be accounted for, so that we choose to
use a Mason number based on average particle magnetization. Also, because average
particle magnetization is a measurable quantity, it can also be used to validate the
assumed particle structure in models of MR fluids.
We also note that the Mason number definition sometimes varies in the litera-
ture. Here we define Mason number using a characteristic separation distance of a
particle diameter, as it gives a useful definition of τ ∗. This leads to a definition of
Mn that is 32 times larger than that in [35].
F.4 Theory
We now seek to relate Mason number to Bingham number, so that microscopic
results can be extended to macroscopic behavior, and vice versa. In order to relate
Bingham number to Mason number, we observe that both are ratios of magnetic and
viscous forces, at bulk and microscopic scales respectively. If we assume that the
ratio of forces scales linearly from microscopic to bulk macroscopic properties, then
191
Bingham (magnetic/viscous) and Mason (viscous/magnetic) number are inversely
related. Assuming that the characteristic shear rate is the same in both Bi and Mn,
solving for γ˙ in (F.10) gives
γ˙ =
F0
σ2
1
3piηc
Mn =
τ ∗
3piηc
Mn.
This can then be substituted into (F.5),
Bi = 3pi
τy/τ
∗
ηpl/ηc
Mn−1, (F.12)
giving us a relationship between Bingham and Mason number, in terms of two ratios.
The first ratio, the ratio of the yield stress of the bulk material to the magnetic
forces between two particles,
τy
τ ∗
=
τy
piµ0M2p/48
(F.13)
is defined as the normalized yield stress. The second is the ratio of suspension
viscosity to carrier fluid viscosity,
ηpl
ηc
=
Suspension Viscosity
Carrier Viscosity
, (F.14)
the normalized viscosity. Normalized yield stress, τy/τ
∗ represents how effective the
magnetic attraction between particles manifests as a yield stress, and ηpl/ηc represents
how the addition of particles effects the suspension viscosity. Most importantly, we
192
assert that these two ratios are constant fluid properties of fundamental interest,
as will be shown below, and these are ratios that can be predicted from existing
literature and theory.
Again, note that the Bingham number is the ratio of magnetically induced
shear stress to viscous shear stress, and that the Mason number is the ratio of viscous
shear stress to magnetically induced interparticle stress. Another useful perspective
is that the product of Bingham number and Mason number is a constant, or
Bi Mn = 3pi
τy/τ
∗
ηpl/ηc
(F.15)
thereby evincing our earlier hypothesis. The term on the right, a constant, acts as
a figure of merit of the fluid-particle recipe, representing the effect the addition of
particles has on yield stress divided by the corresponding increase in viscosity. This
term will be expanded upon further in Section F.5.
F.4.1 Normalized Yield Stress
In order to argue that τy/τ
∗ is a fluid property, we need to show that yield
stress scales roughly proportionally to average particle magnetization across field
strength.For low volume fraction fluids, average particle magnetization as a yield
stress scaling function has been shown to be a successful scaling law [36]. For the high
volume fraction, high yield stress fluids of interest in controllable force applications,
an estimate for τy/τ
∗ can be obtained using existing fits of B vs. H and τy vs. H
for a range fluids, and then solving for τy/τ
∗. Carlson [154] offers the empirical and
193
widely used fits
τy = 271.7Cφ
1.524 tanh(5.04µ0H) (F.16)
B = 1.91φ1.133 [1− exp(−10.97µ0H)] + µ0H (F.17)
designed to cover iron-based fluids, including those sold by Lord Corporation. In
these equations, τMR is in kPa, H is in A/m and the parameter C is 1.0, 1.16 or
0.95 depending on whether the carrier fluid is hydrocarbon oil, water, or silicon oil,
respectively. Neglecting the effect of carrier fluid, we can substitute these equations
into τy/τ
∗ and obtain
τy
τ ∗
=
1.43φ1.257 tanh(5.04µ0H)
[1− exp(−10.97µ0H)]2
. (F.18)
For fields above H=100 kA/M, this equation is largely constant, depending solely
on fluid volume fraction and carrier fluid, as initially required. However in the limit
of H=0, (F.18) yields τy/τ
∗ = ∞, as M drops off faster than yield stress, but an
examination of Carlson’s data shows that his yield stress model overestimates the
experimentally measured yield stress at low field strengths, so this issue can be
neglected. Observing that for 0.2 ≤ φ ≤ 0.5, τy/τ ∗ is within 15% of φ, this allows
the construction of the rule of thumb, τy/τ
∗ ≈ φ.
Beyond developing experimental rules of thumb for yield stress, normalized
yield stress also appears in most particle level modeling work, however, typically in
the form of τy/H
2. Pen and paper models work by assuming a particle structure,
194
allowing the particles to magnetically interact, deforming the structure under shear,
and then computing the interparticle forces to find the shear stress. Computational
models follow a similar structure, but with a dynamic structural deformation[34],
[41], [155]. In order to resolve some of the nonlinear magnetization in the particles,
some models allow regions of the particle to saturate, obtaining τy ∝ φH3/2M1/2s , but
these models fail near Ms [156], [157]. The unsurprising solution is to work in terms
of averaged particle magnetization, as it allows a model to work across the entire
spectrum of applied field, as shown in [158]. The benefit of this nondimensional form
for yield stress is that it takes τy, a field dependent term, and puts it in a form that
is roughly independent of field, depends only on particle type and concentration,
and can be predicted using various analytical methods.
F.4.2 Normalized Viscosity
The normalized viscosity ratio, ηpl/ηc, is the ratio of off-state viscosity to carrier
viscosity. The typical assumptions about plastic viscosity are that it is independent
of applied magnetic field, and can be determined by measurements of the fluid with
no field applied. Since carrier fluid viscosity is independent of field, ηpl/ηc will be a
function only of the addition of the particles, and thus a fundamental property of the
fluid. Since the particles in an MR fluid are traditionally hard spheres, a prediction of
this material property can be determined from the theoretical and empirical relations
developed in the study of hard sphere dispersions [131], for example, the empirical
195
Quemada relation,
η
ηc
=
(
1− φ
φmax
)−2
(F.19)
where φmax is the maximum particle volume fraction. However, these hard sphere
equations must be used with caution, as practical MR fluids use additives which
significantly affect the plastic viscosity, and so generalized forms with more parameters
may be required. Once an appropriate viscosity relationship has been found, Eq.
(F.12) can be expressed as a function solely dependent on φ. The notion that
normalized viscosity is constant can also be used to show that the highly temperature
dependent post yield performance of MR fluids is caused only by changes in the
carrier fluid [40].
However, while the assumption that ηpl is constant is effective in the practical
analysis of MR devices, this assumption often breaks down when examining the
rheogram of practical MR fluids. The plastic viscosity for a particular value of field
is determined, in practice, from the high shear rate asymptote of rheogram or shear
stress vs, shear rate data as shown Fig.F.1. These plastic viscosities for discrete values
of magnetization, while reasonably similar in value, are not necessarily identical to
the off-state viscosity, so that there exists varying levels of shear thinning. The theory
of hard sphere dispersions tells us that viscosity is dependent on particle structure,
so when field is applied and the particles form chains, the applicability of dispersion
theory to determine ηpl is questionable. In an attempt to resolve this dilemma, Berli
and de Vincente developed a structure based model for viscosity [159], and used it
to model traditional MR fluids as well as inverse ferrofluids. Their analysis leads
196
to low and high shear rate viscosity plateaus, which can be well represented for a
typical MR fluid as a Casson plastic,
τ 1/2 = τ 1/2y + (η∞γ˙)
1/2. (F.20)
This introduces a field dependent viscosity term that smooths the transition between
pre- and post-yield behavior, improving the quality of the fit for some fluids, which
typically tend be small particle (thermally influenceable) low yield stress fluids.
However, for consistency with the device engineering literature and for ease of analysis,
the widely accepted Bingham plastic constant (post-yield) viscosity approximation
is sufficient for our purposes.
F.5 Experiment
The Bingham–Mason relationship can be measured directly by examining
apparent viscosity. Apparent viscosity, ηapp = τ/γ˙, is the viscosity of a Newtonian
fluid that would give the measured stress at the current shear rate. For a Bingham
plastic, where
τ = τy + ηplγ˙ (F.21)
the apparent viscosity is
ηapp = τy/γ˙ + ηpl (F.22)
The high shear rate limit of apparent viscosity is denoted as η∞, and for a Bingham
plastic, η∞ = ηpl. Apparent viscosity is often normalized by η∞, which for a Bingham
197
10
-2
10
0
Mason Number, Mn ∝ γ˙/M2
10
0
10
1
10
2
η
a
pp
/η
∞
131.26
126.19
120.40
113.88
107.32
100.24
 91.43
 81.91
 69.18
 49.20
 35.54
  3.83
Figure F.3: Normalized apparent viscosity for Lord MRF-140CG
plastic yields,
ηapp
η∞
=
ηpl
ηpl
+
τy
ηplγ˙
= 1 + Bi. (F.23)
Thus a measurement of apparent viscosity leads directly to a measurement of Bingham
number. For MR fluids, a well known result is that when apparent viscosity is plotted
against Mason number, a set of curves of apparent viscosity coalesce to a single
master curve [35], [159]. Therefore, by using normalized apparent viscosity plotted
against Mason number, we obtain a single curve that allows us to directly relate
Bingham number to Mason number for a particular MR fluid being measured.
Figure F.3 shows a measurement of apparent viscosity using the commercially
available Lord MRF-140CG fluid, demonstrating the collapse of apparent viscosity
curves across Mason number. This 20 ◦C temperature controlled data set was collected
using a custom-built high shear rate (γ˙ < 10 000 s−1) Searle cell magnetorheometer
[37]. High shear rates are achieved by using a narrow (d = 0.25 mm) active gap,
198
and the concentric cylinder geometry ensures a uniform shear rate. Using γ˙  0
avoids nonlinearities associated with near yield flow, and ensures that the data is
well modeled as a Bingham plastic. In this experimental procedure, a servomotor
rotates the inner cylinder using an ascending staircase velocity profile, while the
outer cylinder is connected to a fixed base by a 0.706 N m angular load cell. The
applied magnetic field was simulated via a 2D axisymmetric FEM analysis and
validated using sensing coil measurements of flux density, and the fluid magnetization
was calculated using the characterized field and manufacturer supplied B-H data.
Since the carrier fluid formulation for Lord fluids is proprietary, the carrier fluid
was obtained by allowing a well mixed sample to separate, and then decanting the
sediment-free upper layer. Carrier fluid viscosity was measured on an Anton Paar
Phyisca MCR-300 rheometer, with the average of three runs to yield a viscosity value
of ηc = 9.9 mPa s at 25
◦C.
For magnetorheological fluids, apparent viscosity is typically fitted to a curve
of the form
ηapp
η∞
= 1 +KMn−1, (F.24)
where K is a fitted parameter. In Fig. F.3, we show such a fit, with K = 0.078, and
η∞ = 0.59 Pa s. Since MR fluids are well described by the Bingham plastic model,
from (F.23), ηapp/η∞ = 1 + Bi, and from our experiment ηapp/η∞ = 1 + KMn−1,
then Bi ∝ Mn−1, validating (F.12).
The fitted parameter K is also known as the critical Mason number Mn∗ [39],
[148], [159], and corresponds to the Mason number where the low Mn asymptote of
199
ηapp/η∞ intersects ηapp/η∞ = 1, or the Mason number where Bi = 1. From (F.12)
the critical Mason number can be expressed as
Mn∗ = 3pi
τy/τ
∗
ηpl/ηc
, (F.25)
which is also the product of Mason and Bingham numbers (F.15). The critical
Mason number serves two purposes: it acts as a conversion factor between Mason
number and Bingham number, and also acts as a fluid figure of merit. Critical
Mason number converts a nondimensionalized input condition (Mason number) to
nondimensionalized output condition (Bingham number), so that these two numbers
can be easily related. Critical Mason number acts as a controllable force figure of
merit with respect to the addition of particles, and can be thought of the benefit
the addition of particles has had on yield stress to the penalty paid as in increase
in viscosity. Fluids with a large Mn∗ possess a larger controllable force ratio, while
Mn∗ decreases with increased volume fraction. Finally, because τy/τ ∗ and ηpl/ηc are
analytically accessible ratios, we now have a way to determine critical Mason number
during the fluid design process.
However, note that Eq. F.24 is limited to fluids well described by the Bingham
plastic model, a description not suited to all electrically and magnetically responsive
fluids. In some of these fluids, the fluid scales between Mn−2/3 and Mn−1, where
fluids with particles influenced by thermal motion depart from Mn−1 scaling [129],
[160]–[162]. However these fluids are typically low yield stress fluids measured at low
field strengths, and this is attributed to the effects of Brownian forces on the particle
200
0 0.002 0.004 0.006 0.008 0.01 0.012
Mason Number, Mn ∝ γ˙/M 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
τ
/
τ
∗
=
τ
/
(pi
µ
0
M
2 p
/
4
8
)
0.84
0.82
0.80
0.78
0.76
0.73
0.70
0.66
0.60
0.49
0.40
0.08
10
-4
10
-2
10
0
0
1
2
3
4
Figure F.4: Normalized stress vs Mason number, with high Mason number data as
inset. The black line corresponds to a linear least squares fit, and shows how the low
Mason based fit agrees with the inset high Mason number data.
[163]. However, in the context of controllable force applications, where high yield
stress fluids are used, thermal effects on the particles are relatively small, and the
Bingham plastic model and Mn−1 scaling are a proven success [35]. These effects
manifest at high Mn, with experimental apparent viscosity data decreasing slower
than (F.24) predicts. However for the purpose of relating Bi and Mn, typically at
that point Bi is small (Bi < 2), and thus negligible for controllable force applications.
Such errors can potentially be resolved through the use of Casson plastic type models
[159].
Finally, it should be noted that fitting a line to the apparent viscosity curve
can be problematic, especially when high Mn data is unavailable, as there is no
clear value for η∞. An alternate approach is to plot normalized stress, τ/τ ∗ against
Mason number, shown in Fig. F.4. The normalized stress plot demonstrates, like
201
the apparent viscosity plot, that stress values can be collapsed across varying field
strengths through Mason number based analysis, while retaining plastic viscosity
information at low Mason numbers, unlike apparent viscosity, offering an alternate,
improved master curve. This makes determining τy/τ
∗ and ηpl/ηc much easier, as
for a Bingham plastic, normalized stress is given by
τ
τ ∗
=
τy
τ ∗
+
ηpl/ηc
3pi
Mn. (F.26)
Then a simple linear least-squares fit gives τy/τ
∗ = 0.499 and ηpl/ηc = 59.9, from
which we generated the previously stated values for K and η∞. This procedure has
the benefit that it separates τy/τ
∗ and ηpl/ηc, allowing them to be fit separately,
instead of confounding the two terms into K. This fitting procedure works even
when there is no high Mason number data, as shown in Fig. F.4. The benefits of
plotting in this form is that it clearly shows the successful normalization of yield
stress, as well as that ηpl/ηc is independent of field, and separates them into discrete
elements for easy linear fitting, and serves as a complement to typical stress-shear
rate graphs and normalized apparent viscosity graphs. The downside of this plot
is that since it is no longer log-log, it much more clearly shows errors and noise in
experimental data, and can be quite sensitive to errors in characterization of the
magnetic field.
202
F.6 Discussion
The primary purpose of our analysis is to develop a nondimensional scaling
relationship to be used to relate the performance of different devices. In traditional
device design, several output parameters (τy, ηpl and Bi) must be accounted for
across a large input space (magnetic field strength, shear rate and temperature),
requiring extensive experimental characterization. We want to avoid this, and here
Mason number analysis shines – it allows the performance of MR fluid data to be
reduced to one single master curve, dependent solely on Mason number. Then to
obtain the desired output quantity of controllable force, Eq. F.12, yields the Bingham
number. We also show that τy can be made independent of magnetization as τy/τ
∗,
and plastic viscosity can be temperature independent in the form of ηpl/ηc. This
nondimensionalization allows us to take low shear rate, low field data measured on a
small fluid sample, and extend it to a large scale energy absorbing device with high
fields and high shear rates with a high level of confidence. This is demonstrated for
a large scale energy absorbing device operating at γ˙ > 25 000 s−1 in [38].
For pen and paper analysis, there exists the notion of a Mason number above
which no chains can form in the fluid, posing an upper limit on the existence of the
MR effect. However, if this limiting Mason number occurs at a low Bingham number,
the loss in yield stress will not be an issue for experimental devices. In [54], [55],
the limiting Mason number was found to be Mn ≈ 1. Using (F.12), we can use our
previously determined values of τy/τ
∗ and ηpl/ηc to find that at Mn = 1, Bi = 0.08,
which indicates that this effect occurs at such low controllable force levels that it
203
will have an insignificant effect on MR device performance.
The relation between Bingham and Mason number can also be used to inform
the MR fluid design process. For example, let us assume we have a damper with
a given device geometry, and we seek to design a fluid such that the damper has a
high maximum damping force at maximum field (τy/τ
∗), a large Bingham number
(Mn∗), and a sufficient sedimentation time [52]. For maximizing τy/τ ∗, we can
seek to increase yield stress by increasing M or by raising τy/τ
∗ itself. To increase
M , one option is to use particles made out of novel, highly magnetic materials,
or alternatively to replace the carrier fluid with ferrofluids, with the intention of
increasing the particle magnetization [164], [165]. To raise τy/τ
∗, we can either
increase volume fraction, or more interestingly, use novel particle formulations, such
as fluids with nanowires, nonmagnetic particles or differently shaped particles [46],
[166]–[169]. Once the particle geometry and type are chosen, choosing the volume
fraction is a matter of managing the trade off between yield stress and viscosity, as
raising volume fraction raises ηpl/ηc faster than τy/τ
∗, reducing Mn∗. So if we seek
to maximize controllable force, the only remaining free variable in (F.12) is ηc in
Mason number. The final constraint, settling time, is expected to be dependent on
the carrier fluid viscosity and density, and particle geometry, so for our case of fixed
particle formulation, long settling times (large ηc) must be balanced against large
controllable force ratios (small ηc).
204
F.7 Conclusion
We demonstrated that Bi ∝ Mn−1, or Bi×Mn is a constant, in magnetorheo-
logical fluids, both theoretically and experimentally, resulting in Eq. F.12, a simple
algebraic relation. Theoretically, we demonstrated that this behavior arises because
microscopic forces and macroscopic forces are linearly related, and experimentally
validated this relationship through measurements of apparent viscosity on a high
shear rate Searle cell-type rheometer. It was shown that that the relationship between
Bingham number and Mason number depends on two nondimensional quantities,
τy/τ
∗ and ηpl/ηc, and we demonstrated that these ratios represent fundamental
fluid properties that are experimentally and analytically accessible. These ratios
also define the critical Mason number, Mn∗, through Eq. F.25. In order to identify
these quantities, fluid stress measurements were placed in the form of τ/τ ∗ vs. Mn,
allowing for easy identification of τy/τ
∗ and ηpl/ηc.
Finally, this relation was used to examine a Mason number corresponding to
a theoretical upper limit on the performance of MR fluids, and showed that this
Mason number corresponded to Bingham numbers outside the operational range for
experimental results. Finally, the MR fluid design process was looked at through the
lens of Bi ∝ Mn−1, showing how current approaches for novel fluids fall within the
recommendations made by the analytical methods in this paper.
205
F.8 Acknowledgements
S.G. Sherman was supported by a National Science Foundation Graduate
Research Fellowship under Grant No. DGE1322106 during the course of this research.
A.C. Becnel was supported by a National Defense Science and Engineering Fellowship.
206
Appendix G: The high shear rate chain limit
This is a reproduction of “Performance of magnetorheological fluids beyond
the chain based shear limit” by Stephen G. Sherman and Norman M. Wereley, and
published in the Journal of Applied Physics [54].
G.1 Introduction
Magnetorheological (MR) fluids consist of magnetizable particles suspended
in a carrier fluid. Upon the application of the magnetic field, the fluid develops a
field controllable resistance to stress, the MR effect. This rapidly alterable, field
controllable resistance to stress has made MR fluids popular for use in semi-active
damping applications. This ‘MR’ effect is typically attributed to the formation of
particle chains in the fluid.
Martin and Anderson [55] observed that the length of stable chains is limited
by the shear rate, so that above a critical shear rate, no stable chains exist in an MR
fluid. Note that for a typical high volume-fraction MR fluid, the equilibrium state of
the particles is sheets and not chains. But if we accept that chain formations are the
appropriate model, and that particle chains generate resistance to stress in the fluid,
then the MR effect would be mitigated, or even nonexistent, for regimes in which no
207
stable chains can exist. Such a critical shear rate would provide an upper bound on
the shear performance for an MR fluid. However, this critical shear rate, determined
from a critical Mason number, is quite high and would correspond to a shear rate of
γ˙ = 200 000 s−1 for a typical MR fluid at 20% of its saturation magnetization.
Measurements of fluid performance at high shear rates are quite difficult. Flow
mode rheometers (e.g. a capillary rheometer or slit rheometer) can achieve shear
rates in excess of 250,000s−1, but these devices induce a varying shear rate across the
gap, and at high flow velocities, the particles can exit the magnetized flow volume
before the chains have had a chance to form, thereby masking the existence of chain
restructuring [12]. Rotating parallel disk rheometers induce a varying shear rate in
the MR fluid along the disk radius, and have the tendency to expel fluid from the
device at high shear rates. Searle-cell magnetorheometer measurements have been
limited to shear rates of 25 000 s−1 [37].
In this study, we examine the behavior of MR fluids in this high shear rate
region through a high performance simulation model [19], [41] based on the work of
Klingenberg [34], allowing simple direct measurements of particle structure and fluid
stress response. The critical Mason number for a finite-length chain of particles in
shear flow is derived, with corrections for long-range forces and the use of the finite-
stiffness collision model which is used in our simulations. Using direct simulations,
observations are made on how the shear limit affects the static and dynamic yield
stress for a range of low volume-fraction fluids.
208
Figure G.1: Reference frames for the particle system.
G.2 Theory
The MR fluid model follows Martin and Anderson [55] and Klingenberg et al.
[34], where uniformly sized spherical particles act as point dipoles aligned with the
external field, H. Each particle is a sphere with a diameter σ, and corresponding
moment m = 1
6
piMσ3, with magnetization M = 3βH. The force on dipole i from j
is
Fmij = F0
1
(rij/σ)4
[(3 cos2 θij − 1)er + sin(2θij)eθ], (G.1)
where F0 = (3µ0mimj)/(4piσ
4), and the other quantities are shown in Fig. G.1. This
force can be turned into a reference stress, τ ∗ = F0/σ2 = piµ0M2/48.
The particles are assumed to be moving through an implicit carrier fluid with
viscosity ηc. The fluid is moving with velocity vc = γ˙yex, where γ˙ is the shear
rate. Particles experience drag Fd = 3piσηc(vc − vi). This drag can be used in the
simulation code to neglect particle accelerations, such that vi = Fi/3piηcσ+v∞. The
209
particle component of stress on the fluid is
τxy = − 1
V
N∑
i=1
(Fi · ex)(yi − ymid). (G.2)
At this point it is useful to define the Mason number[35], the ratio of particle
viscous forces to particle magnetic forces, here defined as
Mn =
3piηcσ
2γ˙
F0
= 144
ηcγ˙
µ0M2
. (G.3)
Chains rupture when shear forces exceed the magnetic forces holding the chain
together. As in [55], it is assumed that (1) chains are linear, (2) particles are held
separate by an infinitely stiff collision force, and (3) long range magnetic forces are
negligible. For a finite chain of total length S, the interparticle forces reach their
maximum value at the chain center. Solving for the force equilibrium yields
2 tan θ = fMn (G.4)
3 cos2 θ − 1
sin θ cos θ
≥ fMn, (G.5)
where f =S2/8 when S is even and f = (S2 − 1)/8 when S is odd. This equation
has a solution of a maximum angle of θ= cos−1(
√
3/5) = 39.23◦, and for each S a
limiting Mason number above which no chains of that length can form. The S=2
limiting Mason number, that is, the critical Mason number, above which no stable
chain exists, is Mn = 3.266.
210
However, in order to simulate these particles we need a repulsion force charac-
terized by a finite stiffness to separate the particles. The standard repulsion force is
Fij = −2F0 exp
[
−k
(rij
σ
− 1
)]
er. (G.6)
Solving for the force equilibrium yields,
2 tan θ = fMn r5 (G.7)
cos θ =
√
3
5
+
2
5
r4 exp[−k(r − 1)], (G.8)
which can be solved for the equilibrium positions of the particles. Two solutions
exist to these equations, a close solution and a far solution, with the close solution
representing the stable particle chain equilibrium point. For a value of k=50, with
the equilibrium point occurring at θc= 36.43◦ and rc = 1.046σ, the critical Mason
number for S = 2 is Mnc = 2.364, which is 72% of the infinitely stiff value.
For chains of length S > 3, the model must be corrected for long range forces
on the particles. Accounting for the forces from the upper half of the chain on the
lower half, we obtain the long range correction factor,
g(S) =
floor(S/2)∑
i=1
ceil(S/2)∑
j=1
1
(i+ j − 1)4 . (G.9)
Let ζ be the Reimann zeta function. In the limit of large S, g approaches ζ(3), as
seen in [55]. Then, for longer chains, the limiting Mason number for a given chain
length S is given by Mn∗(S) = gMnc. Another higher-order effect is that particle
211
chains are not straight lines; the nonlinear shape increases the hydrodynamic force
on the chain, reducing the critical Mason number. While Martin [55] developed
a correction for this change in shape, simulation indicates that this correction is
unnecessary for short chains.
G.3 Results
To evaluate what occurs as the stable particle chain size decreases, the conditions
of the simulation framework developed in Klingenberg [34] are used. The simulation
is conducted for k = 50, in a cubic volume of height h = 62.5σ. To ensure that
chains form in the simulation, not sheets, low fluid volume fractions, ranging from
φ = 0.005 to 0.05, are used. Force interactions are neglected for particle seperations
of r≥ 4σ. To capture the effects the high shear chain shortening, the static and
dynamic yield stresses of the fluid are measured for Mason numbers ranging from
0.13 to 5.2, corresponding to a range of Mn numbers from the S=13 limit to more
than twice that of the S= 2 limit. Static stress τs is the smoothed peak value of
stress, and dynamic stress τd is the steady state average.
The static yield stress is plotted versus Mason number in Fig. G.2. Vertical
lines denote the critical Mason numbers for S=2, 3 and 4. At each of these critical
Mason numbers, discontinuities in stress can be observed, with the stress decreasing
across the discontinuity as Mason number increases. These discontinuities also
support the conclusion that chains in the fluid support the resistance to stress, as
if the chains become shorter, and thus weaker, the ability of the MR fluid to resist
212
0 1 2 3 4 5
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Mason Number
S
ta
ti
c
S
tr
e
ss
,
τ
s/
τ
∗
 
 
φ
0.005
0.01
0.02
0.03
0.04
0.05
S=4
S=3 S=2
Figure G.2: Static stress with lines indicating the limiting Mason number for S=2,
3 and 4.
stress decreases. Changes in stress are not seen for S > 4, as the long-distance
strengthening factor g(4) ≈ 0.95ζ(3), so that any further strengthening will be small
and hard to observe from the data.
Dynamic yield stress is plotted versus Mason number in Fig. G.3. The results
are similar to those of the static stress for very low volume fractions, φ ≤ 0.01. Step
reductions in dynamic stress can be observed at the limiting Mason numbers as
Mason number increases. However, at higher volume fractions, the behavior is quite
different, with stress increasing up to the S = 3 limit, and no distinct changes in the
stress profile at the various limiting Mason numbers. It should also be noted that
as the Mason number approaches the S = 2 value, the static yield stress is rapidly
falling and approaching the value of the dynamic yield stress, until at S = 2, the
dynamic and static yield stresses are nearly identical.
To gain insight into these behaviors, Fig. G.4 plots the mean particle cluster
213
0 1 2 3 4 5
0
0.005
0.01
0.015
0.02
0.025
0.03
Mason Number
D
y
n
a
m
ic
S
tr
es
s,
τ
d
/
τ
∗
 
 
φ
0.005
0.01
0.02
0.03
0.04
0.05
S=2S=3S=4
Figure G.3: Dynamic stress with lines indicating Mn∗ for S=2, 3 and 4.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51
2
3
4
5
6
10
20
30
40
50
60
100
Mason Number
C
lu
st
e
r
S
iz
e
,
S
 
 
φ
0.005
0.01
0.02
0.03
0.04
0.05
S=4 S=3 S=2
Figure G.4: Steady state chain length with dashed vertical lines indicating Mn∗ for
S=2, 3 and 4. The solid line indicates the predicted value of cluster size.
214
size S versus Mason number. Particles i and j are considered to be in a cluster
if rij < 1.05σ. For the very low volume fraction fluids, φ < 0.01, there are clear
steps in the cluster size at at each of the limiting Mason number values for S=2,
3, and 4, confirming that chains above the limiting size are breaking. The mean
cluster size is below its predicted value because of a significant number of solitary
particles in the simulation. There are also longer, unstable chains that form and
break, with more of these unstable chains occurring as the volume fraction increases.
At φ = 0.02, the number of smaller and larger chains is roughly balanced, and the
results closely match our prediction. For φ > 0.02, the mean cluster size is larger
than what the chain-based model predicts. Here, visual inspection reveals that
instead of the particles forming chains in the fluid, they are forming small, sparsely
filled sheets. These sheet formations allow a larger cluster size, and explain why,
for particle volume fractions φ ≥ 0.02, both cluster size and stress measurements
diverge from predictions based on particle-chain models.
G.4 Conclusions
Changes that occur in stress and particle cluster size when the Mason number
exceeds the limit for stable chains were investigated. Above the critical Mason
number and very low volume fractions φ ≤ 0.01, sudden or step reductions in
the fluid stress and particle cluster size occurred at analytically predicted limiting
Mason numbers. These results validated both theoretical predictions [55] and the
simulation developed in this study. For volume fractions ranging from 1–5%, high
215
shear-rate reductions of shear stress were observed, but no step changes occurred at
the predicted intervals, which can be attributed to the formation of particle sheets,
as opposed to particle chains, in the fluid. Reductions were observed in mean particle
cluster size at the predicted limiting Mason numbers. However, it is cautioned that
the step changes in shear stress at discrete limiting Mason numbers only occur for
low-volume-fraction fluids in which the particles form chains. In higher particle
volume fraction fluids, where the particles instead form sheets, insights from particle
chain formation may be misleading.
216
Appendix H: Chain dynamics response time
H.1 Introduction
Magnetorheological fluid devices are used as controllable vibration and shock
absorbers, and in these scenarios, rapid device response is needed. The time for
an actuator to respond to a control input is the response time, and typical magne-
torheological devices have response times of milliseconds, but this is a system level
property, comprised of many elements. The precise response time of the fluid itself is
not specifically known, but Laun and Gabriel [64] performs the best measurements
available, and find response times of tr = 1− 3 ms.
Most response-time measurements are of devices and systems, and thus in-
clude many confounding factors. The largest component is the response time of the
magnetic circuit, caused by its inductance. This response time can be significantly
reduced through a power supply with feedback control [106]. There is also delay
between the current applied to the magnetic circuit and the magnetic-field response,
caused by eddy currents [64]. This delay can be estimated with a magnetics FEA
program, and mitigated through material selection. Koo et al. [62] show the im-
portance of device compliance in response time, and that increasing piston speeds
will eliminate the effects of compliance. They also measure an increasing circuit
217
response time with increasing applied current, but a decreasing force response time
with applied current. Sahin et al. [63] conducts benchtop measurements of flow
mode response time to a step current at a constant flow rate, and finds response
times from 10 to 20 ms, with response time decreasing with increased applied current.
This much larger than the values of Laun, but we show here that once the fluid
dynamic response time is included, the response time falls into line with shear mode
measurements.
Shear-mode response-time measurements offer a near direct measurement of
the rheological response time. Kikuchi et al. [170] examine the response time to
a step current, reporting results in terms of response shear, γr = trγ˙, and Mason
number, an approach we adopt. Despite opting to include magnetic circuit delays,
they report a response shear of γr = 1− 5. An alternative is to obtain response time
from a sinusoidally varying field. Tests on fluids with nanoparticles [171] estimate
tr < 5 ms at γ˙ = 28 s
−1 and B = 0.05 T, or Mn ≈ 10−2, γr = 0.14. Highly refined
parallel-plate measurements were conducted by Laun and Gabriel [64], considering
both sinusoidal and step response times at B = 1 T, γ˙ = 100 s−1, and found 63.2%
response times of 3.2 ms under a sinusoidal current, 2.8 ms to an applied step current,
and 1.8 ms when current is removed, corresponding to Mn ≈ 3× 10−4 and γr ≈ 0.3.
In this paper, we consider two elements of the response time: the rheological
response time of the fluid, and the response time caused by the fluid dynamics. The
rheological response time is straightforward: it is the time it takes for a stress to
develop in response to an applied field. In flow mode, the change in rheology changes
the velocity profile, causing a fluid dynamic response time. This fluid-dynamic
218
response time is largely uninvestigated for controllable yield stress fluids, so we use
the existing Bingham plastic literature [115] to develop estimates for various flow
regimes. Switching focus to the rheological response time, we use a particle-dynamics
simulation to show that the rheological response time is primarily dependent on the
shear rate, and will have response shear γr < 1 for any Mason number. This fixed
shear strain response allows us to neglect rheological time domain effects for most
large displacement engineering problems, a useful result.
H.2 Background
Magnetorheological fluids are typically modeled as Bingham plastics. In the
typical Bingham plastic model for MR fluids, τ = τy+η∞γ˙, where τy is magnetic field
dependent, and η∞ is constant. The fundamental nondimensional number governing
the bulk flow of Bingham plastics is the Bingham number,
Bi =
τy
η∞γ˙c
=
τy h
η∞v0
. (H.1)
The characteristic shear rate, γ˙c = v0/h, is an inverse characteristic time scale, where
v0 and h are the characteristic speed and length scale of the system.
To capture a rheological response time, we need a time dependent rheological
model. The most basic time dependent rheological model is the Maxwell model [172],
which when expressed in a fixed frame is
τ + λ
∂τ
∂t
= −ηγ˙
219
with a fixed relaxation time scale λ. However, correctly handling time dependence is
complicated and we wish to avoid it. Our basic observation is that in steady shear
experiments, the response time does not appear to be a constant fluid property nor
does it scale with the particle dynamic time scale – instead the response time appears
to scale with γ˙−1. This allows us to treat it as a response strain, a spatial effect.
But before we determine the fundamental rheological response time, we need
to characterize the response time of the fluid system.
H.3 Fluid Dynamics
The key feature of fluid dynamic response times is that they are independent
of velocity. For Newtonian fluids, Schlichting [114, p. 85] showed that fluid response
time has a dimensionless time scale, Tr, where Tr = ηtr/ρh
2, with tr the physical
time scale. From Schlichting, startup of Newtonian shear flow, the fluid will reach
90% convergence to steady state when Tr ≈ 0.15. The Newtonian solution forms an
upper bound for the controllable fluid response time, and here using typical values
for MR fluid, η∞ = 0.1 Pa s, ρ = 3 g/cm3, h = 1 mm, indicates that tr < 5 ms. For
Bingham plastics, a reasonable estimate for response time is Tr = 0.15/(1 + Bi),
which is the Newtonian result using η instead of η∞.
However, the relevant model problem for controllable fluid response time is not
startup shear flow, but a fluid under steady shear which has a field suddenly applied.
In this case, the steady state velocity profile is the same both with and without a
yield stress, so there is no response time associated with a change in velocity profile.
220
Then, any measured response time will be the rheological response time, the subject
of the next section.
In flow mode, there are two benchmark problems, a stationary flow which is
suddenly accelerated, and the case of a flow that suddenly has magnetic field applied.
For the impulsively accelerated flow, the response time can be approximated by
classic startup flow, where a stationary Bingham plastic has a constant pressure
applied. This has been worked extensively [115], and we take the analytical 1-D
solutions and solve for a 90% midline velocity response time. Using their solutions,
response time can be fit Tr = 0.235/1 + 0.2Bi. For a typical high speed MR fluid
device, Bi > 10, Tr < 0.1, so tr < 3 ms.
We can also consider the response time of a steady pressure driven Newtonian
flow that suddenly develops a yield stress. Here, the fluid dynamic response time is
non-zero, as the steady state profile changes with the imposition of yield stress. To
our knowledge, no correct analysis exists for this response time. However, Newtonian
startup flow should be a conservative estimate for the flow mode step-field response
time, and will be generally applicable.
For experimental validation of the startup-flow response time estimate, Sahin
et al. [63] measure response times from 22 ms to 12 ms, when corrected for magnetic
field effects, with fluid response time decreasing as field strength is increased. Using
Tr = 0.235/(1 + 0.2 Bi), Bi ≈ 30 and h = 3 mm, tr ≈ 10 ms, around half the fluid
response time. However, minimal fluid properties are reported, so this can only serve
as a loose estimate. However, our claim is that once the fluid dynamic response time
is accounted for, flow mode measurements begin to approach shear mode response
221
times, which it does here.
H.4 Rheology
The basic idea of a controllable fluid rheological response time is that it takes
some period of time for the yield stress inducing particle structure to form, and this
microstructure response time causes the macroscopic stress response time. Here, we
claim that the rheological response time does not scale with the particle dynamics
response time, but with the shear rate.
In the classic acceleration free Klingenberg model [34], here modified for
particles with non-uniform sizes, the dynamics are governed by:
D0σ
F0
Di
D0
x˙i
σ
=
N∑
j=1
j 6=i
[
Fmij + F
c
ij
]
+ Mn
Di
D0
yi
σ
ex
Fmij =
Fm
F0
(3 cos2 θij − 1)er + sin 2θijeθ
(rij/(ai + aj))4
Fcij = −2
Fm
F0
exp
[
−k
(
rij
ai + aj
− 1
)]
er
Fm =
3µ0mimj
4pi(ai + aj)4
F0 =
pi
48
µ0M
2σ2
Di = 6piηcai D0 = 3piηcσ
mi = 4/3pia
3
iM τyx =
N∑
i=1
(Fi · ex)yi
V
.
Here, σ is the characteristic length scale, which is chosen to be the median
particle diameter, k is the collision stiffness factor, ηc is the carrier fluid viscosity,
individual particle radius ai, rij the distance vector between particles i and j, and
M the magnetization. In this formulation, the Mason number is Mn = D0σγ˙/F0 =
222
144ηcγ˙/µ0M
2 and there is an associated particle dynamics time scale, D0σ/F0 =
144ηc/µ0M
2. Then, if the microstructure causes the yield stress, and then rheological
response time will scale with the particle dynamics time scale or ∝ 1/M2. A brief
examination of the literature shows no direct investigations of the field dependence
of response time,[63], [170], but the measurements we do have indicate only weak
field dependence once fluid dynamic terms are corrected for. Again, for direct shear
measurements, Laun and Gabriel [64] performs all measurements between 0 and 1 T
for both sinusoidal and step currents, and demonstrate that the transient can be
described very accurately by one fixed response time, indicating that the measured
response time is field independent. This is expected, as the particle dynamics response
time is very short, as 144ηc/µ0M
2 ≈ 0.5 µs for a typical fluid with ηc = 0.01 Pa s
and particles near saturation M = 1500 kA/m. But if we consider that the response
time is caused by the other time scale in the fluid, γ˙−1, the dimensionless time
scale becomes a shear, γ = trγ˙, and we observe that all high quality shear-mode
response-time experiments have response shears with γ < 1.
This argument is made more powerful, as a response shear instead of time has
been predicted by classic particle chain models. These models work by assuming
the chains form instantaneously, and then are deformed under shear, with the shear
deformation causing the shear stress response. The assumption of instantaneous
chain formation can be justified from the Mason number, as typical experiments have
Mn < 10−2, so the particle dynamics response time is necessarily much shorter than
the shear rate, and the instantaneous chain formation concept is thus reasonable.
Typical analytical chain models will give shear stress as a function of chain angle, and
223
where the angle is equivalent to a shear strain. The maximum in this relationship
defines a response strain. Ginder and Davis [65] looks at partially magnetically
saturated particle chains, and find γr ≈ .45. Bossis et al. [163] reports γr ≈ 0.1 using
FEM models and experimental measurements of single width particle chains. Martin
and Anderson [55] show maximum stress occurs at γ ≤√2/5 or γ ≤ 0.63 for dipolar
single width chains.
This response time as shear strain argument is bolstered by experiments showing
the onset of yielding occurring at a fixed strain. Small amplitude oscillatory shear
(SAOS) measurements can be used to formulate a Maxwell like fluid model with
compliance at small strains [173], and such a model will contain a response time. An
investigation into yielding process of MR fluids under SAOS through a variety of
methods[174] concludes that flow and steady state shear stresses start at strains of
γr ≈ 1.
If we want to go beyond simple chain models, we can directly examine the
rheological response time for complex structures by performing a particle dynamics
simulation undergoing simultaneous structure formation and shearing. To do this,
we use the previously described dynamics, with force interactions cut off at 4(ai+aj).
The upper and lower walls are given hard wall boundary conditions, and the sides are
given periodic boundary conditions. The only inputs to the system are the Mason
number Mn, collision stiffness k, and the initial particle distribution. The particles are
given a log-normal radius distribution, with all cases having a distribution parameter
of σLN = 0.2. The use of a log normal distribution instead of a uniform particle
distribution will affect the onset of yielding behavior, but is a more realistic choice
224
[41], [42]. These equations are integrated using an adaptive third order integration
scheme [175], with integration error constrained to 10−3σ, which improves solver
stability and accuracy at large time steps.
We start the particles in random positions, relax the configuration until all
particles all well separated, and then instantly apply a magnetic field and shear rate.
At this point the particles begin to form chains and, from here we record the stress
and use the stress history to identify the response time. We define the response time
as when shear stress, τyx reaches 90% of the maximum recorded stress value.
Figure H.1a plots simulated response time scaled by the particle dynamics time
scale for volume fractions φ = 0.2 to φ = 0.4 in a cubic volume of L = 50σ at k = 100,
and shows a strongly decreasing response time with increasing Mason number. We
also see a sudden drop in response time at Mn ≈ 2, near the critical Mason number
where no chains can form, Mn=3.2 [54], [55]. Newtonian fluids should have a very
fast response time, so this result confirms that once there are no chains, the fluid
becomes Newtonian. However, at large Mason numbers, where magnetic forces are
small, the isolated particle drag model means that the experimental relevance of
these results is questionable.
Alternatively, Fig. H.1b plots the same data using the characteristic time scale
of the fluid dynamics, the shear rate, γ˙−1, which gives us a response shear, γr = trγ˙.
This presentation allows validation against the previously discussed results [55], [64],
[65], [163], [171], where reported values of γr have been scaled to a 90% response time.
Then in this time scale perspective, we observe that 0.2 < γr < 1 for five orders of
magnitude in variation of Mason number, indicating that it holds roughly constant.
225
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
Mason Number
10
-4
10
-2
10
0
10
2
10
4
R
es
p
o
n
se
T
im
e
t r
/
(1
4
4
η
/
µ
0
M
2
)
φ=0.20
φ=0.25
φ=0.30
φ=0.35
φ=0.40
(a) Tr
10
-4
10
-2
10
0
10
2
Mason Number
0
0.2
0.4
0.6
0.8
1
1.2
R
es
p
o
n
se
S
h
ea
r
S
tr
a
in
,
γ
r
φ=0.20
φ=0.25
φ=0.30
φ=0.35
φ=0.40
Laun (2007)
Ginder (1994)
Martin (1996)
Kormann (1996)
Bossis (2002)
(b) γr
Figure H.1: Simulated dimensionless response time as a function of volume fraction.
226
10
-4
10
-2
10
0
10
2
Mason Number
0
0.2
0.4
0.6
0.8
1
1.2
R
e
sp
o
n
se
S
h
e
a
r
S
tr
a
in
,
γ
r
L=30, k=100
L=40, k=100
L=50, k=100
L=60, k=100
L=50, k=200
Pinned
Figure H.2: Effects of simulation parameters on γr.
On this same graph, we can validate against the previously listed experimental and
theoretical results, which have 0.1 < γr < 0.7. While there is significant spread,
measurements and models of response time are difficult to make, but critically, all
results indicate that γr < 1.
We also note that at low Mason numbers, response time is significantly affected
by choice of simulation parameters. Figure H.2 shows the effects of model parameters:
collision stiffness k, total simulation length scale L, and boundary conditions. For
stiffer values of k, response shear shows no change. However, wall slip boundary
conditions and the size of the simulation volume has a substantial effect at low Mason
numbers. The response strain has a peak at low Mason numbers, with the peak
moving towards lower Mason numbers with increasing simulation size. At Mason
numbers below the peak, the response strain rapidly falls off, as does shear stress.
Martin and Anderson show a similar shear angle falloff for isolated chains with
interacting with frictionless walls [55], caused by the walls limiting chain length, and
227
thus shear stress. They show that by switching to boundary conditions with friction,
they show that at low Mn, friction raises the maximum stable chain angle/shear.
We simulate friction through pinned boundary conditions at L = 50σ and k = 100,
and we with pinned and unpinned results identical at Mn > 10−2, but at low Mn,
pinned simulations have a constant response strain. Since we use 105 particles versus
the typical 103, this also indicates that pinning the particles is essential in small
simulations for eliminating simulation size dependence. We also note that using
pinned particles keep wall shear stresses constant from Mn = 10−4 to 100. Overall,
the parameter study confirms our basic result: response time scaling with shear rate,
not the particle dynamics time scale.
Static chain models, dynamic particle models and experimental measurements
all agree that maximum force will be reached by γ < 1. This implies that rising device
speeds will cause a commensurate decrease in rheological response time, indicating
that response time won’t pose an issue for high speed devices. Instead, a fix response
strain means an apparent shortening of the duct, as the fluid will require some
displacement to develop a stress.
H.5 Conclusion
We claim that magnetorheological fluids have no response time, but instead a
response shear. First we show that fluid flow has an associated fluid dynamic response
time, and removing the fluid dynamic response time brings flow mode measurements
into line with shear mode experiments. We then examined the rheological response
228
time by simulating the particle dynamics under instantly applied shear and field,
and showed that the simulated response time scales with γ˙−1, and showed that a
response shear of γr < 1 agrees with both experimental measurements and basic
chain models. This response shear means that devices can operate at short time
scales, as long as the characteristic time scale of the fluid dynamics, γ˙−1c , is less than
the desired actuation time scale.
229
Bibliography
[1] A. Kareem, T. Kijewski, and Y. Tamura, “Mitigation of motions of tall
buildings with specific examples of recent applications,” Wind and Structures,
vol. 2, no. 3, pp. 201–251, 1999.
[2] D. Hrovat, P. Barak, and M. Rabins, “SemiActive versus Passive or Active
Tuned Mass Dampers for Structural Control,” Journal of Engineering Me-
chanics, vol. 109, no. 3, pp. 691–705, 1983. doi: 10.1061/(ASCE)0733-
9399(1983)109:3(691).
[3] G. W. Housner, L. A. Bergman, T. K. Caughey, A. G. Chassiakos, R. O.
Claus, S. F. Masri, R. E. Skelton, T. T. Soong, B. F. Spencer, and J. T. P.
Yao, “Structural Control: Past, Present, and Future,” Journal of Engineering
Mechanics, vol. 123, no. 9, pp. 897–971, 1997. doi: 10.1061/(ASCE)0733-
9399(1997)123:9(897).
[4] B. Spencer and M. Sain, “Controlling buildings: a new frontier in feedback,”
IEEE Control Systems Magazine, vol. 17, no. 6, pp. 19–35, 1997. doi: 10.
1109/37.642972.
[5] J. D. Carlson and M. R. Jolly, “MR fluid, foam and elastomer devices,”
Mechatronics, vol. 10, no. 4-5, pp. 555–569, 2000. doi: 10.1016/S0957-
4158(99)00064-1.
[6] W. M. Winslow, “Induced fibration of suspensions,” Journal of Applied
Physics, vol. 20, no. 12, pp. 1137–1140, 1949. doi: 10.1063/1.1698285.
[7] J. Rabinow, “The Magnetic Fluid Clutch,” Transactions of the American
Institute of Electrical Engineers, vol. 67, no. 2, pp. 1308–1315, 1948. doi:
10.1109/T-AIEE.1948.5059821.
[8] J. D. Carlson, D. M. Catanzarite, and K. A. S. Clair, “Commercial Magne-
torheological Fluid Devices,” International Journal of Modern Physics B, vol.
10, no. 23-24, pp. 2857–2865, 1996. doi: 10.1142/S0217979296001306.
[9] A. L. Browne, J. D. McCleary, C. S. Namuduri, and S. R. Webb, “Impact
Performance of Magnetorheological Fluids,” Journal of Intelligent Material
Systems and Structures, vol. 20, no. 6, pp. 723–728, 2008. doi: 10.1177/
1045389X08096358.
[10] G. J. Hiemenz, Y.-T. Choi, and N. M. Wereley, “Semi-Active Control of Ver-
tical Stroking Helicopter Crew Seat for Enhanced Crashworthiness,” Journal
of Aircraft, vol. 44, no. 3, pp. 90–90, 2007. doi: 10.2514/1.26492.
230
[11] F. D. Goncalves, “Characterizing the Behavior of Magnetorheological Fluids
at High Velocities and High Shear Rates,” PhD thesis, Viriginia Polytechnic
Institute, 2005, p. 103.
[12] F. D. Goncalves, M. Ahmadian, and J. D. Carlson, “Investigating the magne-
torheological effect at high flow velocities,” Smart Materials and Structures,
vol. 15, no. 1, pp. 75–85, 2006. doi: 10.1088/0964-1726/15/1/036.
[13] M. Mao, “Adaptive Magnetorheological Sliding Seat System for Ground
Vehicles,” PhD thesis, University of Maryland, 2011, p. 276.
[14] M. Mao, W. Hu, Y. T. Choi, N. M. Wereley, A. L. Browne, and J. C. Ulicny,
“Experimental validation of a magnetorheological energy absorber design
analysis,” Journal of Intelligent Material Systems and Structures, vol. 25, no.
3, pp. 352–363, 2013. doi: 10.1177/1045389X13494934.
[15] H. Sodeyama, K. Suzuki, and K. Sunakoda, “Development of Large Capacity
Semi-Active Seismic Damper Using Magneto-Rheological Fluid,” Journal of
Pressure Vessel Technology, vol. 126, no. 1, p. 105, 2004. doi: 10.1115/1.
1634587.
[16] A. C. Becnel, S. G. Sherman, W. Hu, and N. M. Wereley, “Squeeze strength-
ening of magnetorheological fluids using mixed mode operation,” Journal of
Applied Physics, vol. 117, no. 17, 2015. doi: 10.1063/1.4907603.
[17] M. Ocalan and G. H. McKinley, “Rheology and microstructural evolution in
pressure-driven flow of a magnetorheological fluid with strong particle-wall
interactions,” Journal of Intelligent Material Systems and Structures, vol. 23,
no. 9, pp. 969–978, 2012. doi: 10.1177/1045389X11429601.
[18] D. J. Klingenberg and C. F. Zukoski, “Studies on the steady-shear behavior
of electrorheological suspensions,” Langmuir, vol. 6, no. 1, pp. 15–24, 1990.
doi: 10.1021/la00091a003.
[19] S. G. Sherman, D. A. Paley, and N. M. Wereley, “Parallel Simulation of
Transient Magnetorheological Direct Shear Flows Using Millions of Particles,”
IEEE Transactions on Magnetics, vol. 48, no. 11, pp. 3517–3520, 2012. doi:
10.1109/TMAG.2012.2201214.
[20] H. G. Lagger, C. Bierwisch, J. G. Korvink, and M. Moseler, “Discrete element
study of viscous flow in magnetorheological fluids,” Rheologica Acta, vol. 53,
no. 5-6, pp. 417–443, 2014. doi: 10.1007/s00397-014-0768-0.
[21] H. G. R. Lagger, “Particle-based simulation of sheared magnetorheological
fluids,” PhD thesis, Albert Ludwigs University of Freiburg, 2014.
[22] H. G. Lagger, T. Breinlinger, J. G. Korvink, M. Moseler, A. Di Renzo, F. Di
Maio, and C. Bierwisch, “Influence of hydrodynamic drag model on shear stress
in the simulation of magnetorheological fluids,” Journal of Non-Newtonian
Fluid Mechanics, vol. 218, pp. 16–26, 2015. doi: 10.1016/j.jnnfm.2015.01.
010.
231
[23] D. J. Klingenberg, F. van Swol, and C. F. Zukoski, “The small shear rate
response of electrorheological suspensions. I. Simulation in the pointdipole
limit,” The Journal of Chemical Physics, vol. 94, no. 9, p. 6160, 1991. doi:
10.1063/1.460402.
[24] D. J. Klingenberg, F. van Swol, and C. F. Zukoski, “The small shear rate
response of electrorheological suspensions. II. Extension beyond the point
dipole limit,” The Journal of Chemical Physics, vol. 94, no. 9, pp. 6170–6178,
1991. doi: 10.1063/1.460403.
[25] H. Ly, F Reitich, M. Jolly, H. Banks, and K Ito, “Simulations of Particle
Dynamics in Magnetorheological Fluids,” Journal of Computational Physics,
vol. 155, no. 1, pp. 160–177, 1999. doi: 10.1006/jcph.1999.6335.
[26] K Han, Y. T. Feng, and D. R. J. Owen, “Three-dimensional modelling and
simulation of magnetorheological fluids,” International Journal for Numerical
Methods in Engineering, vol. 84, no. 11, pp. 1273–1302, 2010. doi: 10.1002/
nme.2940.
[27] R. T. Bonnecaze and J. F. Brady, “Dynamic simulation of an electrorheological
fluid,” The Journal of Chemical Physics, vol. 96, no. 3, p. 2183, 1992. doi:
10.1063/1.462070.
[28] M Parthasarathy and D. J. Klingenberg, “Electrorheology: Mechanisms and
models,” Materials Science and Engineering: R: Reports, vol. 17, no. 2, pp. 57–
103, 1996. doi: 10.1016/0927-796X(96)00191-X.
[29] D. J. Klingenberg and J. C. Ulicny, “Enhancing magnetorheology,” Inter-
national Journal of Modern Physics B, vol. 25, no. 07, p. 911, 2011. doi:
10.1142/S021797921105847X.
[30] J. C. Ulicny, K. S. Snavely, M. A. Golden, and D. J. Klingenberg, “Enhancing
magnetorheology with nonmagnetizable particles,” Applied Physics Letters,
vol. 96, no. 23, p. 231 903, 2010. doi: 10.1063/1.3431608.
[31] Nvidia, NVIDIA CUDA C Programming Guide, 2011.
[32] K. W. Yung, P. B. Landecker, and D. D. Villani, “An Analytic Solution for the
Force Between Two Magnetic Dipoles,” Magnetic and Electrical Separation,
vol. 9, no. 1, pp. 39–52, 1998. doi: 10.1155/1998/79537.
[33] M Parthasarathy and D. J. Klingenberg, “A microstructural investigation
of the nonlinear response of electrorheological suspensions I. Start up of
steady shear flow,” Rheologica Acta, vol. 34, no. 5, pp. 417–429, 1995. doi:
10.1007/BF00396555.
[34] D. J. Klingenberg, F. van Swol, and C. F. Zukoski, “Dynamic simulation of
electrorheological suspensions,” The Journal of Chemical Physics, vol. 91, no.
12, p. 7888, 1989. doi: 10.1063/1.457256.
[35] D. J. Klingenberg, J. C. Ulicny, and M. A. Golden, “Mason numbers for
magnetorheology,” Journal of Rheology, vol. 51, no. 5, p. 883, 2007. doi:
10.1122/1.2764089.
232
[36] F Vereda, J de Vicente, J. P. Segovia-Gutie´rrez, and R Hidalgo-Alvarez,
“Average particle magnetization as an experimental scaling parameter for
the yield stress of dilute magnetorheological fluids,” Journal of Physics D:
Applied Physics, vol. 44, no. 42, p. 425 002, 2011. doi: 10.1088/0022-
3727/44/42/425002.
[37] A. C. Becnel, W. Hu, and N. M. Wereley, “Measurement of magnetorheological
fluid properties at shear rates of up to 25 000/s,” IEEE Transactions on
Magnetics, vol. 48, no. 11, pp. 3525–3528, 2012. doi: 10.1109/TMAG.2012.
2207707.
[38] A. C. Becnel, S. Sherman, W. Hu, and N. M. Wereley, “Nondimensional
scaling of magnetorheological rotary shear mode devices using the Mason
number,” Journal of Magnetism and Magnetic Materials, vol. 380, pp. 90–97,
2015. doi: 10.1016/j.jmmm.2014.10.049.
[39] J. de Vicente, D. J. Klingenberg, and R. Hidalgo-Alvarez, “Magnetorheological
fluids: a review,” Soft Matter, vol. 7, no. 8, p. 3701, 2011. doi: 10.1039/
c0sm01221a.
[40] S. G. Sherman, L. A. Powell, A. C. Becnel, and N. M. Wereley, “Scaling
temperature dependent rheology of magnetorheological fluids,” Journal of
Applied Physics, vol. 117, no. 17, p. 17C751, 2015. doi: 10.1063/1.4918628.
[41] S. G. Sherman and N. M. Wereley, “Effect of Particle Size Distribution
on Chain Structures in Magnetorheological Fluids,” IEEE Transactions on
Magnetics, vol. 49, no. 7, pp. 3430–3433, 2013. doi: 10.1109/TMAG.2013.
2245409.
[42] J. C. Ferna´ndez-Toledano, J. A. Ruiz-Lo´pez, R. Hidalgo-A´lvarez, and J. de
Vicente, “Simulations of polydisperse magnetorheological fluids: A structural
and kinetic investigation,” Journal of Rheology, vol. 59, no. 2, pp. 475–498,
2015. doi: 10.1122/1.4906544.
[43] F Vereda, J. P. Segovia-Gutie´rrez, J de Vicente, and R Hidalgo-Alvarez,
“Particle roughness in magnetorheology: effect on the strength of the field-
induced structures,” Journal of Physics D: Applied Physics, vol. 48, 15309
(11p), 2015. doi: 10.1088/0022-3727/48/1/015309.
[44] M. T. Lopez-Lopez, P. Kuzhir, and G. Bossis, “Magnetorheology of fiber
suspensions. I. Experimental,” Journal of Rheology, vol. 53, no. 1, p. 115,
2009. doi: 10.1122/1.3005402.
[45] P. Kuzhir, M. T. Lopez-Lopez, and G. Bossis, “Magnetorheology of fiber
suspensions. II. Theory,” Journal of Rheology, vol. 53, no. 1, p. 127, 2009.
doi: 10.1122/1.3005405.
[46] J de Vicente, J. P. Segovia-Gutie´rrez, E Andablo-Reyes, F Vereda, and R
Hidalgo-Alvarez, “Dynamic rheology of sphere- and rod-based magnetorheo-
logical fluids.,” The Journal of chemical physics, vol. 131, no. 19, p. 194 902,
2009. doi: 10.1063/1.3259358.
233
[47] H. M. Laun, C. Gabriel, and C. Kieburg, “Wall material and roughness effects
on transmittable shear stresses of magnetorheological fluids in plateplate
magnetorheometry,” Rheologica Acta, vol. 50, no. 2, pp. 141–157, 2011. doi:
10.1007/s00397-011-0531-8.
[48] B. Kavlicoglu, F. Gordaninejad, and X. Wang, “A Unified Approach for Flow
Analysis of Magnetorheological Fluids,” Journal of Applied Mechanics, vol.
78, no. 4, p. 041 008, 2011. doi: 10.1115/1.4003346.
[49] W. Li and X. Zhang, “The effect of friction on magnetorheological fluids,”
Korea-Australia Rheology Journal, vol. 20, no. 2, pp. 45–50, 2008.
[50] J. de Vicente and J. Ramı´rez, “Effect of friction between particles in the
dynamic response of model magnetic structures.,” Journal of colloid and
interface science, vol. 316, no. 2, pp. 867–76, 2007. doi: 10.1016/j.jcis.
2007.08.022.
[51] I. B. Jang, H. B. Kim, J. Y. Lee, J. L. You, H. J. Choi, and M. S. Jhon, “Role
of organic coating on carbonyl iron suspended particles in magnetorheological
fluids,” Journal of Applied Physics, vol. 97, no. 10, pp. 10–13, 2005. doi:
10.1063/1.1853835.
[52] J. D. Carlson, “What Makes a Good MR Fluid?” Journal of Intelligent
Material Systems and Structures, vol. 13, no. 7-8, pp. 431–435, 2002. doi:
10.1106/104538902028221.
[53] S. G. Sherman, A. C. Becnel, and N. M. Wereley, “Relating Mason number
to Bingham number in magnetorheological fluids,” Journal of Magnetism and
Magnetic Materials, vol. 380, pp. 98–104, 2015. doi: 10.1016/j.jmmm.2014.
11.010.
[54] S. G. Sherman and N. M. Wereley, “Performance of magnetorheological fluids
beyond the chain based shear limit,” Journal of Applied Physics, vol. 115, no.
17, 17B523, 2014. doi: 10.1063/1.4867963.
[55] J. E. Martin and R. A. Anderson, “Chain model of electrorheology,” The
Journal of Chemical Physics, vol. 104, no. 12, p. 4814, 1996. doi: 10.1063/1.
471176.
[56] H. Sodeyama, K. Sunakoda, H. Fujitani, S. Soda, N. Iwata, and K. Hata,
“Dynamic Tests and Simulation of Magneto-Rheological Dampers,” Computer-
Aided Civil and Infrastructure Engineering, vol. 18, no. 1, pp. 45–57, 2003.
doi: 10.1111/1467-8667.t01-1-00298.
[57] H. M. Laun, C. Gabriel, and C. Kieburg, “Twin gap magnetorheometer using
ferromagnetic steel plates: Performance and validation,” Journal of Rheology,
vol. 54, no. 2, p. 327, 2010. doi: 10.1122/1.3302804.
[58] D. Gu¨th and J. Maas, “Characterization and modeling of the behavior of
magnetorheological fluids at high shear rates in rotational systems,” Journal
of Intelligent Material Systems and Structures, p. 114, 2015. doi: 10.1177/
1045389X15577646.
234
[59] D Gu¨th, A Wiebe, and J Maas, “Design of shear gaps for high-speed and
high-load MRF brakes and clutches,” Journal of Physics: Conference Series,
vol. 412, no. 1, p. 012 046, 2013. doi: 10.1088/1742-6596/412/1/012046.
[60] X. Wang and F. Gordaninejad, “Study of magnetorheological fluids at high
shear rates,” Rheologica Acta, vol. 45, no. 6, pp. 899–908, 2006. doi: 10.1007/
s00397-005-0058-y.
[61] C. Gabriel and H. M. Laun, “Combined slit and plateplate magnetorheometry
of a magnetorheological fluid (MRF) and parameterization using the Casson
model,” Rheologica Acta, vol. 48, no. 7, pp. 755–768, 2009. doi: 10.1007/
s00397-009-0369-5.
[62] J.-H. Koo, F. D. Goncalves, and M. Ahmadian, “A comprehensive analysis of
the response time of MR dampers,” Smart Materials and Structures, vol. 15,
no. 2, pp. 351–358, 2006. doi: 10.1088/0964-1726/15/2/015.
[63] H. Sahin, F. Gordaninejad, X. Wang, and Y. Liu, “Response time of magne-
torheological fluids and magnetorheological valves under various flow condi-
tions,” Journal of Intelligent Material Systems and Structures, vol. 23, no. 9,
pp. 949–957, 2012. doi: 10.1177/1045389X12447984.
[64] H. M. Laun and C. Gabriel, “Measurement modes of the response time
of a magneto-rheological fluid (MRF) for changing magnetic flux density,”
Rheologica Acta, vol. 46, no. 5, pp. 665–676, 2007. doi: 10.1007/s00397-
006-0155-6.
[65] J. M. Ginder and L. C. Davis, “Shear stresses in magnetorheological fluids:
Role of magnetic saturation,” Applied Physics Letters, vol. 65, no. 26, p. 3410,
1994. doi: 10.1063/1.112408.
[66] S. S. Abdali, E. Mitsoulis, and N. C. Markatos, “Entry and exit flows of
Bingham fluids,” Journal of Rheology, vol. 36, no. 2, p. 389, 1992. doi:
10.1122/1.550350.
[67] E. Mitsoulis, S. Marangoudakis, M. Spyratos, T. Zisis, and N. a. Malamataris,
“Pressure-Driven Flows of Bingham Plastics Over a Square Cavity,” Journal
of Fluids Engineering, vol. 128, no. 5, p. 993, 2006. doi: 10.1115/1.2236130.
[68] A. J. Taylor, “Conduit flow of an incompressible, yield-stress fluid,” Journal
of Rheology, vol. 41, no. 1, p. 93, 1997. doi: 10.1122/1.550802.
[69] T. Min, H. G. Choi, J. Y. Yoo, and H. Choi, “Laminar convective heat transfer
of a Bingham plastic in a circular pipeII. Numerical approach hydrodynam-
ically developing flow and simultaneously developing flow,” International
Journal of Heat and Mass Transfer, vol. 40, no. 15, pp. 3689–3701, 1997. doi:
10.1016/S0017-9310(97)00004-5.
[70] G. C. Vradis, J. Dougher, and S. Kumar, “Entrance pipe flow and heat transfer
for a Bingham plastic,” International Journal of Heat and Mass Transfer, vol.
36, no. 3, pp. 543–552, 1993. doi: 10.1016/0017-9310(93)80030-X.
235
[71] E. J. Park, L. F. da Luz, and A. Suleman, “Multidisciplinary design opti-
mization of an automotive magnetorheological brake design,” Computers &
Structures, vol. 86, no. 3-5, pp. 207–216, 2008. doi: 10.1016/j.compstruc.
2007.01.035.
[72] D. a. Bompos and P. G. Nikolakopoulos, “CFD simulation of magnetorhe-
ological fluid journal bearings,” Simulation Modelling Practice and Theory,
vol. 19, no. 4, pp. 1035–1060, 2011. doi: 10.1016/j.simpat.2011.01.001.
[73] E. Gedik, H. Kurt, Z. Recebli, and C. Balan, “Two-dimensional CFD simu-
lation of magnetorheological fluid between two fixed parallel plates applied
external magnetic field,” Computers & Fluids, vol. 63, pp. 128–134, 2012. doi:
10.1016/j.compfluid.2012.04.011.
[74] Z. Parlak and T. Engin, “Time-dependent CFD and quasi-static analysis of
magnetorheological fluid dampers with experimental validation,” International
Journal of Mechanical Sciences, vol. 64, no. 1, pp. 22–31, 2012. doi: 10.1016/
j.ijmecsci.2012.08.006.
[75] W. A. Bullough, R. J. Atkin, S. Urang, T. G. Kum, C. Musch, and T.
Rober, “Two Dimensional Flow of an ESF: Experiment, CFD, Bingham
Plastic Analysis,” International Journal of Modern Physics B, vol. 15, no.
6-7, pp. 745–757, 2001. doi: 10.1142/S0217979201005222.
[76] J. Goldasz and B. Sapinski, “Nondimensional characterization of flow-mode
magnetorheological/electrorheological fluid dampers,” Journal of Intelligent
Material Systems and Structures, vol. 23, no. 14, pp. 1545–1562, 2012. doi:
10.1177/1045389X12447293.
[77] J. Go ldasz and B. Sapin´ski, “CFD Study of the flow of MR fluids,” in
Insight into Magnetorheological Shock Absorbers, Cham: Springer International
Publishing, 2015, pp. 117–130, isbn: 978-3-319-13232-7. doi: 10.1007/978-
3-319-13233-4.
[78] E. Mitsoulis, “Flows of viscoplastic materials: Models and computations,”
Rheology Reviews, vol. 2007, pp. 135 –178, 2007. doi: 10.1016/S0045-
7949(96)00167-8.
[79] T. C. Papanastasiou, “Flows of Materials with Yield,” Journal of Rheology,
vol. 31, no. 5, p. 385, 1987. doi: 10.1122/1.549926.
[80] N. M. Wereley and L. Pang, “Nondimensional analysis of semi-active elec-
trorheological and magnetorheological dampers using approximate parallel
plate models,” Smart Materials and Structures, vol. 7, no. 5, pp. 732–743,
1998. doi: 10.1088/0964-1726/7/5/015.
[81] B. O. A. Hedstro¨m, “Flow of Plastic Materials in Pipes,” Industrial &
Engineering Chemistry, vol. 44, no. 3, pp. 651–656, 1952. doi: 10.1021/
ie50507a056.
[82] G. W. Govier and K Aziz, The Flow of Complex Mixtures in Pipes. New York:
Van Nostrand Reinhold, 1972.
236
[83] D. J. Peel, R Stanway, and W. A. Bullough, “Dynamic modelling of an ER
vibration damper for vehicle suspension applications,” Smart Materials and
Structures, vol. 5, no. 5, pp. 591–606, 1996. doi: 10.1088/0964-1726/5/5/
008.
[84] A. N. Alexandrou, E. Duc, and V. Entov, “Inertial, viscous and yield stress
effects in Bingham fluid filling of a 2-D cavity,” Journal of Non-Newtonian
Fluid Mechanics, vol. 96, no. 3, pp. 383–403, 2001. doi: 10.1016/S0377-
0257(00)00199-3.
[85] D Bresch, E. D. Ferna´ndez-Nieto, I. R. Ionescu, and P Vigneaux, “Augmented
Lagrangian Method and Compressible Visco-plastic Flows: Applications to
Shallow Dense Avalanches,” in New Directions in Mathematical Fluid Me-
chanics, Basel: Birkha¨user Basel, 2009, pp. 57–89. doi: 10.1007/978-3-
0346-0152-8_4.
[86] R. W. Phillips, “Engineering applications of fluids with a variable yield stress,”
PhD thesis, University of California, Berkeley, 1969, p. 205.
[87] N. M. Wereley, J. Lindler, N. Rosenfeld, and Y.-T. Choi, “Biviscous damping
behavior in electrorheological shock absorbers,” Smart Materials and Struc-
tures, vol. 13, no. 4, pp. 743–752, 2004. doi: 10.1088/0964-1726/13/4/012.
[88] S. S. Chen, L. T. Fan, and C. L. Hwang, “Entrance region flow of the Bingham
fluid in a circular pipe,” AIChE Journal, vol. 16, no. 2, pp. 293–299, 1970.
doi: 10.1002/aic.690160224.
[89] V. L. Shah and R. J. Soto, “Entrance flow of a bingham fluid in a tube,”
Applied Scientific Research, vol. 30, no. 4, pp. 271–278, 1975. doi: 10.1007/
BF00386695.
[90] E. B. Bagley, “End Corrections in the Capillary Flow of Polyethylene,” Journal
of Applied Physics, vol. 28, no. 5, p. 624, 1957. doi: 10.1063/1.1722814.
[91] H. G. Weller, G Tabor, H Jasak, and C Fureby, “A tensorial approach
to computational continuum mechanics using object-oriented techniques,”
Computers in Physics, vol. 12, no. 6, pp. 620–631, 1998. doi: 10.1063/1.
168744.
[92] S. Patankar, Numerical heat transfer and fluid flow. CRC Press, 1980.
[93] P. Kuzhir, G. Bossis, V. Bashtovoi, and O. Volkova, “Effect of the orientation
of the magnetic field on the flow of magnetorheological fluid. II. Cylindrical
channel,” Journal of Rheology, vol. 47, no. 6, p. 1385, 2003. doi: 10.1122/1.
1619378.
[94] R. G. Owens and T. N. Phillips, Computational Rheology. London: Imperial
College Press, 2002, p. 417, isbn: 1-86094-186-9.
[95] F. M. White, Viscous Fluid Flow, Second. Boston: McGraw-Hill, 1991.
[96] W. Hanks and D. R. Pratt, “On the Flow of Bingham Plastic Slurries in Pipes
and Between Parallel Plates,” Society of Petroleum Engineering Journal, vol.
7, no. 04, pp. 342–346, 1967. doi: 10.2118/1682-PA.
237
[97] U. Lange, L. Richter, and L. Zipser, “Flow of Magnetorheological Fluids,”
Journal of Intelligent Material Systems and Structures, vol. 12, no. 3, pp. 161–
164, 2001. doi: 10.1106/PF05-DTU2-2QTD-28B6.
[98] H. Fujitani, H. Sodeyama, T. Tomura, T. Hiwatashi, Y. Shiozaki, K. Hata, K.
Sunakoda, S. Morishita, and S. Soda, “Development of a 400kN magnetorhe-
ological damper for a real base-isolated building,” in Smart Structures and
Materials 2003: Damping and Isolation, G. S. Agnes and K.-W. Wang, Eds.,
vol. 5052, 2003, pp. 265–276, isbn: 1111111111. doi: 10.1117/12.483810.
[99] S.-B. Choi, M.-H. Nam, and B.-K. Lee, “Vibration Control of a MR Seat
Damper for Commercial Vehicles,” Journal of Intelligent Material Systems and
Structures, vol. 11, no. December 2000, pp. 936–944, 2001. doi: 10.1106/AERG-
3QKV-31V8-F250.
[100] S.-B. Choi, M.-S. Seong, and S.-H. Ha, “Vibration control of an MR vehicle
suspension system considering both hysteretic behavior and parameter varia-
tion,” Smart Materials and Structures, vol. 18, no. 12, p. 125 010, 2009. doi:
10.1088/0964-1726/18/12/125010.
[101] F Gordaninejad, X Wang, G Hitchcock, K Bangrakulur, S Ruan, and M
Siino, “Modular High-Force Seismic Magneto-Rheological Fluid Damper,”
Journal of Structural Engineering, vol. 136, no. 2, pp. 135–143, 2010. doi:
10.1061/(ASCE)0733-9445(2010)136:2(135).
[102] Z. C. Li and J Wang, “A Gun Recoil System Employing A Magnetorheological
Fluid Damper,” Smart Materials and Structures, vol. 21, no. 10, p. 105 003,
2012. doi: 10.1088/0964-1726/21/10/105003.
[103] Z.-X. Li and L.-H. Xu, “Performance Tests and Hysteresis Model of MRF-04K
Damper,” Journal of Structural Engineering, vol. 131, no. 8, pp. 1303–1306,
2005. doi: 10.1061/(ASCE)0733-9445(2005)131:8(1303).
[104] A. Rodriguez, N. Iwata, F. Ikhouane, and J. Rodellar, “Modeling and Iden-
tification of a Large-Scale Magnetorheological Fluid Damper,” Advances in
Science and Technology, vol. 56, pp. 374–379, 2008. doi: 10.4028/www.
scientific.net/AST.56.374.
[105] J. W. Tu, J. Liu, W. L. Qu, Q. Zhou, H. B. Cheng, and X. D. Cheng, “Design
and Fabrication of 500-kN Large-scale MR Damper,” Journal of Intelligent
Material Systems and Structures, vol. 22, no. 5, pp. 475–487, 2011. doi:
10.1177/1045389X11399942.
[106] G. Yang, “Large-Scale Magnetorheological Fluid Damper for Vibration Miti-
gation,” PhD thesis, University of Notre Dame, 2001, p. 480.
[107] Y.-T. Choi, R. Robinson, W. Hu, N. M. Wereley, T. S. Birchette, A. O.
Bolukbasi, and J. Woodhouse, “Analysis and Control of a Magnetorheological
Landing Gear System for a Helicopter,” Journal of the American Helicopter
Society, vol. 61, no. 3, pp. 1–8, 2016. doi: 10.4050/JAHS.61.032006.
238
[108] M. Mao, W. Hu, Y.-T. Choi, N. M. Wereley, A. L. Browne, J. Ulicny, and
N. Johnson, “Nonlinear modeling of magnetorheological energy absorbers
under impact conditions,” Smart Materials and Structures, vol. 22, no. 11,
p. 115 015, 2013. doi: 10.1088/0964-1726/22/11/115015.
[109] H. J. Singh, W. Hu, N. M. Wereley, and W. Glass, “Experimental validation of
a magnetorheological energy absorber design optimized for shock and impact
loads,” Smart Materials and Structures, vol. 23, no. 12, p. 125 033, 2014. doi:
10.1088/0964-1726/23/12/125033.
[110] H. E. Merritt, Hydraulic Control Systems. New York: John Wiley and Sons,
1967.
[111] R. M. Sadri and J. M. Floryan, “Accurate Evaluation of the Loss Coefficient
and the Entrance Length of the Inlet Region of a Channel,” Journal of Fluids
Engineering, vol. 124, no. 3, p. 685, 2002. doi: 10.1115/1.1493813.
[112] K. Wilson, “Transitional and Turbulent Flows of Bingham Plastics,” Mineral
Processing and Extractive Metallurgy Review, vol. 20, no. 1, pp. 225–237, 2000.
doi: 10.1080/08827509908962474.
[113] M. Malin, “The Turbulent Flow of Bingham Plastic Fluids in Smooth Circular
Tubes,” International Communications in Heat and Mass Transfer, vol. 24,
no. 6, pp. 793–804, 1997. doi: 10.1016/S0735-1933(97)00066-3.
[114] H. Schlichting, Boundary-Layer Theory, 6th ed. McGraw-Hill, 1968.
[115] I. Dapra` and G. Scarpi, “Start-up of channel-flow of a Bingham fluid initially
at rest,” Rendiconti Lincei Matematica e Applicazioni, vol. 15, no. 2, pp. 125–
134, 2004.
[116] Y.-T. Choi and N. M. Wereley, “Comparative Analysis of the Time Response
of Electrorheological and Magnetorheological Dampers Using Nondimensional
Parameters,” Journal of Intelligent Material Systems and Structures, vol. 13,
no. 7-8, pp. 443–451, 2002. doi: 10.1106/104538902028557.
[117] C. T. Crowe, J. D. Schwarzkopf, M. Sommerfeld, and Y. Tsuji, Multiphase
Flow with Droplets and Particles, 2nd. Boca Raton: CRC Press, 2012, isbn:
978-1-4398-4050-4.
[118] ASTM D341-09, “Standard Practice for Viscosity-Temperature Charts for
Liquid Petroleum Products,” ASTM International, West Conshohocken, PA,
Tech. Rep., 2015. doi: 10.1520/D0341-09R15.
[119] S. R. Hong, N. M. Wereley, Y. T. Choi, and S. B. Choi, “Analytical and
experimental validation of a nondimensional Bingham model for mixed-mode
magnetorheological dampers,” Journal of Sound and Vibration, vol. 312, no.
3, pp. 399–417, 2008. doi: 10.1016/j.jsv.2007.07.087.
[120] G. Yang, B. F. Spencer, H.-J. Jung, and J. D. Carlson, “Dynamic Modeling
of Large-Scale Magnetorheological Damper Systems for Civil Engineering
Applications,” Journal of Engineering Mechanics, vol. 130, no. 9, pp. 1107–
1114, 2004. doi: 10.1061/(ASCE)0733-9399(2004)130:9(1107).
239
[121] H. Fujitani, H. Sodeyama, K. Hata, Y. Komatsu, N. Iwata, K. Sunakoda, and
S. Soda, “Dynamic performance evaluation of a 200kN magneto-rheological
damper,” Technical Note of National Institute for Land and Infrastructure
Management, vol. 41, pp. 349–356, 2002.
[122] M. J. Crochet and K Walters, “Numerical Methods in Non-Newtonian Fluid
Mechanics,” Annual Review of Fluid Mechanics, vol. 15, no. 1, pp. 241–260,
1983. doi: 10.1146/annurev.fl.15.010183.001325.
[123] M. Mao, W. Hu, Y.-T. Choi, and N. M. Wereley, “A Magnetorheological
Damper with Bifold Valves for Shock and Vibration Mitigation,” Journal of
Intelligent Material Systems and Structures, vol. 18, no. December, 2007. doi:
10.1177/1045389X07083131.
[124] N. L. Wilson, N. M. Wereley, W. Hu, and G. J. Hiemenz, “Analysis of
a magnetorheological damper incorporating temperature dependence,” In-
ternational Journal of Vehicle Design, vol. 63, no. 2/3, p. 137, 2013. doi:
10.1504/IJVD.2013.056102.
[125] F. Gordaninejad and D. G. Breese, “Heating of Magnetorheological Fluid
Dampers,” Journal of Intelligent Material Systems and Structures, vol. 10,
no. 8, pp. 634–645, 1999. doi: 10.1106/55D1-XAXP-YFH6-B2FB.
[126] K. D. Weiss and T. G. Duclos, “Controllable fluids: the temperature depen-
dence of post-yield properties,” International Journal of Modern Physics B,
vol. 08, no. 20-21, pp. 3015–3032, 1994. doi: 10.1142/S0217979294001275.
[127] K. D. Weiss, J. D. Carlson, T. G. Duclos, and K. J. Abbey, Temperature
independent magnetorheological materials, U.S. Patent, 1997.
[128] H. Sahin, X. Wang, and F. Gordaninejad, “Temperature Dependence of
Magneto-rheological Materials,” Journal of Intelligent Material Systems and
Structures, vol. 20, no. 18, pp. 2215–2222, 2009. doi: 10.1177/1045389X09351608.
[129] E Lemaire, A. Meunier, G. Bossis, J. Liu, D. Felt, P. Bashtovoi, and N.
Matoussevitch, “Influence of the particle size on the rheology of magne-
torheological fluids,” Journal of Rheology, vol. 39, no. 5, p. 1011, 1995. doi:
10.1122/1.550614.
[130] E. Furst and A. Gast, “Dynamics and lateral interactions of dipolar chains,”
Physical Review E, vol. 62, no. 5, pp. 6916–6925, 2000. doi: 10 . 1103 /
PhysRevE.62.6916.
[131] J. Mewis and N. J. Wagner, Colloidal Suspension Rheology. Cambridge:
Cambridge University Press, 2012, p. 393, isbn: 9780521515993.
[132] D. Knezˇevi and V. Savi, “Mathematical modeling of changing of dynamic vis-
cosity, as a function of temperature and pressure, of mineral oils for hydraulic
systems,” Facta Universitatis: Mechanical Engineering, vol. 4, pp. 27–34,
2006.
240
[133] S. L. Vieira, L. B. Pompeo Neto, and A. C. F. Arruda, “Transient behavior
of an electrorheological fluid in shear flow mode,” Journal of Rheology, vol.
44, no. 5, p. 1139, 2000. doi: 10.1122/1.1289282.
[134] S Henley and F. E. Filisko, “Flow profiles of electrorheological suspensions:
An alternative model for ER activity,” Journal of Rheology, vol. 43, no. 5,
p. 1323, 1999. doi: 10.1122/1.551024.
[135] S. Viera, M. Nakano, S. Henley, F. E. Filisko, L. Pompeo Neto, and A. Arruda,
“Transient behavior of the microstructure of electrorheological fluids in shear
flow mode,” International Journal of Modern Physics B, vol. 15, no. 6-7,
pp. 695–703, 2001. doi: 10.1142/S0217979201005179.
[136] J. G. Cao, J. P. Huang, and L. W. Zhou, “Structure of electrorheological fluids
under an electric field and a shear flow: experiment and computer simulation.,”
Journal of Physical Chemistry. B, vol. 110, no. 24, pp. 11 635–9, 2006. doi:
10.1021/jp0611774.
[137] Y. Pappas and D. J. Klingenberg, “Simulations of magnetorheological suspen-
sions in Poiseuille flow,” Rheologica Acta, vol. 45, no. 5, pp. 621–629, 2005.
doi: 10.1007/s00397-005-0016-8.
[138] S. G. Sherman, D. A. Paley, and N. M. Wereley, “Massively Parallel Simula-
tions of Chain Formation and Restructuring Dynamics in a Magnetorheological
Fluid,” ASME Conference Proceedings, vol. 2011, no. 54716, pp. 651–658,
2011. doi: 10.1115/SMASIS2011-5188.
[139] K. von Pfeil, M. D. Graham, D. J. Klingenberg, and J. F. Morris, “Structure
evolution in electrorheological and magnetorheological suspensions from a
continuum perspective,” Journal of Applied Physics, vol. 93, no. 9, p. 5769,
2003. doi: 10.1063/1.1563037.
[140] J Anderson, C Lorenz, and A Travesset, “General purpose molecular dynamics
simulations fully implemented on graphics processing units,” Journal of
Computational Physics, vol. 227, no. 10, pp. 5342–5359, 2008. doi: 10.1016/
j.jcp.2008.01.047.
[141] M Mohebi, N Jamasbi, and J. Liu, “Simulation of the formation of nonequi-
librium structures in magnetorheological fluids subject to an external mag-
netic field,” Physical Review E, vol. 54, no. 5, pp. 5407–5413, 1996. doi:
10.1103/PhysRevE.54.5407.
[142] R. T. Foister, Magnetorheological Fluids, U.S. Patent, 1997.
[143] Y Ido, T Inagaki, and T Yamaguchi, “Numerical simulation of microstructure
formation of suspended particles in magnetorheological fluids.,” Journal of
Physics Condensed Matter, vol. 22, no. 32, p. 324 103, 2010. doi: 10.1088/
0953-8984/22/32/324103.
[144] R. Tarjan, “Depth-first search and linear graph algorithms,” SIAM Journal
on Computing, vol. 1, no. 2, pp. 146–160, 1972.
241
[145] Y.-T. Choi and N. M. Wereley, “Biodynamic Response Mitigation to Shock
Loads Using Magnetorheological Helicopter Crew Seat Suspensions,” Journal
of Aircraft, vol. 42, no. 5, pp. 1288–1295, 2005. doi: 10.2514/1.6839.
[146] N. M. Wereley, Y.-T. Choi, and H. J. Singh, “Adaptive Energy Absorbers
for Drop-induced Shock Mitigation,” Journal of Intelligent Material Sys-
tems and Structures, vol. 22, no. 6, pp. 515–519, 2011. doi: 10 . 1177 /
1045389X10393767.
[147] H. J. Singh and N. M. Wereley, “Adaptive magnetorheological shock isolation
mounts for drop-induced impacts,” Smart Materials and Structures, vol. 22,
no. 12, p. 122 001, 2013. doi: 10.1088/0964-1726/22/12/122001.
[148] L. Marshall, C. F. Zukoski, and J. W. Goodwin, “Effects of electric fields
on the rheology of non-aqueous concentrated suspensions,” Journal of the
Chemical Society, Faraday Transactions 1, vol. 85, no. 9, p. 2785, 1989. doi:
10.1039/f19898502785.
[149] S. Melle, O. Caldero´n, M. Rubio, and G. Fuller, “Microstructure evolution in
magnetorheological suspensions governed by Mason number,” Physical Review
E, vol. 68, no. 4, p. 041 503, 2003. doi: 10.1103/PhysRevE.68.041503.
[150] R. S. Allan and S. G. Mason, “Particle Behaviour in Shear and Electric
Fields. I. Deformation and Burst of Fluid Drops,” Proceedings of the Royal
Society A: Mathematical, Physical and Engineering Sciences, vol. 267, no.
1328, pp. 45–61, 1962. doi: 10.1098/rspa.1962.0082.
[151] G. M. Kamath, M. K. Hurt, and N. M. Wereley, “Analysis and testing of
Bingham plastic behavior in semi-active electrorheological fluid dampers,”
Smart Materials and Structures, vol. 5, no. 5, pp. 576–590, 1996. doi: 10.
1088/0964-1726/5/5/007.
[152] N. M. Wereley, J. U. Cho, Y. T. Choi, and S. B. Choi, “Magnetorheological
dampers in shear mode,” Smart Materials and Structures, vol. 17, no. 1,
p. 015 022, 2008. doi: 10.1088/0964-1726/17/01/015022.
[153] J de Vicente, G Bossis, S Lacis, and M Guyot, “Permeability measurements
in cobalt ferrite and carbonyl iron powders and suspensions,” Journal of
Magnetism and Magnetic Materials, vol. 251, no. 1, pp. 100–108, 2002. doi:
10.1016/S0304-8853(02)00484-5.
[154] J. D. Carlson, “MR fluids and devices in the real world,” International
Journal of Modern Physics B, vol. 19, no. 7-9, pp. 1463–1470, 2005. doi:
10.1142/S0217979205030451.
[155] D. J. Klingenberg, C. H. Olk, M. A. Golden, and J. C. Ulicny, “Effects of
nonmagnetic interparticle forces on magnetorheological fluids.,” Journal of
Physics Condensed Matter, vol. 22, no. 32, pp. 324 101–324 105, 2010. doi:
10.1088/0953-8984/22/32/324101.
242
[156] J. Ginder, L. Davis, and L. Elie, “Rheology of magnetorheological fluids:
models and measurements,” International Journal of Modern Physics B, vol.
10, no. 23n24, pp. 3293–3303, 1996. doi: 10.1142/S0217979296001744.
[157] G Bossis, S Lacis, A Meunier, and O Volkova, “Magnetorheological fluids,”
Journal of Magnetism and Magnetic Materials, vol. 252, pp. 224–228, 2002.
doi: 10.1016/S0304-8853(02)00680-7.
[158] M. R. Jolly, J. D. Carlson, and B. C. Mun˜oz, “A model of the behaviour of
magnetorheological materials,” Smart Materials and Structures, vol. 5, no. 5,
pp. 607–614, 1996. doi: 10.1088/0964-1726/5/5/009.
[159] C. L. A. Berli and J. de Vicente, “A structural viscosity model for magne-
torheology,” Applied Physics Letters, vol. 101, no. 2, p. 021 903, 2012. doi:
10.1063/1.4734504.
[160] T. Halsey, J. Martin, and D. Adolf, “Rheology of electrorheological fluids,”
Physical Review Letters, vol. 68, no. 10, pp. 1519–1522, 1992. doi: 10.1103/
PhysRevLett.68.1519.
[161] B. J. de Gans, N. J. Duin, D. van den Ende, and J. Mellema, “The influence
of particle size on the magnetorheological properties of an inverse ferrofluid,”
The Journal of Chemical Physics, vol. 113, no. 5, p. 2032, 2000. doi: 10.1063/
1.482011.
[162] O. Volkova, G. Bossis, M. Guyot, V. Bashtovoi, and A. Reks, “Magnetorheol-
ogy of magnetic holes compared to magnetic particles,” Journal of Rheology,
vol. 44, no. 1, p. 91, 2000. doi: 10.1122/1.551075.
[163] G Bossis, O Volkova, S Lacis, and A Meunier, “Ferrofluids,” in Ferrofluids,
ser. Lecture Notes in Physics, S. Odenbach, Ed., vol. 594, Berlin, Heidelberg:
Springer Berlin Heidelberg, 2002, pp. 202–230, isbn: 978-3-540-43978-3. doi:
10.1007/3-540-45646-5.
[164] J. D. Carlson and K. D. Weiss, Magnetorheological materials based on alloy
particles, U.S. Patent, 1995.
[165] J. M. Ginder, L. D. Elie, and L. C. Davis, Magnetic Fluid-Based Magnetorhe-
ological Fluids, U.S. Patent, 1996.
[166] D. T. Zimmerman, R. C. Bell, J. A. Filer, J. O. Karli, and N. M. Wereley,
“Elastic percolation transition in nanowire-based magnetorheological fluids,”
Applied Physics Letters, vol. 95, no. 1, p. 014 102, 2009. doi: 10.1063/1.
3167815.
[167] G. T. Ngatu, N. M. Wereley, J. O. Karli, and R. C. Bell, “Dimorphic
magnetorheological fluids: exploiting partial substitution of microspheres
by nanowires,” Smart Materials and Structures, vol. 17, no. 4, p. 045 022,
2008. doi: 10.1088/0964-1726/17/4/045022.
243
[168] L. A. Powell, N. M. Wereley, and J. Ulicny, “Magnetorheological Fluids
Employing Substitution of Nonmagnetic for Magnetic Particles to Increase
Yield Stress,” IEEE Transactions on Magnetics, vol. 48, no. 11, pp. 3764–3767,
2012. doi: 10.1109/TMAG.2012.2202885.
[169] K. Shah, D. Xuan Phu, and S.-B. Choi, “Rheological properties of bi-dispersed
magnetorheological fluids based on plate-like iron particles with application to
a small-sized damper,” Journal of Applied Physics, vol. 115, no. 20, p. 203 907,
2014. doi: 10.1063/1.4879681.
[170] T. Kikuchi, J. Noma, S. Akaiwa, and Y. Ueshima, “Response time of magne-
torheological fluid-based haptic device,” Journal of Intelligent Material Sys-
tems and Structures, vol. 27, no. 7, pp. 3–4, 2015. doi: 10.1177/1045389X15596621.
[171] C. Kormann, H. Laun, and H. Richter, “MR Fluids with Nano-Sized Magnetic
Particles,” International Journal of Modern Physics B, vol. 10, no. 23n24,
pp. 3167–3172, 1996. doi: 10.1142/S0217979296001604.
[172] F. A. Morrison, Understanding Rheology. New York: Oxford University Press,
2001, pp. 188–190.
[173] H. M. Laun, C. Gabriel, and C. Kieburg, “Magnetorheological Fluid in
Oscillatory Shear and Parameterization with Regard to MR Device Proper-
ties,” Journal of Intelligent Material Systems and Structures, vol. 21, no. 15,
pp. 1479–1489, 2009. doi: 10.1177/1045389X09351759.
[174] J. C. Ferna´ndez-Toledano, J. Rodr´ıguez-Lo´pez, K. Shahrivar, R. Hidalgo-
A´lvarez, L. Elvira, F. Montero de Espinosa, and J. de Vicente, “Two-step
yielding in magnetorheology,” Journal of Rheology, vol. 58, no. 5, pp. 1507–
1534, 2014. doi: 10.1122/1.4880675.
[175] P. Bogacki and L. F. Shampine, “A 3(2) pair of Runge - Kutta formulas,”
Applied Mathematics Letters, vol. 2, no. 4, pp. 321–325, 1989. doi: 10.1016/
0893-9659(89)90079-7.
244