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DYNAMICS OF SUSPENSIONS OF ELASTIC
CAPSULES IN POLYMER SOLUTION
by
Pratik Pranay
A dissertation submitted in partial
fulfillment of the requirements for the degree of
Doctor of Philosophy
(Chemical Engineering)
at the
UNIVERSITY OF WISCONSIN – MADISON
2011
i
Acknowledgments
I express my deepest gratitude towards Prof. Michael D. Graham, for constantly
guiding me in my area of research and exposing me to various projects throughout
my PhD work. Without his constant support and motivation it would not have been
possible to work on such diversified and challenging aspects of research. I will always
remember the insightful discussions with him on simple to complicated problems that
we undertook in my PhD work and hope to carry the same basic understanding and
way of undertaking new problems in future.
I thank Profs. Rawlings, Klingenberg, Yin and Chesler for taking time to be on
my thesis committee.
It has been a great experience working with many former and current members
of the Graham group. I thank Samartha G. Anekal, Patrick T. Underhill, Juan P.
Hernandez-Ortiz, Wei Li, Mauricio Lopez, Hongbo Ma and Aslin Izmitli for providing
assistance during my first few years. I have enjoyed interacting with Li Xi who has
been my office-mate for the past four years especially in giving me a good company
at late night hours. Pieter J. A. Janssen has been a good friend and I will miss our
thoughtful discussions and jokes on every topic. I thank Amit Kumar for helping me
in simulations and providing answers to all my questions. I have enjoyed working with
ii
Kushal Sinha both inside and outside the group. Yu Zhang has been a good friend
throughout my PhD life. In the relatively short time I had with the new members,
Rafael Henrıquez-Rivera, Timothy Feyereisen, Shifan Mao, Prof. Shinji Tamano and
Friedemann Hahn have been a good company.
During my stay in the Madison, I have made many good friends, who made my
times in Madison enjoyable. I have been fortunate to have met Deepa Sanwal and I
thank her for her company, love and friendship over the last couple of years. I treasure
my friendship with Rohit Malshe and will miss the unforgettable times spent together.
Vikram Adhikarla and Maneesh Mishra have been good firnds in Madison and I wish
the best for them in future. I will miss running with Profs. Klingenberg and Root.
Manan Chopra, Santosh Reddy, Maneesh Rathi and Murali Rajamani have been
trustworthy friends and have provided me with timely advice and help.
Above all, I am thankful to my parents whose motivation and inspiration have
helped me grow at all stages of my life.
Research projects presented in this dissertation was supported by the National
Science Foundation.
iii
Abstract
Many serious medical conditions, from hemorrhage to coronary artery disease to di-
abetes, are associated with disruptions in blood flow. It has been observed that
potential beneficial effects on hemodynamics arise from addition of low concentra-
tions of high molecular weight long-chain polymer molecules known as drag-reducing
additives (DRAs) to blood. These are so named because of their drag reducing ef-
fects on the turbulent flows, but since blood flow in small vessels is not turbulent, the
effect of these polymers on hemodynamics must have a separate origin that are not
understood. The present work represents an initial step towards gaining fundamental
understanding of these observed effects of DRAs on blood flow, and more broadly,
shedding light on the dynamics of complex multiphase fluids.
The dynamics and pair collisions of fluid-filled elastic capsules during Couette flow
in Newtonian fluids and dilute solutions of high molecular weight (drag-reducing)
polymers are investigated via direct simulation. Capsule membranes are modeled
using either a neo-Hookean constitutive model or a model introduced by Skalak et
al. (R. Skalak, A. Tozeren, R. P. Zarda, and S. Chien, “Strain energy function of red
blood-cell membranes”, Biophysical Journal 13, 245–280 (1973)), which includes an
energy penalty for area changes. This model was developed to capture the elastic
iv
properties of red blood cells. Polymer molecules are modeled as bead-spring trimers
with finitely extensible non-linearly elastic (FENE) springs; parameters were chosen
to loosely approximate 4000 kD poly(ethylene oxide). Simulations are performed
with a novel Stokes flow formulation of the Immersed Boundary Method (IBM) for
the capsules, combined with Brownian dynamics for the polymer molecules. Results
for isolated capsules in shear indicate that at the very low concentrations considered
here, polymers have little effect on the capsule shape. In the case of pair collisions, the
effect of polymer is strongly dependent on the elastic properties of the capsules’ mem-
branes. For neo-Hookean capsules or for Skalak capsules with only a small penalty for
area change, the net displacement in the gradient direction after collision is virtually
unaffected by polymer. For Skalak capsules with a large penalty for area change,
polymers substantially decrease the net displacement when compared to the Newto-
nian case and the effect is enhanced on increasing the polymer concentration. The
differences between the polymer effects in the various cases is associated with the
extensional flow generated in the region between the capsules as they leave the colli-
sion. The extension rate is highest when there is a strong resistance to a change in
membrane area, and is substantially decreased in the presence of polymer.
Suspensions of fluid-filled elastic capsules in a Couette flow in Newtonian fluids and
dilute solutions of high molecular weight (drag-reducing) polymers are investigated.
A simple theoretical model is presented to describe the cross-stream migration of
deformable capsules in suspensions which comes from a balance of shear-induced dif-
fusion and wall-induced migration due to capsule deformability. The model provides
an explicit theoretical prediction of the dependence of capsule-depleted layer thick-
ness on the capillary number. A computational approach is then used to examine
v
the motion of elastic capsules in polymer solutions. Capsule membranes are modeled
using a neo-Hookean constitutive model and polymer molecules are modeled as bead-
spring chains with FENE springs; parameters were chosen to loosely approximate
4000 kD poly(ethylene oxide). Results for an isolated capsule near a wall indicate
that the wall-induced migration depends strongly on the capillary number, expressing
capsule deformability. Numerical simulation of suspensions of capsules in Newtonian
fluid illustrates the inhomogeneous distribution of capsules at steady-state with the
formation of capsule-depleted layer near the walls. The thickness of this layer is found
to be strongly dependent on the capillary number. The shear-induced diffusivity, on
the other hand, show a weak dependence on capillary number. These results indicate
that the mechanism of wall-induced migration is the primary source for determining
the capsule-depleted layer thickness of capsules in suspensions. Numerical simulation
results show that both the wall-induced migration and the shear-induced diffusive
motion of the capsules are suppressed under the influence of polymer. Results on sus-
pensions of capsules illustrate that the net effect of polymers is to reduce the thickness
of the capsule-depleted layer resulting in a redistribution of capsules at steady state.
The results are in qualitative agreement to the experimental observations.
vi
Contents
Acknowledgments i
Abstract iii
List of Figures x
List of Tables xxii
1 Introduction 1
2 Background 6
2.1 Dynamics of blood flow in the microcirculation . . . . . . . . . . . . . 6
2.2 Drag reducing additives in blood flow . . . . . . . . . . . . . . . . . . 9
2.3 Simulations of blood flow . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Models and discretization methods 17
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Capsule Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Polymer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 Fluid velocity calculation . . . . . . . . . . . . . . . . . . . . . . . . . 28
vii
3.5 Volume correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Single capsule dynamics 38
4.1 Method and code validation . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Single capsule in shear . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5 Pair collisions in shear 46
5.1 Newtonian fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2 Polymer solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3 Effect of Ca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.4 Mechanism of polymer effects . . . . . . . . . . . . . . . . . . . . . . 60
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6 Suspensions of capsules in Couette flow 63
6.1 Theory for suspension of capsules in shear flow . . . . . . . . . . . . 63
6.1.1 Wall-induced migration of a single capsule . . . . . . . . . . . 63
6.1.2 Shear-induced diffusion . . . . . . . . . . . . . . . . . . . . . . 65
6.1.3 Model for steady-state distribution . . . . . . . . . . . . . . . 66
6.1.4 Steady-state capsule-depleted layer near a single wall . . . . . 68
6.2 Migration of a single capsule in a Couette Flow . . . . . . . . . . . . 69
6.2.1 Newtonian fluid . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2.2 Validation of dipole approximation . . . . . . . . . . . . . . . 72
6.2.3 Polymer fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.3 Suspensions of capsules in a Couette flow . . . . . . . . . . . . . . . . 84
6.3.1 Newtonian solution . . . . . . . . . . . . . . . . . . . . . . . . 84
viii
6.3.2 Polymer solution . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.3.3 Capsule-depleted layer . . . . . . . . . . . . . . . . . . . . . . 95
6.3.4 Diffusion at steady state . . . . . . . . . . . . . . . . . . . . . 98
6.4 Comparison with the theoretical model . . . . . . . . . . . . . . . . . 102
6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7 Conclusion 107
8 Current work: Suspensions of capsules in a pressure-driven flow 110
8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.2 Migration of an isolated capsule in a pressure-driven Flow . . . . . . 112
8.3 Suspensions of capsules in a pressure-driven Flow . . . . . . . . . . . 116
8.4 Discussion and future work . . . . . . . . . . . . . . . . . . . . . . . . 122
9 Future work 124
9.1 Suspensions of Red Blood Cells . . . . . . . . . . . . . . . . . . . . . 124
9.2 Drug delivery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
9.3 Leucocyte margination under the influence of polymers . . . . . . . . 130
A Single Polymer Dynamics 132
B Comparison of GGEM/IBM with other methods 137
B.1 Overview of Conventional IBM Method . . . . . . . . . . . . . . . . . 137
B.2 IBM and GGEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
B.3 BIM and GGEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
C Implementation Issues-Slit 149
ix
C.1 Different flow profiles: . . . . . . . . . . . . . . . . . . . . . . . . . . 150
C.2 Assignment: putting ρg on the mesh - collocation . . . . . . . . . . . 152
C.3 Solution approach - FFT/finite differences . . . . . . . . . . . . . . . 153
C.3.1 Treatment of periodic boundary condition: . . . . . . . . . . . 154
C.3.2 Proper treatment of k = 0: . . . . . . . . . . . . . . . . . . . . 156
C.3.3 Finite Difference: . . . . . . . . . . . . . . . . . . . . . . . . . 160
C.4 Interpolation: getting ug at the particle positions . . . . . . . . . . . 165
C.4.1 Lagrange interpolation of degree 2 : . . . . . . . . . . . . . . . 166
C.4.2 Interpolation of global velocity: . . . . . . . . . . . . . . . . . 166
x
List of Figures
1.1 Fractional survival vs. time of rats subjected to hemorrhagic shock and
not resuscitated (CON), injected with normal saline (NS), or injected
with the same volume of saline containing 10 ppm of drag reducing
polymer (DRP)1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Schematic of suspensions of fluid-filled elastic capsules in shear flow of
a very dilute polymer solution. . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Schematic of blood flow in the microcirculation, illustrating phenomena
of migration, margination and plasma-skimming. . . . . . . . . . . . . 7
2.2 Experimental data (symbols) on the thickness of cell-free layer as a
function of flow rate for a suspensions of RBCs in Newtonian (Control)
and drag-reducing polymer (DRP) solutions from Kameneva et al.2. . 11
xi
4.1 (a) Deformation parameter as a function of time for single non-prestressed
Neo-Hookean (NH) capsules in shear flow in a cubic box of size 12.5a.
Symbols are simulations from Lacet al.3. Lines are results from present
work. (b) Steady state deformation parameter as a function of Ca
for single non-prestressed Neo-Hookean (NH) capsules in shear flow
in a cubic box of size 12.5a. For comparison, results of Doddi and
Bagchi4(DB), Lac et al.3(LB), Ramanujan and Pozrikidis5(RP) are
also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
(a) D vs. t∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
(b) steady-state D vs. Ca . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Xi Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3 Low Ca Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4 (a) Steady state values of deformation parameter at different Ca for
single non-prestressed NH and SK (C = 10) capsules (shown by bold
lines) and for preinflated NH and SK capsules (shown by dotted lines)
in shear flow. (b) Deformation parameter for preinflated NH and SK
(C = 10) capsules under shear flow in a cubic box of size 12.5a. Images
correspond to capsule shapes taken at t∗ = 10.. . . . . . . . . . . . . . 43
(a) Effect of preinflation on steady state deformation parameter . . 43
(b) Deformation parameter for preinflated NH and SK capsules . . 43
4.5 Difference in (a) deformation parameter and (b) inclination angle for
preinflated capsules in Newtonian fluid and a polymer fluid (β = 0.997)
under shear flow in a cubic box of size 12.5a. Subscript N and P
correspond to Newtonian and polymer case respectively. . . . . . . . 44
xii
(a) DN −DP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
(b) θN − θP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.1 Schematic of pair collisions of fluid-filled elastic capsules in shear flow
of a very dilute polymer solution. . . . . . . . . . . . . . . . . . . . . 47
5.2 Pair collisions of preinflated NH capsules in a Newtonian fluid . . . . 48
(a) verification with Lac et al.6 . . . . . . . . . . . . . . . . . . . . 48
(b) effect of screening parameter α . . . . . . . . . . . . . . . . . . 48
5.3 Relative trajectories for pair collisions of preinflated NH capsules in a
Newtonian fluid at different Ca. Images correspond to snapshots taken
at collision (∆x = 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.4 Relative trajectories for pair collisions of preinflated NH and SK cap-
sules (C = 10) in a Newtonian fluid at different Ca. Images correspond
to snapshots taken at collision (∆x = 0). . . . . . . . . . . . . . . . . 50
5.5 Deformation parameter as a function of relative separation in the stream-
wise direction (∆x) for pair collisions of pre-inflated NH (Ca = 0.142)
and SK (Ca = 0.60, C = 10) capsules in a Newtonian fluid. . . . . . . 51
5.6 Paircollision Polymer Effect . . . . . . . . . . . . . . . . . . . . . . . 53
5.7 Pair collision with different initial Separation . . . . . . . . . . . . . . 54
5.8 Variation of relative trajectories with non-dimensional area dilation
modulus, C, for SK (Ca = 0.60) capsules in Newtonian (thick lines)
and polymeric (thin lines, β = 0.997) fluid. . . . . . . . . . . . . . . . 55
xiii
5.9 Relative trajectories as a function of (1− β) for SK (Ca = 0.60, C =
10) capsules under shear flow in a Newtonian (thick lines) and poly-
meric (thin lines) fluid. Inset shows the difference in ∆y of SK capsules
at ∆x = 8a in Newtonian and polymeric fluid, (∆yN − ∆yP), plotted
against (1− β). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.10 Variation of (a) maximum displacement (∆ymax −∆y0) and (b) net final
displacement (∆yfinal −∆y0) with Ca for NH and SK (C = 10) capsules
in Newtonian (thick lines) and polymeric (thin lines, β = 0.997) fluid.
Note: ∆yfinal is calculated at ∆x = 8a. The figure in the inset of (a)
shows the schematic of these displacements. . . . . . . . . . . . . . . 58
(a) maximum displacement (∆ymax −∆y0) vs. Ca . . . . . . . . . . 58
(b) net final displacement (∆yfinal −∆y0) vs. Ca . . . . . . . . . . . 58
5.11 Largest eigenvalue λmax of the deformation rate at the origin as a func-
tion of ∆x for pair collisions under various conditions. Thick lines
are from Newtonian simulations; thin lines are from simulations with
polymers, β = 0.997. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.1 Schematic of suspensions of fluid-filled elastic capsules in Couette flow. 64
6.2 (a) Schematic of migration of an isolated capsule away from the nearest
wall in a Couette flow. (b) Schematic showing pair collision of the
capsules. The lines shows the trajectory of a capsule undergoing a
collision process in the shear (x− y) plane. . . . . . . . . . . . . . . . 67
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
xiv
6.3 Migration of a capsule in a Newtonian fluid in a Couette flow. (a)
Trajectory of the center of mass of a capsule y as a function of time t∗
in the wall-normal direction. The walls are at y = 0 and y = By = 10a.
(b) Capsule deformation D as a function of center of mass of a capsule
y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.4 (a) Difference between first (N1) and second (N2) normal stress differ-
ences as a function of y. (b) N1 − N2 evaluated at y = 2.5a (quarter
channel height) as a function of Ca. The symbols represent the simu-
lation results and the dashed line represents the exponential fit. . . . 71
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.5 Validation of a point dipole approximation. (a) Trajectory of the center
of mass of an isolated capsule y at Ca = 0.30 as a function of time t∗
in the wall-normal direction for different values of initial condition y0.
The walls are at y = 0 and y = By = 10a. Symbols are simulation
results and lines are the fits using eq. 6.13.(b) Trajectory of a capsule
as a function time for different values of Ca. . . . . . . . . . . . . . . 73
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
xv
6.6 (a) Migration velocity umig as a function of center of mass of a capsule
y. (d) Comparison of the numerical value of the slope k obtained from
simulations (by fitting eq. 6.13) and the theoretical value obtained
using eq. 6.12, at different values of Ca. . . . . . . . . . . . . . . . . . 74
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.7 Migration of a capsule in a Newtonian (solid lines) and polymer (dashed
lines, β = 0.994,Wi = 20) solutions. (a) Trajectory of the center of
mass of a capsule y as a function of time t∗ in the wall-normal direction.
(b) Steady state capsule deformation D as a function of Ca. . . . . . 77
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.8 (a) N1 −N2 evaluated at y = 2.5a as a function of Ca. (b) Migration
velocity umig evaluated at y = 2.5a (quarter channel height) as a func-
tion of N1 −N2. The symbols represent the simulation results and the
dashed line represents the linear fit. . . . . . . . . . . . . . . . . . . . 78
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.9 Effect of (a) polymer concentration expressed as 1 − β at fixed Wi
(= 20) and (b) Wi at fixed β (= 0.994) on the trajectory of an isolated
capsule (Ca = 0.30) in the wall-normal direction of in a Couette flow.
Symbols are simulation results and lines are the fits. . . . . . . . . . . 81
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
xvi
6.10 Migration velocity umig evaluated at y = 2.5a as a function of (c) 1−β
at fixed Wi (= 20) and (d) Wi at fixed β (= 0.994) for an isolated
capsule (Ca = 0.30). . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.11 (a) Snapshots of the suspensions of capsules (Ca = 0.60,φ of 0.10) in
a Newtonian fluid at t∗ = 1 (left) and t∗ = 300 (right) in a Newtonian
fluid in a cubic box of size 10a . (b) Trajectories of the center of mass
of capsules (Ca = 0.60, φ = 0.10) in the wall-normal direction as a
function of time. The walls are at y = 0 and y = By = 10a. . . . . . . 85
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.12 (a) Average distance from the centerline < |y−ycenter| > of suspensions
(φ = 0.10) of capsules in a Newtonian fluid in a a cubic box of size
10a as a function of time t∗. (b) Steady state distribution of capsules
(φ = 0.10) as a function of y. The walls are at y = 0, 10a and y = 5a
is the channel centerline. The walls are at y = 0, 10a and y = 5a is the
channel centerline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
xvii
6.13 (a) Average distance from the centerline < |y−ycenter| > of suspensions
of capsules (φ = 0.10, Ca = 0.60) in a Newtonian fluid in a cubic box
of size 16a as a function of time t∗. (b) Steady state distribution of
capsules (φ = 0.10, Ca = 0.60) as a function of y. The walls are at
y = 0, 16a and y = 8a is the channel centerline. . . . . . . . . . . . . 89
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.14 The effect of volume fraction φ on the steady state distribution of
capsules at Ca = 0.60 as a function of y. The walls are at y = 0, 10a
and y = 5a is the channel centerline. . . . . . . . . . . . . . . . . . . 90
6.15 (a) Snapshots of the suspensions of capsules (Ca = 0.30) at t∗ = 10
in a polymer (β = 0.994,Wi = 20) solutions in a cubic box of size
10a. Polymer molecules are shown as thin black lines (b) Average
distance from the centerline < |y−ycenter| > of suspensions of capsules in
Newtonian (solid line) and polymer(dashed lines, β = 0.994,Wi = 20)
solutions as a function of time t∗. . . . . . . . . . . . . . . . . . . . . 92
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.16 (a) Steady state distribution of capsules as a function of y in Newtonian
(solid line) and polymer (dashed line, β = 0.994,Wi = 20) solutions
in a cubic box of size 10a. (b) Steady state distribution of capsules in
the “bulk” (2.5a ≤ y ≤ 7.5a) region as a function of Ca in Newtonian
(solid line) and polymer(dashed line, β = 0.994,Wi = 20) solutions. . 93
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
xviii
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.17 Steady state distribution of capsules (Ca = 0.60) in Newtonian and
polymer (Wi = 20) solutions with different values of β. . . . . . . . . 95
6.18 (a) Dependence of capsule-depleted layer thickness on Ca for suspen-
sions of capsules in Newtonian and polymer (Wi = 20) solutions with
different values of β in a cubic box of size 10a. Symbols are the sim-
ulation results and lines are the fits. The standard deviation is based
on results from different initial configurations. (b) Experimental data
(symbols) on the thickness of cell-free layer as a function of flow rate
for a suspensions of RBCs from Kameneva et al.7. . . . . . . . . . . . 96
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.19 (a) Mean squared displacement of suspensions of capsules in the wall-
normal direction at steady state in Newtonian and polymer ( β =
0.994,Wi = 20) solutions as a function as a function of time t∗ and the
corresponding short-time diffusivities (b) in the wall-normal direction
as a function of Ca. The standard deviation (error bar) is based on
results from different initial configurations. . . . . . . . . . . . . . . . 99
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.20 Short-time diffusivities in the wall-normal direction as a function of y in
Newtonian (solid line) and polymer(dashed line, β = 0.994,Wi = 20)
solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
xix
6.21 Comparison of the thickness of the capsule free layer for suspensions
(φ = 0.10) of capsules in polymer ( β = 0.994,Wi = 20) solution
obtained from simulations and predicted from theory (eq. 6.16) as a
function of Ca. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.1 Schematic of suspensions of fluid-filled elastic capsules in a pressure-
driven flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.2 Migration of NH capsules in a Newtonian fluid in a pressure-driven
flow. (a) Trajectory of the center of mass of a capsule y as a function
of time t∗ in the wall-normal direction. The walls are at y = 0 and
y = By = 10a. (b) Capsule deformation D as a function of center of
mass of a capsule y. . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.3 (a) Difference between first (N1) and second (N2) normal stress differ-
ences as a function of y. (b) The trajectory of an isolated capsule in a
Newtonian fluid for different channel heights. Solid lines are simulation
results for channel of height By = 10a and dashed are for By = 40a. . 114
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.4 (a) Average distance from the centerline < |y−ycenter| > of suspensions
(φ = 0.10) of NH capsules in a Newtonian fluid in a a cubic box of size
10a as a function of time t∗. (b) Average velocity of the center of mass
of capsules in flow (x)direction as a function of time t∗. The dashed line
represents the average velocity of the undisturbed flow (〈U〉/U = 2/3) 117
xx
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.5 (a) Steady-state distribution of the velocity of capsule’s center (sym-
bols) in flow direction (x) as a function of y in a Newtonian fluid in
a cubic box of size 10a. The dashed lines are the quadratic fits to
the symbols. The solid line (black) represents parabolic profile of the
undisturbed velocity. (b) Steady state distribution of capsules as a
function of y. The walls are at y = 0 and y = 10a. . . . . . . . . . . . 119
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.6 Average distance from the centerline < |y − ycenter| > of suspensions
(φ = 0.10) of NH (Ca = 0.142) capsules in a Newtonian fluid for
different channel heights. . . . . . . . . . . . . . . . . . . . . . . . . . 121
9.1 A schematic showing suspensions of RBCs in the microcirculation. . . 125
9.2 Examples of recently sythesized types of particles with potential for
drug delivery. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
9.3 A schematic showing hypothesized distributions of drug delivery par-
ticles in the microcirculation. . . . . . . . . . . . . . . . . . . . . . . 129
xxi
A.1 (a) Average steady state RMS end-to-end distance < R0 > and (b)
average steady state polymer stress < τ p > as a function of Wi for
a single polymer molecule in an unbounded shear flow. x, y and z
represents flow, gradient and neutral directions respectively. “HI” de-
note simulations including hydrodynamic interactions in the Brownian
term. “FD” represents simulations neglecting hydrodynamic interac-
tions in the Browninan term. . . . . . . . . . . . . . . . . . . . . . . . 135
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
B.1 Schematic of different length scales . . . . . . . . . . . . . . . . . . . 141
(a) length scales associated with conventional IBM method . . . . . 141
(b) length scales associated with IBM/GGEM . . . . . . . . . . . . 141
xxii
List of Tables
1
Chapter 1
Introduction
Many serious medical conditions contribute to or arise from disruptions in normal
blood flow. Examples include: hemorrhage (simple replacement with fluids other
than whole blood leads to short-term improvement but often to severe long-term
consequences whose origin is not well-understood8), sepsis (systemic infection arising
from severe injury or surgical complications9), blood hyperviscosity syndromes (which
accompany many disorders and injuries, including severe burns10), vascular diseases
such as atherosclerosis (which lead to poor circulation and potentially to catastrophic
events such as heart attacks or strokes), and diabetes (one of whose long-term com-
plications is chronic peripheral vascular disease, which results in poor wound healing
in the extremities and may lead to gangrene). Various approaches exist for treatment
of these disorders. For example, many drugs used in prevention and treatment of
atherosclerosis impede formation of plaques in arteries or formation of blood clots,
but these drugs do not directly affect blood flow per se. Vasodilators and vasocon-
strictors control blood flow by inducing blood vessels to change size; vasodilators
2
in particular are commonly used in the treatment of heart disease. However, these
agents only affect the blood flow in large vessels (> 100µm diameter), and they have
no known effect on blood flow distribution at the capillary level (∼ 5µm diameter)
where exchange of nutrients between tissue and blood takes place. There are other
approaches that directly modify the behavior of blood flow, especially at the micro-
circulatory level of vessels having size below 100µm. For example, the addition of a
“plasma expander” for treatment of severe hemorrhage – generally a dextran solution
– to the blood is standard8. This has the effect of increasing blood volume, decreasing
the red blood cell (RBC) concentration and thus the blood viscosity and, because of
the presence of the dextran, approximately maintaining the proper osmotic pressure
in the blood. While having significant short-term benefit, this treatment has long-
term side effects, which can include severe and frequently lethal edema in the lungs8.
For chronic disorders such as peripheral vascular disease, drugs such as pentoxifylline
(Trental R©) are commonly used. Pentoxifylline does not affect plasma viscosity or
RBC concentration, but significantly increases the deformability of the RBC, result-
ing in a decreased blood viscosity and probably also improving the trafficking of the
RBCs in the microcirculation10.
Recently, studies in animals have shown that a radically different approach to mod-
ification of hemodynamics has promising beneficial effects11,12,13,14,15,16,17,18,19,20,21,22? 23,24,2,7,1,25:
In this approach, fluid containing dissolved long-chain water-soluble polymers is added
to blood; the resulting polymer concentration in the blood is as as low as several
ppm. For example, a study on dogs26 reports, upon addition of the polymer ad-
ditive to blood, a 27% increase in cardiac output despite a significant decrease in
arterial blood pressure - from 130/80 to 110/80 mmHg, and pulse, from 180 min−1 to
3
Figure 1.1: Fractional survival vs. time of rats subjected to hemorrhagic shock andnot resuscitated (CON), injected with normal saline (NS), or injected with the samevolume of saline containing 10 ppm of drag reducing polymer (DRP)1.
120 min−1, corresponding to a reduction in overall hemodynamic resistance of about
40%. Two further recent studies with dogs found that that these polymer additives
decreased pressure drop across a coronary stenosis and improved oxygen transport
to heart tissues downstream of the stenosis27,28. Another study examined the effect
of these polymer additives on small-volume fluid resuscitation of rats subjected to
massive hemorrhage7,1. As shown in Fig. 1.1, the survival rate for rats resuscitated
with polymer solution was dramatically higher – more than double – than for rats
treated with normal saline. Additionally, measurements of whole-body O2 consump-
tion and CO2 production showed rates about 50% higher for the polymer solution
treated animals compared to the saline treated ones. There are a number of points
that should be noted while considering the origin of the observed effects of polymers.
4
For a given chemical composition of the polymer, but varying molecular weight, the
physiological effects were absent at low molecular weight but significant at high molec-
ular weight. Additionally, chemically different polymers (e.g. poly(ethylene oxide),
polyacrylamide, certain polysaccharides) yielded very similar results as along as the
molecular weight was sufficiently high (typically > 106 Dalton). Furthermore, the
hemodynamic effects of these additives are virtually immediate, indicating that al-
though the additives are ultimately cleared from the blood by the immune system,
the immune response occurs on a time scale much slower than the hemodynamic one.
These observations indicate that the origin of the physiological effects is physical
rather than chemical.
The polymers used in all these studies are members of a class of molecules known
as “drag reducing additives” (DRAs). When added to simple single-phase fluids
such as water at ppm levels, DRAs significantly reduce drag in turbulent flows29,30.
Flow in small blood vessels is not turbulent, so turbulent drag reduction per se is
not the explanation for the hemorrhage recovery and other results described above.
Nevertheless, during flow, the collisions of the RBCs lead to fluctuations in fluid
velocity that are somewhat analogous to the fluctuations present in turbulent flow.
The effect of these fluctuations on the dynamics of the individual polymer molecules
and the feedback from the polymer dynamics back to the dynamics of the RBCs are
completely unknown.
The aim of the present work is to take first steps toward understanding the influ-
ence of drag reducing polymers on blood flow. We will introduce a theoretical, mod-
eling and computational approach to study the dynamics of suspensions of red blood
cells (RBCs) in a polymer solution. RBCs are modelled as fluid-filled elastic cap-
5
Wall
beads experience Stokes drag tomotion relative relative to
fluid
nodes move with the fluidvelocity
γ
By
x
y
z
2a
Wall
⋅
Figure 1.2: Schematic of suspensions of fluid-filled elastic capsules in shear flow of avery dilute polymer solution.
sules (liquid drops enclosed by a solid elastic membrane) and polymer molecules are
simulated as bead spring chains with parameters chosen to model 4×106 D polyethy-
lene oxide (PEO). We present a novel, highly efficient formulation of the immersed
boundary method for Stokes flow. Specifically, we will highlight the effect of polymer
molecules on a number of important phenomena exhibited by the blood cells in the
microcirculation, with an ultimate aim to shed lights on the dynamics of complex
multiphase fluids. Fig. 1.2 illustrates the basic situation of interest.
6
Chapter 2
Background
2.1 Dynamics of blood flow in the microcirculation
The so called microcirculation is defined by vessels having diameter less than 100µm.
A red blood cell has an average diameter of about 8µm. The viscosity of plasma is
around 1.2cP31,32. For the flow in the microcirculation, with shear rates ranging from
100s−1 to 1000s−1, the Reynolds number is much less than unity. Many important
phenomena are observed which are illustrated in Fig. 2.1. The RBCs migrates away
from the wall towards the center of the vessel. This leads to inhomogeneous distribu-
tion of RBCs – having higher concentration at the core of the vessel and a formation
of several microns thick cell-free layer. The thickness of this layer appears to grow
slowly with increasing flow rate33,2 and is assumed to follow an approximate relation-
ship : thickness ∼ (flowrate)n, with n reported to be in the range of 0.3 − 0.533,34.
The origin of this relationship is not known. The migration of RBCs towards the
center of a vessel is assumed to increase the probability of less deformable leucocytes,
7
red blood cell
marginated leucocytecell−free layer
side branch:plasma−skimminglowers hematocrit
Figure 2.1: Schematic of blood flow in the microcirculation, illustrating phenomenaof migration, margination and plasma-skimming.
such as white blood cells and platelets, to be found near the blood vessel walls, a
phenomenon known as margination. Margination is found to play a key role in the
process of inflammation35,36. Due to the presence of the cell-free layer, fewer blood
cells are drawn from the nearby wall region of large blood vessels in to the smaller
branching capillary and is hypothesized to be the cause for the lower hematocrit in
these capillaries – a phenomenon known as plasma-skimming effect. In suspensions,
the cells collide with each other intermittently. These collisions lead to substantial
velocity fluctuations which drives the diffusive motion of cells and solutes in blood
flow.
A number of mechanisms may contribute to the formation of non-uniform distri-
bution of RBCs. A deformable particle (capsule, drops, vesicle) migrates away from
8
the wall even at zero Reynolds number37,38,39. Migration arises due to the distur-
bance velocity in the fluid caused by the deformable particle as tries to relax to its
equilibrium position. This disturbance velocity is asymmetric due to the presence
of a nearby wall and tends to push the particle away from the wall. A deformable
particle, to the leading order, can be treated as a point dipole and the wall-induced
migration effect arises due to the disturbance velocity in flow caused by the image of
the point dipole on the other side of the wall. In suspensions, the cells do not migrate
continuously towards the center40.
In suspensions, the cells do not migrate continuously towards the center. The
wall-induced migration is balanced by the diffusive motion of the cells, leading to the
development of cell depleted layer. Although there has been considerable progress
towards understanding the mechanisms of wall-induced migration and shear-induced
diffusion of these deformable particles, the balance of these effects that lead to the
development of cell-depleted layer and concentration distribution remain poorly un-
derstood. Migration is also observed in suspensions of rigid particles but it can occur
only through particle interactions and at high concentrations. The net flux comes
from the contribution of gradient in shear rate and the diffusion due to concentration
gradient41,42. This shear-induced particle drift in this case also leads to the blunt-
ing of the velocity profiles43. For the case of dilute polymers, where the Brownian
diffusion is substantial, the polymer depleted layer comes from the balance between
wall-induced migration and Brownian diffusion44. In suspensions of cells, the Brow-
nian diffusion of the cells are negligible where as the shear-induced diffusion, due to
random collisions, is substantial. For the case of dilute suspensions of deformable
drops, analogous to the suspensions of deformable cells, King and Leighton45 fol-
9
lowed by Hudson46 performed a theoretical analysis on the spatial distribution of
drops accounting for wall-induced migration and shear-induced diffusion. Their anal-
ysis predicted a drop-free region analogous to the cell-free layer observed in the case of
blood flow. However, no analytical or asymptotic form was obtained for the drop-free
layer. As part of the thesis, we derive a simple variant of the these theories, which
are based on a fundamental understanding of stochastic processes, to describe the
cross-stream migration of deformable particles in suspensions.
2.2 Drag reducing additives in blood flow
As noted above, the in vivo experiments on animals show potential beneficial effects
on hemodynamics arise from addition DRAs to blood. The effects of DRAs on blood
flow have also been observed in in vitro experiments2,7. Of particular interest is the
work done by Kameneva et al.2, who studied the dependence of cell-free layer with
flow rate in their microchannel experiment of blood. They showed that the addition
of small amount of these polymer additives to the suspensions of RBCs resulted in
a redistribution of RBCs with a significant reduction in the thickness of the cell-
free layer. As shown in Fig .2.2, addition of concentration of drag-reducing polymer
as low as 10ppm can cause significant reduction in the thickness of cell-free layer.
Additionally, addition of these polymer additives also reduce the concentration of
leucocytes in the cell-free layer47.
The effects of DRAs on both turbulent flow and blood flow are determined by
their physical nature as long-chain flexible linear polymers rather than their specific
chemical composition. At sufficiently high molecular weight, concentrations of the
10
order of 10 ppm are sufficient to induce reductions in turbulent drag of 50% or more,
and concentrations at this same level lead to the physiological effects described above.
At these concentrations, the shear viscosity of the fluid (with or without blood cells)
is nearly indistinguishable from that without polymers. In contrast, the extensional
viscosity might be orders of magnitude larger than the polymer contribution to the
shear viscosity and, more importantly, even larger than the viscosity of the solvent.
In particular, the degree of drag reduction in turbulent flow is found to correlate quite
well with the ratio between extensional and shear viscosities, the Trouton ratio. The
origin of the differences between shear and extensional viscosities of DRAs is closely
associated with their long-chain nature48.For example, consider a molecule of 4000
kD poly(ethylene oxide), or PEO. This molecule has a fully extended length L of
about 30 µm. In quiescent solution, however, it exists in a random coil configuration
with an RMS end-to-end distance R0 of only a few hundred nanometers. Since a
red blood cell has dimensions on the order of several microns, we see that there is
no separation of length scales in this situation between the polymers and the cells.
The relaxation time λ, for this molecule, estimated as the time the molecule takes
to diffuse its own radius in water, is of the order of 10 ms. Given a characteristic
strain rate γ for the flow (e.g. an estimate of shear rate or extension rate), one can
define the Weissenberg number Wi. Only when Wi & 1 will polymer chains stretch
significantly in flow. How much the polymer chains stretch depends on the specific
nature of the flow. For an extensional flow with Wi & 1, the Trouton ratio is roughly
proportional to (L/R0)2 , which for 4000 kD PEO is > 104. Thus a solution of this
polymer in which the polymer makes a negligible contribution to the stress in shear
flow can exhibit large stresses in extensional flow.
11
Flow Rate (ml/min)
Cel
l−F
ree
Laye
r (µm
) ControlDRP, 10ppm
Figure 2.2: Experimental data (symbols) on the thickness of cell-free layer as a func-tion of flow rate for a suspensions of RBCs in Newtonian (Control) and drag-reducingpolymer (DRP) solutions from Kameneva et al.2.
These ideas can be applied to blood flow in the microcirculation. If we estimate
the velocity U of blood in a vessel of radius R = 16 µm to be 12 mm/s (using data
from cat mesentery given by Fung32), then the Reynolds number for blood in this
vessel is much less than unity. On the other hand, using U/R as an estimate of γ,
we find for the 4000 kD PEO described above a Weissenberg number of around 10, is
large enough to significantly stretch polymer chains. It is important to note that only
in the microcirculation are strain rates likely to be large enough to significantly stretch
DRAs. Furthermore, local shear rates in the gap between an RBC and an arteriole
or capillary wall may be very large, and collisions between RBCs during flow lead
to substantial velocity fluctuations and in particular to transient extensional flows
in the neighborhood of colliding cells. By analogy with the role of fluctuations in
12
turbulence, we hypothesize that these fluctuations significantly affect the stretching
of the polymer molecules and correspondingly their effects on the flow.
2.3 Simulations of blood flow
As noted above, the Reynolds number for the flow in the microcirculation is small,
hence the flow is governed by the Stokes equation. At the length and time scales of
interest, the details of the molecular structure of a cell can be neglected and the entire
cell can be modeled as a fluid-filled elastic capsule, which is a fluid droplet enclosed
by a solid elastic membrane. There are many computational studies on the dynamics
of capsules in flow5,49,50,51,52,53,54,55,3,56,6,57,58,4,37,59. Most of these studies of capsule
dynamics have focused on the behavior of individual capsules in flow; the work of
Lac et al.3 is of particular relevance to the present work. These authors studied
the dynamics of individual capsules in unbounded shear and extensional flow at low
Reynolds number. Fluid viscosities inside and outside the capsule were matched, with
value ηs. Two models of the membrane elasticity were used: the neo-Hookean (NH)
model, which approximates the behavior of a cross-linked rubber, and a model (SK)
proposed by Skalak and coworkers60 to describe the elasticity of a red blood cell. For
the NH model, the surface shear modulus G completely parameterizes the elasticity,
while for the SK model an additional parameter C appears, which represents the
energy penalty for area change. These models will be described in section 3.2 in
further detail. For cell membranes, C � 1. The ratio of viscous to elastic stresses
is measured by a capillary number Ca = ηsγa/G, where a is the equilibrium capsule
radius. Simulations were performed with the boundary element method51 and values
13
of C between 0.5 and 10 were considered. In an intermediate range of Ca around unity,
the deformations of the capsule were qualitatively similar to those observed for a liquid
drop. Potentially important distinctions arose at lower or higher Ca, particularly in
planar extensional flow. At small Ca, the surface wrinkles due to a buckling instability.
Lac et. al.56 have reported that the instability is due to their numerical method (use
of B-spline polynomials). Li and Sarkar61 also reported that wrinkling is numerical
rather than physical; their simulated wrinkles depend on resolution. This wrinkling
instability can be prevented by slightly inflating the capsule so that there is a small
pre-stress in the membrane even at equilibrium (see Lac and Barthes-Biesel56). At
large Ca, the interface develops regions with very large curvature. In the numerical
method of Lac et al.56, these situations lead to loss of existence of steady solutions. Li
and Sarkar61 also reported stability problems at high Ca. This loss of existence does
not occur in the smoothed spectral boundary element simulations of Dodson and
Dimitrakopoulos58 or in the noval coupling method of Walter et. al.62, indicating
that the loss of existence is numerical rather than physical. At a given value of
Ca, as C increases, the degree of deformation of the membrane decreases, and the
range of Ca where steady solutions can be obtained dramatically increases. Similar
observations are found in simulations of uniaxial extension. In planar extension,
related work by Dodson and Dimitrakopoulos predicts the possibility of multiple
steady state configurations of the capsule58.
The dynamics of pair collisions of capsules in shear has also been studied. Bagchi
et al.63 studied this problem in the context of evaluating models of aggregation of
RBCs. Their simulations were two-dimensional, using a NH model for the mem-
brane, a Lagrangian finite element discretization of the membrane first presented by
14
Charrier et al.64 and a front-tracking immersed boundary method (IBM) for the fluid
dynamics65,66. They predicted that increasing internal viscosity or membrane rigid-
ity – both of which suppress cell deformation – favor aggregation, in agreement with
physiological observations. Doddi and Bagchi4 performed a related study in three-
dimensions to determine the effect of inertia on pair collisions of capsules. Jadhav et
al.67 performed three-dimensional simulations, also using the NH model and IBM for
the fluid dynamics of collisions during shear, focusing on the shape of the “contact
region” between the colliding capsules: i.e. the region where the capsules are in clos-
est proximity. At low capillary number, the two capsules remain convex throughout
the collision and this region is an elongated disk. At high capillary numbers however,
the capsules each display a dimple in the collision region that arises due to the high
pressure generated by the squeezing flow in that region. Correspondingly, the contact
region becomes a distorted annulus. Lac et al.57,6 examined pair collisions of NH
capsules, focusing on the net displacements due to collisions – i.e. the displacements
that lead to shear-induced diffusion in suspensions. The main result of these studies
was that although capsule shapes in shear appear qualitatively similar to those of
drops, the collision dynamics of capsules and drops can be qualitatively different.
Recent advances in numerical methods have enabled researchers to study large
scale simulations of deformable capsules68,69,70. Of particular interest is the work
done by Freund34, who used an accelerated boundary element formulation to study
the margination of leucocytes in presence of deformable red blood cells in a 2-D mi-
crovessel. Through his model, he was able to predict the blunting of velocity profiles
(plug-flow behavior) and observed the formation of a cell-free layer the thickness of
which increased with the flow rate. Accelerated boundary integral formulations69 are
15
gaining considerable interest to be implemented to study multiscale simulations of
deformable particle. Doddi and Bagchi70 used an immersed-boundary formulation
to perform three dimensional simulations of multiple deformable cells (capsules and
RBCs) in microchannel. Through their simulations, they showed the blunting of ve-
locity profiles, the cell-free layer dependence on the hematocrit and the non-monotonic
behavior of effective viscosity on vessel diameter (Farhaeus-Lindqvist effect). Other
methods such as lattice Boltzmann method (LBM) have also been used to study the
suspensions of RBCs71,72,73 but the focus of these studies have been on the rheology
rather than the dynamics of suspensions of cells. Recent numerical simulations of
pair collisions of capsules in polymer solutions of our current work (Pranay et al.74)
showed dramatic effect of DRAs on collision dynamics – polymers suppressed the
net displacement of capsules after the collision. The uniaxial extensional flow gener-
ated in the gap of the colliding capsules stretched the polymers significantly and the
stretching worked against the separation of the departing capsules after the collision.
To our knowledge, no numerical study of the dynamics of suspensions of capsules in
polymer solutions has been performed.
The remainder of this work is organized as follows. Chapter 3 describe the models
and discretization methods used for the capsules and polymer molecules. Chapter 4
reports results for single capsules in shear, as background for Chapter 5, which de-
scribes and discusses pair collisions in Newtonian fluids and polymer solutions. Chap-
ter 6 describes and discusses the dynamics of suspensions of capsules in Newtonian
fluids and polymer solutions in a Couette flow. Specifically in Section 6.1 we describe
the theory for the suspensions of capsules as a balance of two competing mechanisms
and derive the closed-form expression for the capsule-free layer and Chapter 6.2 re-
16
ports results for the migration of a single capsule in a Couette flow in Newtonian
and polymer solutions. Chapter 7 provides a brief summary of the current study.
Chapter 9 shows some preliminary work on pressure-driven flows along with some of
the proposed future work with theory. Appendix A reports the validation for using
an approximate treatment of the Brownian motion of polymer molecules. Appendix
B elucidates the relationship between the present approach to velocity computations
and conventional immersed boundary and boundary integral methods. Appendix C
provides the details of implementation of the stokes flow solver (GGEM) used in the
current study.
17
Chapter 3
Models and discretization methods
3.1 Overview
We consider the motion of liquid capsules and polymer molecules, immersed in a
surrounding fluid of viscosity ηs, under shear flow with strain rate γ between two
parallel plates as shown in Figure 1.2. The simulation box is periodic in the flow
(x) and vorticity (z) directions. The wall-normal direction is y. The lengths of the
domain in x, y and z are Bx, By and Bz, respectively. Time is non-dimensionalized
with 1/γ as t∗ = γt. Each capsule is comprised of a nominally spherical elastic
membrane of radius a, enclosing a Newtonian liquid with density and viscosity equal
to that of the outside fluid. The membrane has a two-dimensional elastic modulus
G and negligible thickness, and we characterize the capsule dynamics as a function
of capillary number Ca = ηsγa/G. As detailed below, two models for the membrane
elasticity will be studied. The polymer molecules are modeled (as detailed below)
as bead-spring chains, with three beads connected together by finitely extensible
18
non-linear elastic (FENE) springs. The maximum end-to-end length of the polymer
molecules is sufficiently large to mimic a high-molecular weight (> 106 Da) polymer,
the details of which will be described in section 3.3. The polymer contribution to
the fluid viscosity is denoted ηp. The relevant dimensionless numbers for the polymer
molecules are the Weissenberg number, Wi, the viscosity ratio β = ηs/(ηs + ηp), and
the extensibility parameter, Ex, defined as the steady state Trouton ratio in the limit
of high extension rate. We consider the case of very dilute long-chain polymers, where
1−β � 1 but Ex> 1, so that polymer effects are negligible in pure shear flow but can
be important in flows with a substantial extensional component. Finally, the relative
sizes of polymer chains and capsules is important. The polymer molecules have a
mean equilibrium end-to-end distance of about 450 nm and fully extended length 30
µm, while the equilibrium radius a of the capsules is 3 µm.
3.2 Capsule Model
We use two different models for the capsule membrane. The first is a neo-Hookean
model, which mimics the behavior of rubber-like materials, and the second is a model
of the red blood cell (RBC) membrane that was originally proposed by Skalak et al.60.
The Skalak model can be parametrized to yield a strong resistance to area change
relative to its resistance for shear deformation. The mathematical details of the two
models and their application will be discussed after we introduce some formalism for
describing the kinematics of membrane deformation.
This formalism is mostly clearly presented for deformations in a plane, which for
the moment we take to be the xy plane. Let (x, y) and (X, Y ) denote respectively
19
the undeformed and deformed coordinates of a material point, with respect to a fixed
set of Cartesian axes. If u and v denote the displacements of the material point in x
and y directions respectively, then
X = x+ u,
Y = y + v.(3.1)
The relation between the position vectors of two points infinitesimally close to each
other, before and after deformation is given by
dX
dY
=
1 + ∂u∂x
∂u∂y
∂v∂x
1 + ∂v∂y
dx
dy
, (3.2)
or compactly as
dX = F · dx. (3.3)
The square of the distance between the two neighboring points after deformation is
given by
dS2 = dX · dX = dx ·G · dx,
G = FT · F,(3.4)
where G is a symmetric, positive definite matrix. The elements of G are given by
the expressions
G11 =
(
1 +∂u
∂x
)2
+
(
∂v
∂x
)2
,
G22 =
(
∂u
∂y
)2
+
(
1 +∂v
∂y
)2
,
G12 = G21 =
(
1 +∂u
∂x
)(
∂u
∂y
)
+
(
1 +∂v
∂y
)(
∂v
∂x
)
.
(3.5)
20
The principal stretch ratios, λ1 and λ2, are defined as the eigen values of G, and are
given by the expressions,
λ21 =
12
(
G11 +G22 +√
(G11 −G22)2 + 4G2
12
)
,
λ22 =
12
(
G11 +G22 −√
(G11 −G22)2 + 4G2
12
)
.(3.6)
For a thin membrane that displays no resistance to bending, the strain energy
density W of the membrane is a function of λ1 and λ2. Following Barthes-Biesel et
al.55, for a neo-Hookean model the strain energy density function is given by
WNH =G
2
[
I1 − 1 +1
I2 + 1
]
. (3.7)
Here G is the two-dimensional shear modulus for the membrane, having units of force
per unit length. The two invariants, I1 and I2 are given by
I1 = λ21 + λ2
2 − 2, I2 = λ21λ
22 − 1. (3.8)
The Skalak model60 has the strain energy density
WSK =G
4
[(
I21 + 2I1 − 2I2)
+ CI22]
. (3.9)
The Skalak model contains a shear modulus G and an additional parameter C associ-
ated with the energy penalty for area change; the area dilation modulusK is related to
the shear modulus G (using the nomenclature of Skalak et. al.60) as K = 2G(1+2C).
Typically, C � 1 indicating approximate area-incompressibility.It has been shown55
that under a simple uniaxial deformation, results for the Skalak model reach an
21
asymptotic value for C ≥ 10. We will be considering C = 0, 1 and 10 below, focus-
ing on the case C = 10. We adopt the finite element method developed by Charrier
et al.64 to describe the surface of the deformable particle, and a variant of the im-
mersed boundary method75 to describe the fluid-structure interaction. These choices
result in a discretized description that is first-order accurate in the element size. In
this approach, the membrane is discretized into flat triangular elements, in which the
strain is uniform and which are assumed to remain flat even after deformation. The
element corners, or nodes, are taken to move with the local fluid velocity as required
by the no-slip boundary condition. That is,
dxci
dt= u (xc
i) , (3.10)
where xci is the position of the ith node and u (xc
i) the fluid velocity evaluated at that
node. This expression is integrated with a second-order Adams-Bashforth method.
Results presented here are computed with a time step ∆t of 5·10−3. As is conventional
in immersed boundary descriptions of fluid-structure interaction75, the forces exerted
by the fluid on the membrane and vice versa are taken to be localized at these
nodes. Since the inertia and Brownian fluctuations of the membrane are taken to
be negligible, the total force exerted on any point on the membrane is equal to zero
at any instant. Thus, the elastic membrane forces f ci on each node of the capsule have
to be balanced by the hydrodynamic forces fhi exerted on that node:
f ci + fhi = 0, (3.11)
22
The reaction force exerted by the node on the fluid is thus given by
fhf (xci) = −fhi = f ci . (3.12)
The force exerted by the nodes on the fluid enters into the computation of the fluid
velocity field, as described in Section 3.4.
The description of the capsule motion is completed by the computation of the
nodal forces f ci . In the Charrier et al.64 approach, these forces are determined by
computing displacements of the vertices of the deformed elements with respect to the
undeformed elements and applying the principle of virtual work. To facilitate the
determination of the principal stretch ratios λ1 and λ2, each element in the deformed
state and the corresponding un-deformed element are transformed to a plane by
rigid body rotations, using a transformation matrix Rα, for each element α. The
deformation of each element is then calculated using the positions of the nodes in the
deformed state relative to their positions in the undeformed state. The deformation
at any point inside the element is calculated by interpolating linearly from the nodes.
The principal stretch ratios can then be calculated from the nodal displacements.
Having chosen a suitable membrane strain energy density function, and having
calculated the nodal displacements, the forces at node i of an element along two per-
pendicular directions in the plane of the element are calculated using the expressions
fLx,i = −Ae
[
∂W∂λ1
∂λ1
∂ui+ ∂W
∂λ2
∂λ2
∂ui
]
,
fLy,i = −Ae
[
∂W∂λ1
∂λ1
∂vi+ ∂W
∂λ2
∂λ2
∂vi
]
.(3.13)
Here, fLx and fL
y are the nodal forces along the two perpendicular directions, x and
23
y, the “local” or transformed co-ordinate system, and Ae is the undeformed area of
the element. After computing the deformation forces in the transformed co-ordinate
system, the global components of the nodal forces for element can be calculated by
transforming back to the global Cartesian coordinates using the expression fαi =
(Rα)T · fLi . The resulting forces are the nodal forces with respect to a fixed Cartesian
(global) coordinate system. The total elastic force on a capsule node is calculated as
the sum of forces resulting from the deformations of triangular elements surrounding
that node, and is given by
f ci =∑
α
fαi , (3.14)
where the summation is over all triangular elements to which the node belongs.
3.3 Polymer Model
Each polymer molecule in our simulations is described using a coarse-grained model-
ing approach appropriate for highly flexible polymer molecules76,77. In this approach,
each molecule is represented as a string of beads connected by springs. Each bead
represents a large number of polymer segments and has a given Stokes law friction
coefficient ζ ; corresponding drag and Brownian forces are exerted on it by the fluid.
The springs connecting the beads reflect the connectivity of the polymer chain and the
effect of entropy in driving the chains toward an equilibrium random coil conforma-
tion. The simplest nontrivial bead-spring chain model (and the least computationally
expensive) is a dumbbell model with only two beads. This captures the longest time
and length scales of the chain and in many cases is adequate for qualitative prediction
24
of flow properties of polymer solutions. However, in the situation under consideration
at present, the fully extended length of a polymer chain is substantially larger than
a capsule – if we used a dumbbell model in this case, it would be highly probable
that although the two beads of the dumbbell remain outside the capsules, the spring
connecting the beads would pass right through a capsule. To minimize the possibility
of such an unphysical event, we use a three-bead chain. We now turn to specific
aspects of the polymer model.
The polymer is very loosely modeled after a 4×106 D poly(ethylene oxide) (PEO)
molecule. The number density of molecules is n. We take the contour length of
each molecule, LC , to be 30 µm, with a Kuhn length, l = 1 nm corresponding to
nk = LC/l = 3×104 Kuhn segments per polymer molecule. The spring force exerted
on each bead by its nearest neighbor(s) is given by the so called FENE (Finitely
Extensible Nonlinear Elastic) force law77:
fpi,i±1 = −Hqi,i±1
1−( qi,i±1
L
)2 , (3.15)
where H is the spring constant, qi,i±1 = xpi±1 − xp
i is the vector connecting the two
beads, and L = LC/2 is the contour length of each chain segment. The spring constant
H is calculated using the relation
H =3kBT
R20,s
, (3.16)
where kb is Boltzmann’s constant, T is temperature and R0,s is the average end to end
distance of each chain segment at equilibrium. Thus the total spring force exerted on
25
each bead is
fpi = fpi,i+1 + fpi,i−1. (3.17)
For beads at either end of the chain, one of these terms will be missing. Assuming
good solvent conditions, which is valid for PEO in water, the average end to end
distance is estimated using:
R0,s ≈(
nk
Nb − 1
)35
l = 320 nm, (3.18)
where Nb is the number of beads in the chain (here it is always three). For a three-
bead chain this expression yields an RMS end-to-end distance R0 =√
2R20,s = 452
nm. The bead radius ab is taken to be 130 nm, and the corresponding Stokes drag
coefficient is
ζ = 6πηsab. (3.19)
For chains with such small numbers of beads, it is reasonable to estimate dynamic
properties with the Rouse model77. The polymer contribution to the viscosity is
estimated from this model to be
ηp =2nkBTζ
3H, (3.20)
and the stress relaxation time as
λ =ηp
nkBT= 6.7ms. (3.21)
26
The Weissenberg number is given by
Wi = λγ, (3.22)
where γ is the shear rate. In our polymer simulations, Wi = 5. The parameter
β =ηs
ηs + ηp(3.23)
measures the ratio of the viscous stress to the total stress in simple shear. The
polymer concentration is proportional to 1 − β. The extensibility parameter Ex
measures the steady state ratio between polymer stresses and viscous stress in uniaxial
extensional flow in the limit of high extension rate. For a three-bead FENE chain,
with hydrodynamic interactions between beads neglected, it is straightforward to find
a closed form expression for this parameter:
Ex =nζL2
C
12ηs(3.24)
In our simulations with polymers, we use β = 0.9985, 0.997 and 0.994, which
correspond to number density n of 0.0235µm−3, 0.0470µm−3 and 0.0940µm−3, or a
mass fraction of 0.157 ppm, 0.314 ppm and 0.628 ppm respectively. The numbers of
polymer chains in the simulation domain for these three cases are 2300, 4600 and 9200,
respectively. These are very low concentrations, much smaller than typically used in
experiments. These were chosen to keep the cost of the computations reasonable,
while still allowing polymer effects to be substantial. In particular, 1− β � 1 for all
situations studied here, indicating that in shear flow, the stresses due to the polymer
27
chains will be very small. On the other hand, Ex ranges from 4.3 to 17.2, indicating
that in extensional flow, polymeric extensional stresses can be much larger than the
viscous extensional stresses. So although the polymer concentration is very low, we
are still in the regime of primary interest for drag reducing polymer solutions: the
situation where the shear viscosity of the solution is barely changed by the polymer
additives, but the extensional viscosity is changed substantially.
The equations of motion for each bead of the chain is determined by the balance
of Stokes drag between the bead and the surrounding fluid, the spring forces acting
between neighboring beads, and the Brownian forces exerted by the fluid on the
bead. It is important to emphasize that, in contrast to the situation for nodes on
the surface of the capsules, the beads are not taken to move with the fluid velocity.
In other words, we do not aim to resolve the details of the fluid motion on the scale
of each polymer bead, but rather use a standard coarse-grained description of the
interaction between the polymer and the surrounding fluid. As we see below, this
difference between the treatment of the capsule-fluid and polymer-fluid interaction
leads to some subtleties in the computation of the fluid velocity (see Figure 1.2)
Turning to the Brownian force, the fluctuation-dissipation theorem implies a non-
trivial coupling between the Brownian motion and the configurations of the polymer
molecules and capsules. This coupling is expensive to compute78. In the present sim-
ulations we are interested only in dilute solutions at high Weissenberg number, where
Brownian motion is relatively unimportant (Weissenberg number is proportional to
particle Peclet number). Therefore, we do not compute the full Brownian term, but
rather apply to each bead the Brownian displacement it would exhibit in isolation
in an unbounded domain. Under this approximation, the force balance leads to the
28
following evolution equation79 for polymer bead positions xpi :
dxpi =
[
u (xpi ) +
1
ζfpi
]
dt+
√
2kBT
ζdW, (3.25)
where u (xpi ) is the fluid velocity at the position of bead i and dW is a vector of
independent random variables, each chosen from a Gaussian distribution with zero
mean and variance dt. This equation is integrated with the stochastic explicit Euler
method79 . The same time step is used for this equation and the evolution equation
for capsule node positions.
3.4 Fluid velocity calculation
The fluid velocity u (x) is driven by the imposed velocities of the top and bottom
walls and by the forces that the capsules and polymer beads exert on the fluid. The
forces exerted by the polymer beads are localized at the bead positions and the forces
exerted by the capsule are localized at the nodal positions. In both cases we will
treat these localized forces as regularized delta functions as is conventional in polymer
dynamics79 and in immersed boundary methods for fluid-structure interactions65,75,80.
The method now described for determining the interaction between capsules and
fluid motion is a variant of the immersed boundary method that takes advantage
of a recently-developed algorithm78 for efficiently computing Stokes flow driven by
regularized point forces in arbitrary geometries. This approach can also be formulated
starting from the boundary integral equation for a deformable particle in flow; the
relationships between the present method and the conventional immersed boundary
and boundary integral methods are described in the Appendix B.
29
As mentioned above, we describe the forces exerted by the membranes and poly-
mers as a distribution of regularized point forces: i.e. as a force density distribution
ρ(x) in the fluid given by:
ρ(x) =nodes∑
i=1
δc (x− xci) f
ci (x
ci) +
beads∑
i=1
δp (x− xpi ) f
pi (x
pi ) , (3.26)
with regularized delta functions that have a quasi-Gaussian form:
δc (r) = ξ3cπ3/2 e
(−ξ2cr2) [5
2− ξ2c r
2]
,
δp (r) =ξ3p
π3/2 e(−ξ2pr
2) [52− ξ2pr
2]
.(3.27)
Here f ci and fpi are the forces exerted by nodes (at positions xci) and beads (at positions
xpi ) respectively, and ξc and ξp are the corresponding regularization parameters for
the delta functions; their reciprocal represents the length scale over which the force
is spread.
Consider first ξc, the regularization parameter for points on the capsule surface.
The approach that we are using for the capsule dynamics is essentially a Stokes
flow/Green’s function-based variant of the IBM developed by Peskin65,75,80. In this
method, force distributions at moving interfaces or membranes are discretized as
distributions of regularized point forces, where the length scale for smoothing the
delta function scales as the grid spacing used in the simulation. That is, if hc is a
characteristic node spacing (or element size) on the surface, then we choose
ξc ∼ h−1c (3.28)
This ensures that the force density associated with each node is spread over the
30
length scale of the associated elements, thereby preventing fluid from penetrating the
membrane surface, and also preventing the unphysically large fluid (and thus node)
velocity that would be present at the node if ξc is made too large (in which case δc(r)
approaches a true delta function). As with the conventional IBM, the simulation
results are insensitive to the choice of the regularization parameter provided its value
lies within a range where ξch = O (1) . Computations to determine the specific value
of ξc are presented in the results section.
For the polymer beads, the choice for ξp is straightforward and scales as the size of
the bead radius, ξp ∼ a−1p . Further, the pair mobility tensor will be positive definite
for ξ−1p ≥ 3ap/
√π. We take this value here.
Having specified the form of the force density associated with the capsule and
polymer molecules, we turn to determining the velocity field driven by that density.
This velocity u is the solution of Stokes’ equations,
−∇P (x) + η∇2u(x) + ρ(x) = 0,
∇ · u(x) = 0(3.29)
subject to periodic boundary conditions in x and z and a no-slip boundary condition
on y = ±By/2:
u(y = ±By/2) = (±γBy/2) ex. (3.30)
Eq. C.1 needs to be solved at each time step with a different force density. To
accomplish this, we use a particle-particle/particle-mesh method that has recently
been developed for Stokes flow problems in nonperiodic geometries, called GGEM
(General Geometry Ewald-Like Method)78,81. We will denote the combination of
31
GGEM with our immersed boundary treatment of the fluid-surface interaction as
IBM/GGEM. Its specific implementation for the present situation is now described.
In what immediately follows, no distinction is made between nodes on a capsule and
polymer beads. The differences between the two will be described in detail later.
GGEM starts with splitting the regularized force density, Eq. 3.26 into two parts,
ρ(x) = ρl(x) + ρg(x),
ρl(x) =∑
i
[δγi (x− xi)− g (x− xi)] fi,
ρg(x) =∑
i
g (x− xi) fi,
(3.31)
where the sums are over both polymer beads and capsule points, so γi = c or p
depending on whether point i is a capsule node or polymer bead. The “screening
function”, g (x) is a regularized delta function that is used to split the force density
into a “local” and “global” part denoted by the subscripts l and g such that the
velocity field driven by ρl(x) decays very rapidly away from the location of each
point force. This function g (x) takes the same mathematical form as the regularizing
functions and is given by
g (x) =α3
π3/2e(−α2r2)
[
5
2− α2r2
]
, (3.32)
where α is the so-called screening parameter and r = |x|. Formally, this parameter
is arbitrary; it is chosen to yield a computationally efficient algorithm. By linearity,
the flow field is a superposition of flow driven by ρl(x) and ρg(x):
u(x) = ul(x) + ug(x). (3.33)
32
Somewhat analogously to Ewald summation methods (see e.g. Sierou and Brady82),
the general idea in GGEM is to separately compute the velocity fields due to the
two force densities. The total velocity is then simply the sum of the individual
contributions.
Consider first the local velocity ul(x), which results from the force density ρl(x).
This is computed analytically using solutions to Stokes’ equations in an unbounded
domain – the error incurred by assuming an unbounded domain will be cancelled
out in the global solution described below. One can show that the modified force
distribution, Eq. (B.8) yields a velocity field given by
ul(x) =∑
i
GR,γil (x− xi) · fi, (3.34)
with the local or screened Green’s function GR,γl given by
GR,γl (x) =
1
8πηs
[
δ +xx
r2
]
[
erf (ξγr)
r− erf (αr)
r
]
− 1
8πηs
[
δ − xx
r2
]
[
2α
π1/2e−α2r2 − 2ξγ
π1/2e−ξ2γr
2
]
, (3.35)
where as previously, γ = c or p. Because of the presence of the screening function
in the local force density, this function decays rapidly to zero, as exp(−α2r2). The
calculation of the local velocity filed at a given point x begins by first identifying
point forces located within a sphere of fixed radius 4/α. Any force outside this cut-off
radius is ignored. The choice of α is discussed below, and is constant for a given
simulation. For any point x in the fluid which is not on the membrane surface or a
33
polymer bead, the local contribution to the velocity is given by
ul (x) =
n∑
i=1
GR,cl (x− xc
i) · f ci +m∑
i=1
GR,pl (x− xp
i ) · fpi . (3.36)
Here n and m respectively are the number of capsule nodal points and polymer beads
which lie within the cut-off sphere with center at x. For a nodal point xcj on the
capsule, the local velocity can be written as
ul
(
xcj
)
=n
∑
i=1,i 6=j
GR,cl (x− xc
i) · f ci +m∑
i=1
GR,pl (x− xp
i ) · fpi + limx→0
GR,cl (x) · f cj
=n
∑
i=1,i 6=j
GR,cl (x− xc
i) · f ci +m∑
i=1
GR,pl (x− xp
i ) · fpi +1
8πηs
[
4ξc√π− 4α√
π
]
· f cj .(3.37)
This approach avoids problems that arise in naive evaluation of Eq. B.15 when
r = 0. For a point xpj corresponding to a polymer bead, the local velocity is given by
ul
(
xpj
)
=
n∑
i=1
GR,cl (x− xc
i) · f ci +m∑
i=1,i 6=j
GR,pl (x− xp
i ) · fpi , (3.38)
The difference between Eqs. 3.37 and 3.38 is the exclusion of the “self-term” in the
local velocity calculation for the polymer beads in the latter. (Recall that in contrast
to the capsule nodes, the polymer beads do not move with the fluid velocity, so the
velocity “seen” by the polymer bead should not include the singular velocity that the
bead itself generates as it moves through the fluid.) Eqs. 3.36, 3.37 and 3.38 give the
local contribution to the velocity at any point in the system.
We now proceed to the calculation of the global contribution to the velocity, ug (x),
which is due to the global part of the force distribution, ρg (x) in Eq. B.8 and the
34
imposed boundary conditions for velocities on top and bottom walls (Eq. 3.30). If
ρg = 0 then ug = u∞. In GGEM the solution to the Stokes equations with a forcing
function ρg (x) is calculated numerically, requiring that the total velocity u (x) satisfy
appropriate boundary conditions. At points xW on the bounding walls, we therefore
have that
ug (xW) = −ul (xW) + u (xW) , (3.39)
where ul (xW) is the known local velocity field evaluated at the wall and u (xW) is
the boundary condition which comes from imposing velocities on top and bottom
walls (Eq. 3.30). Using the known local velocity field at the wall in the boundary
condition, we cancel out the error introduced by using unbounded-geometry solutions
for the local problem. For periodic boundary conditions (in the streamwise and
vorticity directions in the current system), we discretize using Fourier collocation.
Accordingly, the error in the solution in the periodic directions scales exponentially
with mesh resolution. However, it should be noted that the adoption of the Fourier
methods is merely for convenience and accuracy – in principle could be replaced by
other methods such as finite difference or finite elements. In the wall-normal direction,
we use a second-order finite difference scheme – the error decays quadratically with
mesh spacing in y. We choose the number of mesh points in each directions Mi
according to
Mi =√2αBi ; i = x, y, z, (3.40)
where the Bi’s are the box lengths in the simulation. It is important to note that
in traditional immersed boundary methods, the mesh spacing for the fluid solver is
determined by the regularization parameter for the capsule nodes. In contrast, with
35
the present method, the mesh spacing is set by the choice of α. See Appendix ?? for
further discussion of this point. Unless otherwise noted, the computations reported
here use αa = 1. All of the small scale motions are part of the local computation,
which is done analytically so there is no discretization error. The global contribution
to the velocity is calculated at the mesh points and interpolated to any point, x, in
the system using quadratic Lagrange polynomials. It is this interpolation step that
ultimately controls the order of accuracy of the solution.
Finally, we note that, as with the local velocity calculation, the global velocity
calculation at a polymer bead needs to exclude the self-interaction term. The global
velocity field associated with this term is the free-space velocity driven by the bead’s
contribution to ρg(x). This velocity field is determined by the free-space regularized
Stokeslet
GR,α∞ (x) =
1
8πηs
[
δ +xx
r2
]
(
erf (αr)
r
)
+1
8πηs
[
δ − xx
r2
]
(
2α
π1/2e−α2r2
)
, (3.41)
which reduces to the Oseen-Burgers tensor as α → ∞. Now the velocity on a polymer
bead can be written as
u(
xpj
)
= ul
(
xpj
)
+ ug
(
xpj
)
− limx→0
GR,α∞ (x) · fpj
= ul
(
xpj
)
+ ug
(
xpj
)
− 1
8πηs
4α√π· fpj . (3.42)
36
3.5 Volume correction
While the regularization of point forces on the capsule membrane reduces fluid move-
ment across the membrane surface, fluid penetration is not completely eliminated at
the level of discretization used in our simulations. This causes the volume of fluid
enclosed by the capsule to vary, which is undesirable. To prevent this, we add a
constraint to the motion of capsule nodes such that the volume remains constant.
Consider a deformed capsule centered at the origin. The volume V of the capsule is
given by
V =
∫∫∫
V
dV =1
3
∫∫∫
V
(∇ · x) dV, (3.43)
where x is the position vector of a point on the surface of the capsule. Using the
divergence theorem we get
V =1
3
∫∫∫
V
(∇ · x) dV =1
3
∫∫
S
(x · n) dS, (3.44)
where n is the outward unit normal to the surface S. Following Freund34, we seek
displacement corrections z that minimize the function
I = Ψ
1
3
∫∫
S
(x + z) · n dS − V0
+
∫∫
S
(z · z) dS, (3.45)
where V0 is the initial volume of the undeformed capsule and Ψ is a Lagrange multi-
plier. The minimization is achieved by equating the variation of I, δI to zero,
δI = δΨ
1
3
∫∫
S
(z · n) dS + (V − V0)
+
∫∫
S
δz ·(
2z+Ψ
3n
)
dS = 0, (3.46)
37
Since δΨ and δz are arbitrary, we get
2z+Ψ
3n = 0,
1
3
∫∫
S
(z · n) dS + (V − V0) = 0.(3.47)
Solving the two equations, we get the corrections z as
z = −3V − V0
An, (3.48)
Where A is the area of capsule membrane. This correction to the nodal displacement
is applied at every time step.
38
Chapter 4
Single capsule dynamics
4.1 Method and code validation
In this section we report validation tests of our methodology and code for behav-
ior of single capsule behavior. For this, we compare results of single capsules in
shear flow with earlier results. In this case the capsule membranes are in a stress-
free state at equilibrium. Fig. 4.1(a) shows plots of Taylor deformation parameter,
D = (Lmax − Lmin) / (Lmax + Lmin), as a function of dimensionless time NH capsules
at various values of Ca, where Lmin and Lmax are the smallest and largest dimensions
of the capsule in the shear plane. The capsule is placed at the center of a simula-
tion box. For this and for all future results, the capsule surface is discretized into
1280 triangular elements, corresponding to 642 nodes. Results from the boundary
integral simulations of Lac et al.3 are also shown; good agreement is found. Figure
4.1(b) shows a comparison of steady state deformation parameter for NH capsules as
a function of Ca with the numerical results of Doddi and Bagchi4, Lac et al.3 and
39
0 2 4 6 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
t∗
D
Ca = 0.075Ca = 0.15Ca = 0.30Ca = 0.45Ca = 0.60
(a) D vs. t∗
0 0.1 0.2 0.3 0.4 0.5 0.60
0.1
0.2
0.3
0.4
0.5
0.6
Ca
D
SimulationDBLBRP
(b) steady-state D vs. Ca
Figure 4.1: (a) Deformation parameter as a function of time for single non-prestressedNeo-Hookean (NH) capsules in shear flow in a cubic box of size 12.5a. Symbols aresimulations from Lacet al.3. Lines are results from present work. (b) Steady statedeformation parameter as a function of Ca for single non-prestressed Neo-Hookean(NH) capsules in shear flow in a cubic box of size 12.5a. For comparison, results ofDoddi and Bagchi4(DB), Lac et al.3(LB), Ramanujan and Pozrikidis5(RP) are alsoshown.
40
0.4 0.6 0.8 1 1.2 1.4 1.610
−5
10−4
10−3
10−2
10−1
1/(hcξ
c)
|D −
DLB
|/DLB
Figure 4.2: Relative deviation of deformation parameter (D) compared to the resultof Lac et al.3 (DLB) plotted as a function of normalized regularization parameter ξchc
for a single non-prestressed NH capsule in shear flow at Ca = 0.30 in a cubic box ofsize 12.5a.
Ramanujan and Pozrikidis5. Again we observe good agreement with the previous
numerical results with closest agreement to those of Lac et al.3. For the neo-Hookean
capsule at Ca = 0.6, a slight overshoot in D is observed before it reaches its steady
state value. A similar overshoot is seen in the orientations and stress for a suspension
of Brownian rigid rods77 for sufficiently large Weissenberg number (which is analo-
gous to Capillary number in the present problem). This arises because at short times,
the capsule surface tends to move affinely with the flow, with relaxation to the steady
shape and motion following at later times.
Continuing the comparison of our results with those of Lac et al.3, Fig. 4.2 shows
the absolute value of the relative error in the steady-state deformation parameter
(assuming for the moment that the solution of Lac et al.3 is exact) as a function of
the regularization parameter ξc for NH capsules at Ca= 0.30. As expected for an
41
0 0.01 0.02 0.03 0.04 0.050
0.02
0.04
0.06
0.08
0.1
Ca
D
NH, SimulationSK, SimulationNH, TheorySK, Theory
Figure 4.3: Steady state deformation parameter at low Ca for single non-prestressedNeo-Hookean (NH) and Skalak (SK) capsules (C = 10) in shear flow in a cubic boxof size 25a. Lines are theoretical predictions from Barthes-Biesel55 and symbols arefrom present simulations.
immersed boundary method, the solution is most accurate when this parameter is
neither too small nor too large. For the remainder of the paper, this parameter is set
to (hcξc)−1 = 0.75, where the relative error for this particular case is 3 × 10−4. As
another validation of our methodology, we compare to analytical results of Barthes-
Biesel et al.55 for the dependence of deformation parameter D on capillary number
for capsules in the small capillary number limit in unbounded shear flow, for both the
NH and SK models. This comparison is shown in Fig. 4.3 – agreement is excellent,
especially considering that the simulations were performed in a bounded domain of
size 25a, rather than the unbounded domain considered in the perturbation analysis.
Further validation of the methodology in the case of pair collisions is described in
Section 5.1.
42
4.2 Single capsule in shear
For single capsules with relaxed membranes at equilibrium, the steady state defor-
mation parameter in Newtonian flow as a function of capillary number is shown for
NH and SK capsules in Fig. 4.4(a). (Unless otherwise noted, all SK results are
with C = 10, representing the case where area changes are strongly penalized.) As
expected, the NH capsule deforms more than the SK capsule for a given capillary
number, because of the presence of substantial energy penalty for area changes in
the SK model. Results for these capsules, however, are complicated by the presence
of wrinkling instabilities at small Ca and large curvatures at higher Ca, as noted in
Section 2.3. The wrinkling instabilities are more severe in the case of pair collisions.
To avoid wrinkling, Lac et al.56,6 suggested preinflating the capsule to ensure that
stresses on the surface remain purely tensile. Therefore, following Lac et al.56, we
also examine D vs Ca for capsules that have been preinflated to have a radius that
is enlarged by a factor of 1.05. The value for preinflation was chosen based on simu-
lations of pair collisions by Lac et al.56,6, who observed that a minimum preinflation
of 5% was required to prevent the appearance of compressive stresses. These results
are also shown in Fig. 4.4(a). Preinflation has a slight negative effect on D for the
NH capsule and a larger negative effect on the SK capsule. From Fig. 4.4(a) it can
be seen that the deformation for the NH capsule is always larger than that for the
SK capsule. For example, the D for the SK capsule at Ca = 0.6 is virtually the same
(0.20) as the D for the NH capsule at Ca = 0.142. We expand on this observation in
Fig. 4.4(b), which shows the time evolution of D for the NH capsule at Ca = 0.142
and the SK (with C = 10) capsule at Ca = 0.60. Not only are the values of D close,
but the transient evolution and the detailed shapes of the capsules (shown in the inset
43
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.1
0.2
0.3
0.4
0.5
0.6
Ca
DNH non−preinflatedSK non−preinflatedNH preinflatedSK preinflated
(a) Effect of preinflation on steady state deformation parameter
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
t*
D
NH, Ca = 0.142SK, Ca = 0.60NH, Ca = 0.60
(b) Deformation parameter for preinflated NH and SK capsules
Figure 4.4: (a) Steady state values of deformation parameter at different Ca for singlenon-prestressed NH and SK (C = 10) capsules (shown by bold lines) and for prein-flated NH and SK capsules (shown by dotted lines) in shear flow. (b) Deformationparameter for preinflated NH and SK (C = 10) capsules under shear flow in a cubicbox of size 12.5a. Images correspond to capsule shapes taken at t∗ = 10..
44
0 2 4 6 8 10−4
−3
−2
−1
0
1
2
3
4x 10
−3
t*
DN
− D
PNH, Ca = 0.142SK, Ca = 0.60NH, Ca = 0.60
(a) DN −DP
0 2 4 6 8 10−4
−3
−2
−1
0
1
2
3
4x 10
−3
t*
(θN
− θ
P)/π
NH, Ca = 0.142SK, Ca = 0.60NH, Ca = 0.60
(b) θN − θP
Figure 4.5: Difference in (a) deformation parameter and (b) inclination angle forpreinflated capsules in Newtonian fluid and a polymer fluid (β = 0.997) under shearflow in a cubic box of size 12.5a. Subscript N and P correspond to Newtonian andpolymer case respectively.
45
of Fig. 4.4(b)) are quite similar. For comparison, results for the NH capsule at Ca
= 0.6 are also shown. For the remainder of this work, preinflated capsules are used,
and when comparisons between NH and SK models are made, we present results at
constant D and at constant Ca.
To study the effect of a very small amount of polymer (β = 0.997) on the defor-
mation of an isolated capsule, the following protocol was used: Initially, the capsules
are held fixed and rigid, while the polymer is subjected to shear flow for a period
of 20 time units. In this time interval, the polymer molecules are stretched to their
steady-state values. After the equilibration of the polymer molecules, the capsules are
allowed to deform. Fig. 4.5, the effect of polymer on the deformation of an isolated
capsule is illustrated, via plots of the difference of D and θ between the Newtonian
(subscript N) and polymer (subscript P) cases. The figure indicates that in the shear-
dominated flow around an isolated capsule, the polymer stretching is not substantial
and the effect of the polymer on the capsule dynamics is completely negligible (the
deviation of D from its Newtonian value is O (10−3) ).
4.3 Discussion
The dynamics of isolated capsules in shear are found to be almost completely un-
affected by the polymer additives at the very dilute concentrations considered here.
This result is a reflection that the flow field in this case is shear-dominated, and thus
does not lead to substantial polymer stretching.
46
Chapter 5
Pair collisions in shear
5.1 Newtonian fluid
We turn now to pair collisions between capsules in shear flow. Fig. 5.1 shows a
schematic of the simulation box for pair collisions. For the present work we consider
the case where the initial coordinates of the capsule centers are (−4a, 0.25a, 0) and
(4a, −0.25a, 0), with the center of the box located at (0, 0, 0). Thus the initial stream-
wise separation between the capsule centers, ∆x0 is −8a and the initial wall-normal
separation, ∆y0 is −0.5a. The initial displacement in the neutral (z) direction is zero
for all cases studied here. Note that the initial position of the second capsule can
be found from the initial position of the first particle by rotating by π around the
z-axis. A slight complication arises in the simulations because, as is well-known, a
deformable particle in shear flow near a wall will generally migrate away from the
wall37,40,83,84. Migration is minimized by using a large wall-normal domain size. The
effect of migration on capsules was quantified by placing a single capsule with its
47
x
yFlow
Wall
Wall
∆x
By
Bx
Bz
z 2a
∆y
Figure 5.1: Schematic of pair collisions of fluid-filled elastic capsules in shear flow ofa very dilute polymer solution.
center at one of the initial positions of the pair collisions and then subjecting it to
shear flow for a period of time corresponding to the time of collision. The net verti-
cal distance traveled by the capsule in this time interval is the displacement due to
migration. This displacement was less than 1% of the capsule radius.
As a validation for the pair collision code, Fig. 5.2(a) shows the relative trajecto-
ries (i.e. ∆y vs. ∆x) of colliding NH capsules at Ca = 0.45 in the absence of polymers
in a cubic box of side 25a, along with the boundary integral result (in an unbounded
domain) of Lac et al.6. We observe good agreement. To illustrate the insensitivity of
the results to the choice of α, Fig. 5.2(b) shows the relative trajectories (i.e. ∆y vs.
48
−8 −6 −4 −2 0 2 4 6 80
0.5
1
1.5
2
∆x/a
∆y/a
simulationLac et. al.
(a) verification with Lac et al.6
−8 −6 −4 −2 0 2 4 6 80
0.5
1
1.5
2
∆x/a
∆y/a
4/(α a) = 34/(α a) = 44/(α a) = 6
(b) effect of screening parameter α
Figure 5.2: Pair collisions of preinflated NH capsules in a Newtonian fluid in a cubicbox of size 25a. Relative separation of the two capsules in y direction ∆y is plottedas a function of relative separation in x direction ∆x: (a) NH capsules at Ca = 0.45.Symbols are simulation result of Lac et al.6 and lines are present simulation; (b) effectof varying screening parameter α on collision dynamics of NH capsules at Ca = 0.30.
49
−8 −6 −4 −2 0 2 4 6 8 100
0.5
1
1.5
2
∆x/a
∆y/a
NH,Ca = 0.10
NH,Ca = 0.30 NH,Ca = 0.60
NH, Ca = 0.10NH, Ca = 0.30NH, Ca = 0.60
Figure 5.3: Relative trajectories for pair collisions of preinflated NH capsules in aNewtonian fluid at different Ca. Images correspond to snapshots taken at collision(∆x = 0).
∆x) of colliding NH capsules at Ca = 0.30 at different values of α keeping the reso-
lution of the mesh fixed. The figure shows that the results converge for 4/(αa) ≥ 3.
For all remaining simulation results this parameter is fixed at αa = 1. The effect
of dimensionless time step ∆t∗ and capsule mesh size hc on the numerical method
was studied for collision dynamics of NH (Ca = 0.142) and SK (Ca = 0.60, C =
10) capsules. The numerical method was found to converge linearly with time step
(the linear convergence with time step was also obtained by Walter et. al.62 using
Runge-Kutta second order method) and linearly with capsule mesh size as expected
from an immersed boundary method.
As expected from prior studies of pair collisions between particles, the vertical
distance between the particles is larger after the collision than before. The maximum
displacement decreases with increasing Ca. At low Ca, the net final displacement
50
−8 −6 −4 −2 0 2 4 6 8 100
0.5
1
1.5
2
∆x/a
∆y/a
NH, Ca = 0.142SK, Ca = 0.60NH, Ca = 0.60
Ca = 0.60 (NH)
Ca = 0.142 (NH)
Ca = 0.60 (SK)
Figure 5.4: Relative trajectories for pair collisions of preinflated NH and SK capsules(C = 10) in a Newtonian fluid at different Ca. Images correspond to snapshots takenat collision (∆x = 0).
increases with increasing Ca and at high Ca, the net final displacement decreases
with increasing Ca57. In particular, for Ca ≤ 0.10 for NH capsules, the net final
displacement is found to decrease with Ca. The focus of our study corresponds
to the regime of high Ca (Ca > 0.10) where stiffer capsules results in larger net
final displacements. The effect of Ca on maximum and net final displacements is
discussed in section 5.3. Fig. 5.3 shows the relative trajectories of colliding NH
capsules at various Ca in the absence of polymers, as well as shapes of the capsules
upon “impact”, defined when ∆x = 0. For this and all future results, we take
Bx = 25a, By = 20a, Bz = 10a. For all collisions in a Newtonian fluid, we observed
that the symmetry of the initial capsule configuration was conserved – the entire
collision process appears identical if the system is rotated by π around the z-axis. It
should be noted that with unit viscosity ratio we are always in the tank-treading as
51
−8 −6 −4 −2 0 2 4 6 8 100
0.05
0.1
0.15
0.2
0.25
0.3
∆x/a
D
NH, Ca = 0.142SK, Ca = 0.60
Figure 5.5: Deformation parameter as a function of relative separation in the stream-wise direction (∆x) for pair collisions of pre-inflated NH (Ca = 0.142) and SK (Ca= 0.60, C = 10) capsules in a Newtonian fluid.
opposed to tumbling regime.
Fig. 5.4 compares relative trajectories of the NH and SK models (with C = 10) in
the absence of polymer. Recalling that the SK model at Ca = 0.60 and the NH model
at Ca = 0.142 have the same value ofD (= 0.20), results at these conditions are shown.
For comparison, the result for the NH capsule at Ca = 0.60 is also shown. Despite
the virtually identical shapes and transient evolution of capsule deformability of the
NH and SK capsules in isolation at D = 0.20 (see Fig. 4.4(b)), there are nontrivial
differences in the collision dynamics: the SK capsule displays a larger displacement
than the NH capsule throughout the collision process, and the final displacement is
larger by about 0.15a. This is a reflection of the fact that the SK capsule is stiffer
than the NH capsule. In Fig. 5.4, we show snapshots of the two cases during collision;
for the SK capsules (middle), the squeeze flow region between the colliding capsules
52
is smaller than for the NH capsules (left), an indication that the SK capsules are
deforming less in response to the increased pressure found in that region during the
collision.
We further examine the difference between the NH and SK pair collisions by
observing the deformation parameter during the collision. Fig. 5.5 shows plots of
deformation parameter for a collision of NH capsules at Ca = 0.142 and SK capsules
(with C = 10) at Ca = 0.6. Since the Newtonian pair collision is symmetric with
respect to the two capsules involved, the capsules display identical dynamics so only
one curve is shown for each simulation. The deformation parameter D shows larger
variations in deformation for the NH capsule than for the SK capsule, particularly
in the time period when the capsules are approaching one another (∆x < 0). Even
though the two capsules display the same value of D in isolation during shear flow,
their responses clearly differ substantially during collisions. This fact will play an
important role in the effects of polymers on the collision.
5.2 Polymer solution
We turn now to the effect of added polymer on the dynamics of the pair collision
process. We will be exploring the effects of membrane model, initial displacements
(∆x0,∆y0) and polymer concentration 1−β on this process. To begin, Fig. 5.6 shows
the collision trajectories for the conditions of Figure 5.4 (thick lines), along with col-
lision trajectories for capsules in the polymer solution (thin lines) with β = 0.997.
As in the case of single capsules under shear flow in the polymer fluid, the polymer
molecules are first equilibrated for 20 time units during which the capsules remain
53
−8 −6 −4 −2 0 2 4 6 8 100
0.5
1
1.5
2
∆x/a
∆y/a
NH, Ca = 0.142SK, Ca = 0.60NH, Ca = 0.60
Figure 5.6: Relative trajectories for pair collisions of preinflated NH and SK capsules(C = 10) in a Newtonian fluid (thick lines) and polymeric fluid (thin lines, β = 0.997).
unaltered. After the equilibration, the capsules are allowed to deform and trans-
late according to the flow. Because of the stochastic nature of the simulations with
polymers, several realizations with difference initial polymer positions and different
random number sequences were run. The differences between realizations were neg-
ligible. In particular, the standard deviation of the relative separation of capsules in
y−direction (∆y ) for the realizations was less than 0.4% of capsule radius at the
time of collision (∆x = 0) and less than 3.8% of capsule radius after the collision
process (∆x = 8a). The primary observation is that polymers reduce the total dis-
placement for all three cases, although the difference between the Newtonian and the
polymer fluid is very small for the NH capsules. Indeed for the two NH cases, the
trajectories without and with polymers are very similar until ∆x ≈ 2, by which time
the collision process is almost complete. In contrast, for the SK capsule at Ca = 0.6,
54
−8 −6 −4 −2 0 2 4 6 8 10
0
0.5
1
1.5
∆x/a
∆y/a
− ∆
y 0/a
NH, Ca = 0.142SK, Ca = 0.60
∆y0/a = 0.75
∆y0/a = 0.50
∆y0/a = 1.00
Figure 5.7: Relative trajectories for pair collisions of NH and SK (C = 10) capsuleswith different initial separations ∆y0 in Newtonian (thick lines) and polymeric (thinlines, β = 0.997) fluids. The initial y-separation is subtracted off to facilitate com-parison of the results. Note: Newtonian and polymeric results for the NH cases aretoo close to each other to be distinguishable on the plot.
the discrepancy is substantial (0.16a) and begins just before ∆x = 0, while the par-
ticle centers are still moving apart vertically (i.e. just before the maximum in ∆y).
Additionally, this collision breaks the rotation symmetry of the Newtonian collisions
– the ultimate magnitudes of the y-positions of the two capsules differ by about 3%.
This symmetry-breaking was not the origin of the change in relative displacement,
however, a simulation with the rotation symmetry enforced yielded a very similar
relative trajectory.
Fig. 5.7 shows the collision trajectories of NH and SK (with C = 10) capsules
with different initial separations in Newtonian (thick lines) and polymeric (thin lines)
solutions with β = 0.997. Unsurprisingly, the collision dynamics are strongly affected
55
−8 −6 −4 −2 0 2 4 6 80
0.5
1
1.5
2
∆x/a
∆y/a
C = 0C = 1C = 10
Figure 5.8: Variation of relative trajectories with non-dimensional area dilation mod-ulus, C, for SK (Ca = 0.60) capsules in Newtonian (thick lines) and polymeric (thinlines, β = 0.997) fluid.
by initial separation – smaller initial separation leads to higher maximum displace-
ment (around ∆x = 0) and higher net final displacement after the collision. As
observed in the previous results, the displacement of SK capsules is larger than NH
ones throughout the collision process – an illustration of the fact that SK capsules
are stiffer than NH capsules. The effect of polymers on different initial separations is
interesting. The figure shows that effect of polymer (thin lines) depends not only on
the model but also on the proximity of the capsules at the time of collision. The ef-
fect of polymers on NH capsules is not significant for any initial separation, consistent
with the previous results (Fig. 5.6). However the effect on SK capsules is nontrivial –
viscoelastic effects strongly depend on initial separations. As the separation between
the capsules increases the effect of polymer diminishes and the behavior of capsules
is similar to that in the absence of polymers.
56
−8 −6 −4 −2 0 2 4 6 80
0.5
1
1.5
2
∆x/a
∆y/a
0 0.004 0.0080
0.2
0.4
0.6
0.8
(1 − β)
(∆y N
− ∆
y P)/a
Newtonianβ = 0.9985β = 0.9970β = 0.9940
Figure 5.9: Relative trajectories as a function of (1− β) for SK (Ca = 0.60, C = 10)capsules under shear flow in a Newtonian (thick lines) and polymeric (thin lines)fluid. Inset shows the difference in ∆y of SK capsules at ∆x = 8a in Newtonian andpolymeric fluid, (∆yN −∆yP), plotted against (1− β).
One of the significant differences between NH and SK capsule models is the pres-
ence of an explicit area penalty in the SK model. For all previous results of SK
capsules we set C = 10. Figure 5.8 shows the effect of C on the collision dynamics at
Ca = 0.60. Consider the Newtonian case (thick lines) first. The collision dynamics
are strongly affected by the parameter C. It is observed that the capsules with high
values of C (= 10) have more relative displacement from the initial state than the
capsules with low value of C (= 0, 1). This is consistent with the previous observa-
tions that stiffer capsules have higher relative displacements throughout the collision
process (Fig. 5.3 and Fig. 5.4).
Turning to the effect of polymer on the collision dynamics of SK capsules with
various values of C (thin lines on Fig. 5.8), we see that polymers suppress the net
57
displacement for C = 10 while there is almost no effect at lower values (C = 0, 1).
The weak effect of polymer on SK capsules at low C values is consistent with the
observations with NH capsules, which also do not have a strong resistance to area
changes. These results indirectly indicate that during pair collisions, the resistance
to area change of the capsules leads to relatively stronger polymer stretching, a point
revisited in section 5.4.
Now we briefly examine the effect of polymer concentration on the collision dy-
namics. Results will be reported in terms of of 1 − β, which is a linear function
with concentration in the dilute regime considered here. Fig. 5.9 shows the effect
of polymer concentration on the collision dynamics of SK capsules (with C = 10)
at Ca = 0.60 with values of concentration that are half and double the value used
above, corresponding to β values of 0.9985 and 0.9940, respectively. (Results for
Newtonian collisions and collisions with β = 0.997 are repeated here for comparison.)
The effect of polymer concentration is substantial. The inset shows the change in net
displacement from the Newtonian case as a function of 1 − β, illustrating that the
effect of polymer concentration on collision dynamics is nonlinear even at very low
concentrations.
5.3 Effect of Ca
We turn now to the effect of Ca on collision dynamics of capsules in Newtonian
and polymeric fluid. Fig. 5.10(a) shows the maximum displacement during collision
(∆ymax −∆y0) as Ca varies for NH and SK (C = 10) capsules in Newtonian (thick
lines) and polymeric (thin lines, β = 0.997) fluid. As Ca increases, the maximum
58
0 0.5 1 1.50.8
0.9
1
1.1
1.2
1.3
1.4
1.5
Ca
(∆y m
ax −
∆y 0)/
a
NHSK
(∆yfinal
−∆y0)
(∆ymax
−∆y0)
(a) maximum displacement (∆ymax −∆y0) vs. Ca
0 0.5 1 1.50
0.2
0.4
0.6
0.8
1
Ca
(∆y fin
al −
∆y 0)/
a
NHSK, C = 10
(b) net final displacement (∆yfinal −∆y0) vs. Ca
Figure 5.10: Variation of (a) maximum displacement (∆ymax −∆y0) and (b) net finaldisplacement (∆yfinal −∆y0) with Ca for NH and SK (C = 10) capsules in Newtonian(thick lines) and polymeric (thin lines, β = 0.997) fluid. Note: ∆yfinal is calculated at∆x = 8a. The figure in the inset of (a) shows the schematic of these displacements.
59
displacement decreases monotonically – the capsules become less of an obstruction
to each other with increasing Ca, and accordingly the maximum displacement due to
collision decreases. The effect of polymer is minimal on the maximum displacement
for both NH and SK capsules indicating that the effect of polymer is not significant
when the capsules collide (∆x ≈ 0).
The effect of Ca and polymer addition on the net final displacement of capsules
is more complex. Fig. 5.10(b) shows this quantity as a function of Ca for NH
and SK (C = 10) capsules. At low Ca, where the deformations are small, the net
final displacement decreases as Ca decreases. This is because at low Ca, as Ca
decreases, the capsules approach the rigid particle limit and the net displacement
of rigid particles is zero due to the reversibility of Stokes flow. At high Ca, where the
deformations of the capsules are significant, each capsule becomes more elongated in
x but less so in y as Ca increases. Thus the effective “collision cross section” between
the particles decreases with increasing Ca, and accordingly the net displacement due
to collision decreases. The effect of polymer on net final displacement is substantial,
indicating that polymer effect becomes significant when the capsules begin to leave
each other after the collision – a point which will be revisited in section 5.4. The figure
also shows that the effect of polymer is more sensitive at low Ca regime for both NH
and SK capsules – the effect of polymer is stronger for stiffer capsules. However, the
effect of polymers on SK capsules are larger than that on NH capsules, consistent
with the previous observations.
60
−8 −6 −4 −2 0 2 4 6 80
0.5
1
1.5
2
2.5
3
3.5
∆x/a
λ max
(0,0
,0)
Thick Lines − NewtonianThin Lines − Polymer
NH, Ca 0.142SK, Ca = 0.60, C = 0SK, Ca = 0.60, C = 10
Figure 5.11: Largest eigenvalue λmax of the deformation rate at the origin as a functionof ∆x for pair collisions under various conditions. Thick lines are from Newtoniansimulations; thin lines are from simulations with polymers, β = 0.997.
5.4 Mechanism of polymer effects
To gain an understanding of mechanism of the polymer effects on the pair collisions,
we examined the spatial dependence of the largest eigenvalue λmax of the rate of strain
tensor Γ = 12
(
∇u+ (∇u)T)
. In simple shear, λmax = 0.5; during the pair collisions
λmax exceeded 1 only for some of the constitutive models and then only during the
period when 2 . ∆x . 4, the final stage of the collision process. In all cases, the
spatial position where λmax reaches its largest value is the origin. Fig. 5.11 shows
λmax at the origin during pair collisions with ∆y0 = 0.5 under various conditions.
For the C = 0 SK model, there is only a modest increase in λmax in the interval
2 . ∆x . 4 and almost no change when polymers are present. For the NH model,
there is a distinct peak in this interval, but again only a small change in the presence
61
of polymers. For the C = 10 SK case, however, there is a very large increase in λmax
in this interval, and the increase is dramatically reduced (to the level found in the NH
case) when polymers are present. From these results we can see that (1) the largest
deformation rates arise as fluid is drawn into the region between capsules as they
move apart from one another after colliding, (2) the effect is largest for the nearly
area-incompressible case, and (3) this phase of the capsule interaction process is the
one that most strongly stretches the polymers and is accordingly the one that is most
affected by their presence. As the capsules leave the collision, they draw fluid and
chains into the growing gap between them. The uniaxial extension generated in this
region stretches the polymer chains, and the stretching works against the separation
of the capsules as they move around one another. This resistance allows the capsules
to “roll” around one another more than they would in the absence of the polymer
and thus leads to a smaller net relative displacement in y.
5.5 Discussion
At sufficient concentration, the interactions of capsules play become important. In
suspensions, capsules collide with each other leading to substantial velocity fluctu-
ations that drive the diffusive motions of cells and solutes. The dynamics of pair
collision of cells are strongly affected by the presence of drag reducing polymer ad-
ditives, for reasons that are not understood. Results for pair collisions show that
at high Ca, for neo-Hookean capsules or for Skalak capsules with a small penalty for
area change, there is almost no effect of polymer additives on the collision trajectories,
while for Skalak capsules with a substantial energy penalty for area change, there is a
62
substantial effect – in this case the net displacement of the capsules in the wall-normal
(gradient) direction is substantially decreased from the Newtonian case. This effect
strongly depends on the proximity of the capsules at the time of collision – the closer
the capsules are at the time of collision, the effect of polymer gets more substantial.
This effect of polymer is found to depend on the area incompressibility of SK cap-
sules. SK capsules that have strong resistance to area change show substantial effect
of polymers. This effect originates in the increased rate of deformation in the gap
between departing capsules that is observed with membranes that strongly resist area
change. The uniaxial extension generated in this region stretches the polymer chains,
and the stretching works against the separation of the capsules as they move around
one another. At low Ca, where the deformations are small, the effect of polymer on
the collision dynamics of NH and SK capsules are substantial. These observations
are concrete indications that the presence of small amounts of polymer additives can
substantially change the flow of suspensions of model cells, and that the membrane
properties of the cells play an important role in the observed changes.
63
Chapter 6
Suspensions of capsules in Couette
flow
6.1 Theory for suspension of capsules in shear flow
In this section, we present a simple theory for the suspensions of capsules subjected
to a simple shear flow. We consider capsules of radius a confined between two parallel
walls separated by a distance By and subjected to a shear flow with shear rate γ as
shown in Fig. 6.1. The direction of flow is taken to be along x with y and z as
the wall-normal and the vorticity direction respectively. The walls are at y = 0 and
y = By. The volume fraction of the capsules is denoted by φ.
6.1.1 Wall-induced migration of a single capsule
In a bounded flow, a deformable particle (capsule, droplet, vesicle) exhibits cross-
stream migration away from the nearest wall. Fig. 6.2(a) shows the schematic of a
64
γ⋅
y = By
y
xz y = 0
2a
wall
wall
Figure 6.1: Schematic of suspensions of fluid-filled elastic capsules in Couette flow.
deformable capsule near a wall. To the leading order, the particle can be described
by a point dipole44. The migration of a deformable particle in shear flow away from
a single wall can be described as
umig = Kγa(N1 −N2)
ηγa2
(
1
y2
)
, (6.1)
where umig is the migration velocity, y is the distance from the wall, N1 and N2 are
the first and second normal stress differences of an individual capsule and K is a
coefficient given as
K =3
64πa3n, (6.2)
65
where n is the number density of the capsule. The above expression includes the
effect of only the nearest wall. The effects of both walls on the migration of a capsule
can be approximated as
umig = Kγa(N1 −N2)
ηγa2
[
1
y2− 1
(By − y)2
]
. (6.3)
Note that eq. 6.3 is a superposition of the individual effect of both walls. It should
be noted that the expressions for the migration velocity (eqs. 6.1 and 6.3) assumes
the particle to be a point dipole and does not consider the finite size of the particle.
Hence the assumption is strictly valid for y/a � 1.
6.1.2 Shear-induced diffusion
For suspensions of capsules, the interactions of capsules become important. Colli-
sion of capsules is asymmetric and the capsules depart on streamlines that are more
widely separated than the ones before the collision74. Fig. 6.2(b) shows a schematic
of a trajectory of a capsule undergoing a collision process in the shear (x− y) plane.
As observed form the figure, the capsule gets displaced from its original streamline
after the collision. Random collisions lead to random motions perpendicular to the
streamlines. From dimensional and physical point of view, self-diffusivity of a di-
lute suspension of capsules is expected to be proportional to the product of collision
frequency (proportional to γφ, where φ is the local volume fraction) and the mean
squared displacement per collision (proportional to a2). The self-diffusivity Dself can
be written as
Dself = γφa2fself, (6.4)
66
where fself is a coefficient that depends on the deformability, size and shape of capsules.
6.1.3 Model for steady-state distribution
In suspensions, shear-induced diffusion and wall migration work against one another,
and a nonuniform distribution of capsules is expected at steady state. Consider an
initially homogeneous distribution of capsules subjected to shear flow. The local flux
of capsules in the wall-normal direction (y-direction) is
jy = umigφ− ∂ (Dsφ)
∂y, (6.5)
where Ds is the short time diffusivity of the particles and takes the same functional
form as Dself in eq. 6.4 i.e. Ds = γφa2fs. Note that this equation does not take the
form of Fick’s law. This is because in general the diffusivity is position dependent.
The evolution equation of the volume fraction of capsules can be stated as
∂φ
∂t= −∂jy
∂y= − ∂
∂y
(
umigφ− ∂ (Dsφ)
∂y
)
. (6.6)
This equation is a Fokker-Planck equation and is equivalent to writing the evolu-
tion equation for the probability of finding a capsule at a given y−position in the
dilute limit; with the assumption that the motion of capsules is a continuous Markov
process85,86 and is a good approximation for time scales longer than about γ−1.
At steady state, the net flux is zero and the diffusive and convective (due to
migration) fluxes balance each other. The volume fraction balance becomes
∂ (Dsφ)
∂y= umigφ. (6.7)
67
⋅γ By
y
wall
wall
(a)
net finaldisplacementafter thecollision
γ⋅ y x
(b)
Figure 6.2: (a) Schematic of migration of an isolated capsule away from the nearestwall in a Couette flow. (b) Schematic showing pair collision of the capsules. The linesshows the trajectory of a capsule undergoing a collision process in the shear (x− y)plane.
68
6.1.4 Steady-state capsule-depleted layer near a single wall
For simplicity we consider the case of semi-infinite domain where the effect of one
wall is only important. Starting from eq. 6.7 and substituting eq. 6.1 gives
φ∂φ
∂y=
(
K(N1 −N2)
ηγ
a
2fs
)
φ
(
1
y2
)
. (6.8)
There exists two solutions of eq. 6.8. One solution is φ = 0, describing a capsule
free layer near the walls, denoted by y = δ ; δ being the thickness of the capsule free
layer. The other solution can be found by the integration of eq. 6.8 as
φ = φb
(
1− δ
y
)
, (6.9)
where
δ =K
2fsφb
(N1 −N2)
ηγa, (6.10)
and φb is the bulk volume fraction. The equation provides a closed-form analytical
expression for the thickness of the capsule-depleted layer. An important observation
from the above equation is that the thickness of the capsule-depleted layer scales as
δ ∼ φ−1. The above equation also shows a direct relationship of δ with the parameters
of wall-induced migration (δ ∼ K (N1−N2)ηγ
) and inverse relationship with the parameter
of shear-induced diffusion (δ ∼ f−1s ) – indicating that wall-induced migration favors
the formation capsule-depleted layer where as shear-induced diffusion opposes it. In
terms of Ca dependance, both (N1−N2)ηγ
and fs are functions of Ca and so δ will depend
on Ca – the nature of this dependence will be determined by which of these two
competing mechanisms has a stronger relationship with Ca. We will show, through
69
our numerical simulations in sections 6.2 and 6.3, an explicit dependance of (N1−N2)ηγ
,
fs and δ on Ca and then compare our results with the above prediction in section 6.4.
6.2 Migration of a single capsule in a Couette Flow
6.2.1 Newtonian fluid
In this section we report the results of wall-induced migration of an isolated NH cap-
sule in a Couette flow in a Newtonian fluid as shown in Fig. 6.2(a). The capsule is
initially placed at a distance of 2a from the bottom wall (y = 0) and is then subjected
to the shear flow. The effect of box length Bx in the flow direction on single capsule
migration was studied and the results are found to converge for Bx ≥ 10a. For all
the simulations of single capsule migration we use the simulation domain to be a box
of size 16a, 10a and 10a in x, y and z directions respectively. The dimensionless
time step used in all our simulations is 5 × 10−3. The capsule surface is discretized
into 1280 triangular elements corresponding to 642 nodes. Fig. 6.3(a) shows the
trajectory of the center of mass of a capsule in the wall-normal direction as a func-
tion of time at different values of Ca. The figure shows that the capsule migrates
away from the wall towards the center of the channel with the rate of migration
increasing with an increase in Ca. As Ca increases, the capsule becomes more de-
formable – the capsule gets more stretched in the x−direction than in the y−direction.
The shape of the capsule can be characterized by the Taylor deformation parameter,
D = (Lmax − Lmin) / (Lmax + Lmin) where Lmin and Lmax are the smallest and largest
dimensions of the capsule in the shear (x − y) plane. Note that D is zero when the
shape of the capsule is spherical and increases as the shape of the capsule gets more
70
0 50 100 150 200 250 300
0
1
2
3
4
5
t∗
y/a
Center
Wall
Ca = 0.10Ca = 0.142Ca = 0.30Ca = 0.60
(a)
2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
y/a
D
Ca = 0.10Ca = 0.142Ca = 0.30Ca = 0.60
(b)
Figure 6.3: Migration of a capsule in a Newtonian fluid in a Couette flow. (a)Trajectory of the center of mass of a capsule y as a function of time t∗ in the wall-normal direction. The walls are at y = 0 and y = By = 10a. (b) Capsule deformationD as a function of center of mass of a capsule y.
71
0 50 100 150 200 250 3000
0.005
0.01
0.015
0.02
0.025
t∗
(N1−
N2)/
(ηγ)
Ca = 0.10Ca = 0.142Ca = 0.30Ca = 0.60⋅
(a)
0.1 10.01
0.1
1
Ca
(N1−
N2)/
(ηγ)
∼ Ca0.60
⋅
(b)
Figure 6.4: (a) Difference between first (N1) and second (N2) normal stress differencesas a function of y. (b) N1 − N2 evaluated at y = 2.5a (quarter channel height) asa function of Ca. The symbols represent the simulation results and the dashed linerepresents the exponential fit.
72
deformed. Fig. 6.3(b) shows plots of Taylor deformation parameter as a function of
channel height for a capsule at various values of Ca. As the flow starts, D attains its
steady state value within a short time interval (t∗ ≤ 5) implying that capsules attain
their equilibrium shape quickly before moving significantly in the flow. To study the
effect of capsule mesh resolution on single capsule migration, a simulation with higher
capsule mesh resolution (5120 triangular elements) was performed for a capsule at
Ca = 0.60 and compared with the results of capsule with 1280 triangular elements.
The deviation in the deformation parameter for the latter was within ±1.3% of that
of the former. For all simulations of single capsule migration, we use 1280 triangular
elements to discretize the capsule surface.
Fig. 6.4(b) shows the variation of (N1 − N2) with y for a capsule at various
Ca. Here N1 and N2 are the first and second normal stress differences defined as
N1 = τxx − τyy; N2 = τyy − τzz. Here τ is the extra stress due to the presence of
the capsule. The more deformable a capsule (higher Ca), the larger (N1 − N2) it
can exhibit and higher is its tendency to migrate away from the wall. In the limit
of small deformations (low Ca), one expects the normal stress difference to vary as
(N1 −N2)/(ηγ) ∼ Ca1.0, however, in the intermediate range of Ca (between 0.1 and
1.0), our simulation results suggest (N1−N2)/(ηγ) ∼ Ca0.60 as shown in Fig. 6.4(b).
6.2.2 Validation of dipole approximation
As noted above we presented a simple theoretical expression (section 6.1.1) for the
migration velocity of an isolated capsule assuming the capsule to be a point dipole.
This assumption is only valid when the capsule is far away from the wall i.e. y/a � 1.
73
0 20 40 60 80 1000
1
2
3
4
5
t*
y/a
y0 = 1.1a
y0 = 1.2a
y0 = 1.5a
y0 = 2.0a
y0 = 3.0a
(a)
0 50 100 150 200 250 3002
2.5
3
3.5
4
4.5
t∗
y/a
Ca = 0.10Ca = 0.142Ca = 0.30Ca = 0.60
(b)
Figure 6.5: Validation of a point dipole approximation. (a) Trajectory of the centerof mass of an isolated capsule y at Ca = 0.30 as a function of time t∗ in the wall-normal direction for different values of initial condition y0. The walls are at y = 0and y = By = 10a. Symbols are simulation results and lines are the fits using eq.6.13.(b) Trajectory of a capsule as a function time for different values of Ca.
74
2 2.5 3 3.5 4 4.5 50
0.01
0.02
0.03
0.04
0.05u m
ig/(
γa)
y/a
Ca = 0.10Ca = 0.142Ca = 0.30Ca = 0.60
(a)
0 0.2 0.4 0.6 0.80
0.1
0.2
0.3
0.4
0.5
k
Ca
SimulationTheory
(b)
Figure 6.6: (a) Migration velocity umig as a function of center of mass of a capsule y.(d) Comparison of the numerical value of the slope k obtained from simulations (byfitting eq. 6.13) and the theoretical value obtained using eq. 6.12, at different valuesof Ca.
75
In this section we report a validation of this approximation. Rewriting eq. 6.3 as
umig =dy
dt= k[
1
y2− 1
(By − y)2] , (6.11)
where
k = Kγa(N1 −N2)
ηγa2 =
3(N1 −N2)
64πηn. (6.12)
Eq. 6.11 can be solved analytically to obtain
1
4(y3−y30)−
1
8By(y4−y40)−
By
16(y2−y20)−
B2y
16(y−y0)−
B3y
32ln (
By − 2y
By − 2y0) = kt , (6.13)
Note that eq. 6.13 is implicit in y and explicit in t. Fig. 6.5(a) shows the trajectory
of the center of mass of a capsule at Ca = 0.30 in the wall-normal direction as a
function of time for different values of initial condition y0. Symbols are the results
from simulation and lines are the predictions obtained by fitting eq. 6.13. The figure
shows that the dipole approximation of the capsule holds good for y0/a ≥ 2.0. For
reference we also show the simulation results (symbols) of the trajectories of capsule
at different Ca from Fig. 6.3(a) along with the fits (lines) obtained from eq. 6.13.
Again a good match is observed. Fig. 6.6(a) also shows the migration velocity umig of
a capsule as a function of as a function of center of mass of a capsule y at various Ca.
Symbols are the results from simulation and the lines are the fits obtained from first
fitting eq. 6.13 to obtain k and then using k in eq. 6.11. The results show an excellent
match between our simulation results and the fits obtained from the assumption. It
should be noted that the theoretical predictions do not predict a zero velocity of the
capsule at t = 0 which is actually the case with our simulations and hence does not
76
match with the simulation results at initial times.
Finally, Fig. 6.6(b) shows a comparison between the numerical value of the slope
k, obtained from fitting eq. 6.13, and theoretical value of k, obtained from using eq.
6.12, at different Ca. The results show that the at low Ca, where the deformations are
small, there is a good match between the numerical and the theoretical values of k but
at high Ca, where the deformations are substantial, the corresponding values deviate
from each other – the theory over predicts the simulation results. This indicates
that the capsule is not a true dipole and higher order singularities are also present.
These singularities become significant at high Ca and must be considered to make
quantitative predictions.
6.2.3 Polymer fluid
We now turn to the effect of polymer on the migration of an isolated capsule. For all
the simulations of suspensions of capsules in polymer solution we use the simulation
domain to be a box of size 16a, 10a and 10a in x, y and z directions respectively. The
results for the polymer solution are found to converge for the box of size Bx ≥ 16a.
In our simulation with polymers, the polymer molecules are first equilibrated for a
period of 50 time units during which the capsules are held fixed and rigid, while the
polymer molecules are stretched to their steady-state values. After the equilibration,
the capsules are allowed to deform and translate according to the flow. The polymer
molecules are initially randomly uniformly distributed in the simulation domain. For
polymers, we use β = 0.9985, 0.997, 0.994 and 0.988, which correspond to number
density n of 0.0235µm−3, 0.047µm−3, 0.094µm−3 and 0.188µm−3, or a mass fraction
of 0.157 ppm, 0.314 ppm, 0.628 ppm and 1.256 ppm of PEO respectively. The numbers
77
0 50 100 150 2002
2.5
3
3.5
4
4.5
t∗
y/a
NewtonianPolymer (β = 0.997, Wi = 20)
Ca = 0.10Ca = 0.142Ca = 0.30Ca = 0.60
(a)
0 0.2 0.4 0.6 0.80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Ca
D
NewtonianPolymer (β = 0.994, Wi = 20)
(b)
Figure 6.7: Migration of a capsule in a Newtonian (solid lines) and polymer (dashedlines, β = 0.994,Wi = 20) solutions. (a) Trajectory of the center of mass of a capsuley as a function of time t∗ in the wall-normal direction. (b) Steady state capsuledeformation D as a function of Ca.
78
0 0.2 0.4 0.6 0.80
0.005
0.01
0.015
0.02
Ca
(N1−
N2)/
(ηγ)
NewtonianPolymer (β = 0.994, Wi = 20)
⋅
(a)
0 0.005 0.01 0.015 0.020
0.005
0.01
0.015
0.02
0.025
0.03
(N1−N
2)/(ηγ)
u mig
| (y =
2.5
a)/(
γa)
NewtonianPolymer (β = 0.994, Wi = 20)
⋅
⋅
(b)
Figure 6.8: (a) N1 − N2 evaluated at y = 2.5a as a function of Ca. (b) Migrationvelocity umig evaluated at y = 2.5a (quarter channel height) as a function of N1−N2.The symbols represent the simulation results and the dashed line represents the linearfit.
79
of polymer chains in the simulation domain for these cases are 1016, 2032 4064 and
8128 respectively. Ex ranges from 4.3 to 34.4. We use Wi = 5, 10 and 20 for the
polymers. Fig. 6.7(a) shows the trajectory of the center of a capsule in wall-normal
direction in Newtonian (circles) and polymer (squares, β = 0.994,Wi = 20) solutions
in a Couette flow. Symbols are simulation results and lines are the fits using eq. 6.13.
Because of the stochastic nature of the simulations with polymers, several realizations
with different initial polymer positions and different random number sequences were
run. The differences between realizations were negligible. In particular, the standard
deviation of the center of mass of a capsule in y−direction evaluated at y = 2.5a for
the realizations was less than 3.0% of capsule radius. The figure shows that polymer
slows down the migration and the effect gets substantial for more deformable capsules
(high Ca). For example, the difference in the capsule’s center between the Newtonian
and the polymer case at t∗ = 300 is 0.5a at Ca 0.10 and gets as big as 1a at Ca
= 0.60. It should be noted that the effect of polymer does not change the final steady
state position of the capsule i.e. the capsule in the polymer case still has a tendency
to migrate towards the center but at a slower rate than in the Newtonian case. The
figure also shows a comparison between the simulation results (symbols) and the fits
(lines) obtained from using the point dipole approximation of capsule (eq. 6.13) for
the polymer case. Although a small discrepancy is observed at initial times (t∗ ≤ 5),
the dipole assumption shows a good match with the numerical simulation results for
later times (t∗ > 5) – indicating that the dipole approximation holds good for the
polymer case as well. Fig. 6.7(b) shows the steady state deformation parameter of the
capsule as a function of Ca in a Newtonian (solid lines) and polymer (dashed lines,
β = 0.994,Wi = 20) solutions. The primary observation is that polymer results in
80
changing the equilibrium shapes of the capsule – the capsule in the polymer solution
is less deformed than in the Newtonian fluid i.e. the capsule gets less stretched in
the shear plane. This is also reflected in Fig. 6.8(a) which shows that the polymer
case results in lower normal stress differences of the capsules than in the Newtonian
case. As noted above, the effect of polymer on (N1 − N2) and D increases for more
deformable capsules however the difference is small to probably affect the dynamics
of single capsule. Fig. 6.8(b) shows the migration velocity umig evaluated at y = 2.5a
as a function of the normal stress differences of the capsules (at different Ca) in
Newtonian (black) and polymer (blue) solutions. The migration velocity is obtained
using point dipole assumption (eq. 6.11). The figure shows a linear relationship of
umig with (N1 − N2) for both Newtonian and polymer solutions. Interestingly, the
slope of the linear fit between migration velocity and (N1 −N2) for the polymer case
is significantly lower in value than the Newtonian case the origin of which is not
understood at present. The effect of polymer concentration 1 − β at fixed Wi of
20 on the trajectory and migration velocity umig|y=2.5a of a capsule (Ca = 0.30) is
shown in Fig. 6.9(a) and Fig. 6.10(a) respectively. The polymer concentration is
expressed in terms of 1 − β, which is a linear function of concentration in the dilute
regime considered here. The figures show that the effect of polymer concentration
on migration of an isolated capsule is substantial even at such low concentrations.
The effect of WI at fixed polymer concentration β = 0.994 on the trajectory and
migration velocity umig|y=2.5a of a capsule (Ca = 0.30) is shown in Fig. 6.9(b) and
Fig. 6.10(b) respectively. The figure shows that the effect of polymer gets substantial
when the polymer chains stretch significantly in flow (higher Wi). For all simulations
of suspensions of capsules, we use the value of β to be 0.994 and 0.988 and the value
81
0 50 100 150 2002
2.5
3
3.5
4
4.5
t∗
y/a
NewtonianPolymer (β = 0.997, Wi = 20)
Newtonian β = 0.997 β = 0.994 β = 0.988
(a)
0 50 100 150 2002
2.5
3
3.5
4
4.5
t∗
y/a
NewtonianPolymer (β = 0.997, Wi = 20)
NewtonianWi = 5Wi = 10Wi = 20
(b)
Figure 6.9: Effect of (a) polymer concentration expressed as 1−β at fixed Wi (= 20)and (b) Wi at fixed β (= 0.994) on the trajectory of an isolated capsule (Ca = 0.30)in the wall-normal direction of in a Couette flow. Symbols are simulation results andlines are the fits.
82
0 0.005 0.01 0.0150
0.005
0.01
0.015
0.02
0.025
0.03
1− β
u mig
| (y =
2.5
a)/(
γa)
⋅
(a)
0 5 10 15 200
0.005
0.01
0.015
0.02
0.025
0.03
Wi
u mig
| (y =
2.5
a)/(
γa)
⋅
(b)
Figure 6.10: Migration velocity umig evaluated at y = 2.5a as a function of (c) 1−β atfixed Wi (= 20) and (d) Wi at fixed β (= 0.994) for an isolated capsule (Ca = 0.30).
83
of Wi to be 20. This is done so as to keep the computation cost reasonable while still
allowing the effects of polymer to be substantial.
It should be noted that the phenomenon of wall-induced migration in shear flows
is also observed in dilute polymer solutions and for highly extensible polymers it can
lead to depletion layer at steady state that can be much larger than the size of the
polymer molecule44 (depletion layer thickness scales as ∼ Wi2/3). However, the time
to reach a steady state is much larger than the time period considered here (it is
several orders of magnitude of the polymer relaxation time). We have studied the
temporal variation of the distribution of polymers in our simulations of single capsule
migration and found the variation in the distribution to be negligible – polymers were
uniformly distributed throughout the domain. The observed effects of polymer on
single capsule migration are in contrast to the theoretical predictions38,87 on migration
of deformable drops (analogous to deformable capsules) in viscoelastic fluids – their
predictions suggest that viscoelasticity enhances single particle migration away from
the walls in a Couette flow. It should be noted that the theoretical predictions are
based on the ideal limit where there is separation of length scales between polymer
molecules and drops so that the polymeric fluid can be treated using a continuum
approach ( second-order fluid). Additionally, the predications are derived in the limit
of Wi � 1 i.e. the fluid is weakly non-Newtonian. However the situation under
consideration is qualitatively different – here there is no separation of length scales
between polymer chains and capsules. Furthermore, we are interested in the regime
of high Wi where there is significant stretching of polymer molecules by the flow.
Interestingly, numerical simulation results of a single rigid sphere in viscoelastic fluid
in Couette flow by Avino et al.88 showed that viscoelasticity (at Wi ∼ 1) induce
84
particle motion towards the wall. They made a simple heuristic argument made that
the viscoelasticity leads to normal stress differences that are asymmetric under the
presence of a wall and this asymmetry leads to the cross-stream migration towards
the wall.
6.3 Suspensions of capsules in a Couette flow
6.3.1 Newtonian solution
We now consider the dynamics of dilute suspensions of capsules in a Couette flow
in a Newtonian fluid in a Couette flow. Simulation box of two different sizes are
considered – a cubic box of size 10a and 16a respectively. The volume fraction of
capsules considered is φ = 0.10 and φ = 0.20 which corresponds to simulations of 25
to 100 capsules. The capsules are initially distributed randomly in the computational
domain and are then subjected to a shear flow for a time period of 400 to 500 time
units. Previous relevant numerical studies have either considered two-dimensional34,89
or have considered three-dimensional simulations of capsules at high mesh resolution
for smaller time period70,69. In our case, we are interested in simulations of capsules
for a time period that is much larger than the previous studies which when coupled
with the simulation of large number polymer molecules becomes computationally too
expensive to obtain results in a reasonable time window. To study the effect of capsule
mesh resolution on suspension dynamics, a simulation of suspensions of capsules at
Ca = 0.142 with a mesh resolution of 1280 triangular elements was performed in a
Newtonian flow in a cubic box of size 10a for a time period of 200 units and results
were compared with that of capsules with 320 triangular elements. The deviation in
85
(a)
0 100 200 300 400 5000
2
4
6
8
10
t∗
y/a
Wall
Wall
(b)
Figure 6.11: (a) Snapshots of the suspensions of capsules (Ca = 0.60,φ of 0.10) ina Newtonian fluid at t∗ = 1 (left) and t∗ = 300 (right) in a Newtonian fluid in acubic box of size 10a . (b) Trajectories of the center of mass of capsules (Ca = 0.60,φ = 0.10) in the wall-normal direction as a function of time. The walls are at y = 0and y = By = 10a.
86
the average distance of center of the capsules from the channel center in wall-normal
direction < y−ycenter > for the latter was within±5% of that of the former indicating
reasonable accuracy of the latter. For all simulations of suspensions of capsules, we
use 320 triangular elements to discretize the capsule surface. These parameters of
capsules were chosen to keep the computational cost moderate while still capturing
the suspension dynamics with a reasonable accuracy.
Fig. 6.11(a) shows the snap shot of the suspensions (φ = 0.10) of capsules (Ca
= 0.30) in a cubic box of size 10a at time t∗ = 1 and t∗ = 300. Instantaneous shapes
and distribution of the capsules can be observed from the figure. Since the flow has
a constant shear rate the shapes of the capsules are similar. Fig. 6.11(b), shows the
trajectories of the center of mass of capsules (φ = 0.10, Ca = 0.60) in the wall-normal
direction as a function of time. It should be noted that due to the finite size of
the capsules, the initial distribution of the center of mass of capsules lie in a range
1a < y < 9a. Starting with a uniform distribution, the capsules are found to migrate
towards the center of the channel. This is also observed in the snapshot of capsules
at t∗ = 300. One can also observe fluctuations in the trajectories which are due to
the hydrodynamic interactions due to finite concentration of capsules. Unlike a single
capsule, the capsules in suspensions do not migrate continuously towards the center
but reach a steady state showing accumulation of the capsules in core of the channel
and a capsule-depleted layer near the walls as observed from the figure.
The net motion of the capsules towards the center of the channel can be further
quantified by examining the time evolution of the average distance of the capsule’s
center from the center of the channel < |y − ycenter| >. Fig. 6.12(a) shows < |y −
ycenter| > for the capsules as a function of time in a Newtonian fluid at different
87
0 100 200 300 4001.6
1.7
1.8
1.9
2
2.1
2.2
2.3
t∗
<|y
−y ce
nter|>
/a
Ca = 0.08Ca = 0.142Ca = 0.30Ca = 0.60
(a)
0 1 2 3 4 50
0.01
0.02
0.03
0.04
y/a
frac
tion
NH, Ca = 0.142NH, Ca = 0.30NH, Ca = 0.60
(b)
Figure 6.12: (a) Average distance from the centerline < |y − ycenter| > of suspensions(φ = 0.10) of capsules in a Newtonian fluid in a a cubic box of size 10a as a functionof time t∗. (b) Steady state distribution of capsules (φ = 0.10) as a function of y.The walls are at y = 0, 10a and y = 5a is the channel centerline. The walls are aty = 0, 10a and y = 5a is the channel centerline.
88
Ca. The value of < |y − ycenter| > initially decreases and then reach a steady state
indicating initial migration of capsules towards the center of the channel leading
to an accumulation of the capsules in the core of the channel at steady state. We
consider the time at which the steady state is reached as the time after when the
change in average distance from the center is less than 5%. For all simulations of the
suspensions of capsules, we take this time as t∗ ≥ 200 as observed from the figure.
The figure also shows that the extent to which the capsules move towards the center
of the channel increases with an increase in Ca indicating that as the deformability
of the capsules increases, the capsules have a greater tendency to migrate away from
the walls and move closer toward the center of the channel. The collective motion of
capsules towards the center of the channel is also observed in a large channel size as
shown in Fig. 6.13(a).
Fig. 6.12(b) shows the steady state distribution of capsules, represented as the
areal fraction of capsules, at various values of Ca as a function of channel height.
The fraction is calculated by dividing the channel height into bins of equal size and
finding the fraction of triangular elements, that discretize the membrane surface,
in those bins. The bin size was taken to be 0.2a. Since we assume these triangular
elements to be of equal size and mass, the fraction of triangular elements is equivalent
to the areal fraction of the capsules. The distribution was extracted by averaging over
a large time window over which the suspensions remain statistically stationary. We
take this time window to be 200 ≤ t∗ ≤ 400. Furthermore, to avoid the influence
of initial configuration of capsules on the steady state, 5 simulations with different
initial configurations were used and the steady state distribution was averaged over all
those configurations. The standard deviation, due to different initial configurations,
89
0 100 200 300 400 5003.25
3.3
3.35
3.4
3.45
3.5
t∗
<|y
−y ce
nter|>
/a
Ca = 0.142
(a)
0 2 4 6 80
0.01
0.02
0.03
0.04
y/a
frac
tion
NH, Ca = 0.142
(b)
Figure 6.13: (a) Average distance from the centerline < |y − ycenter| > of suspensionsof capsules (φ = 0.10, Ca = 0.60) in a Newtonian fluid in a cubic box of size 16a asa function of time t∗. (b) Steady state distribution of capsules (φ = 0.10, Ca = 0.60)as a function of y. The walls are at y = 0, 16a and y = 8a is the channel centerline.
90
0 1 2 3 4 50
0.01
0.02
0.03
0.04
y/a
frac
tion
φ = 0.10φ = 0.20
Figure 6.14: The effect of volume fraction φ on the steady state distribution of cap-sules at Ca = 0.60 as a function of y. The walls are at y = 0, 10a and y = 5a is thechannel centerline.
in the steady-state distribution was observed to be less than 4%. Fig. 6.12(b) shows
that at steady state the distribution of the capsules is inhomogeneous – capsules have
higher concentration near the center of the channel and a capsule-depleted region
near the walls. The thickness of the depleted layer as well as the concentration
near the center of the channel increases with an increase in Ca indicating that more
deformable capsules move farther away from the walls and accumulate closer towards
the channel center. The figure also shows the layer formation of capsules near the
walls, as shown by the off-center peaks in the distribution. The layer formation has
also been observed by Li and Pozrikidis89 in their simulations of suspensions of liquid
drops in wall bounded flows. They observed that the layer formation was a result of
using confined channel size. We have studied the effect of box size on the steady state
91
distribution. Fig. 6.13(b) shows the steady state distribution of capsules (Ca = 0.60)
in a cubic box of size 16a. The figure shows that formation of layers of capsules near
the walls is diminished indicating that layer formation is a result of using small size
of the channel.
The effect of volume fraction φ of capsules (at Ca = 0.60 ) on the steady state
distribution is shown in Fig. 6.14. The steady state distribution profiles at differ-
ent volume fraction are qualitatively similar to each other – higher concentration of
capsules near the center of the channel and a capsule-depleted region near the walls.
However the figure shows that volume fraction has a strong effect on the capsule
depleted region i.e. increasing the concentration of capsules decreases the thickness
of the capsule-depleted region near the walls.
6.3.2 Polymer solution
We turn our attention towards the effect of polymer on the dynamics of suspensions
(φ = 0.10) of capsules in a Couette flow. All the simulations of suspensions of capsules
in polymer solutions, we use the value of β to be 0.994 and 0.988 and the value of
Wi to be 20. It is based on our results from section ?? which showed substantial
effect of polymer on the dynamics of single capsule migration with these parameters.
As in the case of single capsule migration under shear flow in the polymer fluid, the
polymer molecules are first equilibrated for 50 time units during which the capsules
remain unaltered. After the equilibration, the capsules are allowed to deform and
translate according to the flow. The polymers are randomly uniformly distributed in
the entire simulation domain. Fig. 6.15(a) shows the snap shot of the suspensions
(φ = 0.10) of capsules (Ca = 0.30) in a polymer solution (β = 0.994,Wi = 20) in a
92
(a)
0 100 200 300 4001.5
2
2.5
t∗
<|y
−y ce
nter|>
/a
NewtonianPolymer (β = 0.994, Wi = 20)
Ca = 0.142Ca = 0.30Ca = 0.60
(b)
Figure 6.15: (a) Snapshots of the suspensions of capsules (Ca = 0.30) at t∗ = 10 in apolymer (β = 0.994,Wi = 20) solutions in a cubic box of size 10a. Polymer moleculesare shown as thin black lines (b) Average distance from the centerline < |y−ycenter| >of suspensions of capsules in Newtonian (solid line) and polymer(dashed lines, β =0.994,Wi = 20) solutions as a function of time t∗.
93
0 1 2 3 4 50
0.01
0.02
0.03
0.04
y/a
frac
tion
NewtonianPolymer (β = 0.994, Wi = 20)
Ca = 0.142Ca = 0.30Ca = 0.60
(a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.02
0.022
0.024
0.026
0.028
0.03
Ca
frac
tion
NewtonianPolymer (β = 0.994, Wi = 20)
(b)
Figure 6.16: (a) Steady state distribution of capsules as a function of y in Newtonian(solid line) and polymer (dashed line, β = 0.994,Wi = 20) solutions in a cubic boxof size 10a. (b) Steady state distribution of capsules in the “bulk” (2.5a ≤ y ≤ 7.5a)region as a function of Ca in Newtonian (solid line) and polymer(dashed line, β =0.994,Wi = 20) solutions.
94
cubic box of size 10a at time t∗ = 10. The distribution of the capsules and polymer
molecules (thin black lines) can be observed from the figure. Since Wi of the polymers
is high, the flow causes significant stretching of the polymer molecules as shown in
the figure. As noted above, the parameters of polymers are so chosen that there
is no separation of length scales – the relative size of stretched polymer chains and
capsules is comparable. As noted in the case of single capsule migration under shear
flow in the polymer fluid, here also we find the distribution of polymer molecules to
be uniform throughout the entire time of simulation.
Fig. 6.15(b) shows the average distance of the capsules from the center of the
channel at different Ca as a function of time in Newtonian (solid lines) and polymer
(dashed lines, β = 0.994,Wi = 20) solutions in a cubic box of size 10a. The primary
observation is that polymer suppresses the net movement of capsules towards the
center of the channel leading to a steady state value of < |y − ycenter| > that is larger
than that in the Newtonian case. This suggests that under the influence of polymer,
capsules accumulate less in the central region of the channel at steady state. This
difference in the steady state values increases with an increase in Ca indicating that
the effect of polymer gets significant as the capsules become more deformable.
The effect of polymer (β = 0.994,Wi = 20) on the steady state distribution of
capsules is shown in Fig. 6.16(a). The figure shows that polymer changes the distri-
bution of capsules. To better understand the effect of polymer, we show the average
distribution (fraction) of capsules in the “bulk” region, denoted by 2.5a ≤ y ≤ 7.5a
in Fig. 6.16(b) in Newtonian and polymer solutions. The average concentration of
the capsules in this region is lower than the corresponding Newtonian case. These
results indicate that polymer results in redistribution of capsules having lower con-
95
0 1 2 3 4 50
0.01
0.02
0.03
0.04
y/a
frac
tion
NewtonianPolymer (β = 0.994)Polymer (β = 0.988)
Figure 6.17: Steady state distribution of capsules (Ca = 0.60) in Newtonian andpolymer (Wi = 20) solutions with different values of β.
centration of capsules in the central region and correspondingly, by conservation of
capsules, higher concentration of capsules in the wall region. Fig. 6.3.2 shows the
effect of polymer concentration, expressed in terms of 1− β, on the steady state dis-
tribution of NH capsules (Ca = 0.60). As noted in the figure the effect of polymer
on steady state distribution gets enhanced on increasing the concentration i.e. it
results in higher and lower concentration of capsules in the wall and the bulk region
respectively.
6.3.3 Capsule-depleted layer
The existence of the cell-free layer in the flow of blood has been observed both in in
vitro 33,90 and in vivo 91,92,93 experiments. The existence of this layer is the cause of
96
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
1
1.2
Ca
δ/a
δ/a = 0.005+1.364×(Ca)0.397
δ/a = 0.015+0.957×(Ca)0.341
δ/a = 0.010+0.808×(Ca)0.291
NewtonianPolymer (β = 0.994)Polymer (β = 0.988)
(a)
0 0.1 0.2 0.3 0.4 0.50
1
2
3
4
5
flow rate (ml/min)
cell−
free
laye
r (µm
)
δ = 0.50 + 4.55×(flow rate)0.37374
Kameneva et al., 2004
(b)
Figure 6.18: (a) Dependence of capsule-depleted layer thickness on Ca for suspensionsof capsules in Newtonian and polymer (Wi = 20) solutions with different values of βin a cubic box of size 10a. Symbols are the simulation results and lines are the fits.The standard deviation is based on results from different initial configurations. (b)Experimental data (symbols) on the thickness of cell-free layer as a function of flowrate for a suspensions of RBCs from Kameneva et al.7.
97
lower apparent viscosity of blood in small vessels. The thickness of this layer is of the
same order as the cell dimension. Fig. 6.18(a) shows the dependence of the capsule-
depleted layer thickness δ as a function of Ca for suspensions of capsules Newtonian
and polymer (β = 0.994,Wi = 20) solutions in a cubic box of size 10a. The capsule-
depleted layer thickness is calculated from the steady state distributions of capsules
by finding bins with a non-zero value closest to the top and bottom wall and then
averaging over their corresponding distances from the walls. The standard deviation
in δ is based on averaging over different initial configurations. Symbols are the results
from the simulations and lines are the fits. Consider the Newtonian result first. The
figure shows that the capsule-depleted layer thickness follows a scaling relationship
with Ca as δ ∼ (Ca)n, where n is around 0.40 for the Newtonian case. The power-
law dependence of δ with flow rate (∝ Ca) has been observed experimentally33,90
with n reported to between 0.30 and 0.50. Of particular interest is the work done
by Kameneva et al.7, who studied the dependence of cell-free layer with flow rate
in their microchannel experiment of blood. Fig. 6.18(b) shows their experimental
data (symbols) along with our fit (line) to their data. Interestingly, the fit shows
the scaling of cell-free layer with flow rate as δ ∼ (flow rate)0.37 which is close to our
Newtonian results. Freund34, in his 2-D simulations of RBCs in microvessel showed
a similar scaling behavior as δ ∼ (flow rate)0.43. These observations along with our
numerical results suggest that the relationship of the cell-free layer thickness with flow
rate (∝ Ca) weakly depends of the shape of cell and the geometry of the vessel. The
existence of the capsule-depleted layer can also be observed from the visualizations of
the suspensions of capsules (Ca = 0.60) in Fig. 6.11(a) which shows that a uniformly
distributed capsules (snapshot on the left) migrates towards the channel center leading
98
to an inhomogeneous distribution (snapshot on the right) with a capsule-depleted
layer near the walls.
The effect of polymers on the capsule-depleted layer is shown in Fig. 6.18(a). The
primary observation is that the polymer reduces the thickness of the capsule-depleted
layer significantly compared to the Newtonian case and the discrepancy increases
with the increase in Ca. In this case, δ also scales with Ca as δ ∼ (Ca)n with a value
of n that is smaller than that of the Newtonian case and decreases on increasing
the polymer concentration (∝ 1 − β). This is in a qualitative agreement to the
experimental observations of Kameneva et al.7 who showed that the addition of small
amount of polymer (PEO) in blood resulted in a significant reduction in the cell-free
layer thickness and the reduction increased with increasing the flow rate (∝ Ca). It
should be noted that the concentration of polymer used in their experiments (10 ppm)
is order of magnitude larger than the one we are simulating (0.63 ppm). Nevertheless,
the deviation of the results of the polymer case from the Newtonian one observed from
our simulations is a reflection of the fact that the effect of polymer is substantial even
at such low concentrations.
6.3.4 Diffusion at steady state
Having studied the migration of a single capsule and behavior of suspensions capsules
in Newtonian and polymer solutions, what remains to complement the theoretical
prediction with simulation results is to study the diffusion of the capsules. Here
we address the effect of Ca and polymers on the diffusive behavior of the capsules.
We calculate the mean squared displacement of the capsules after the steady state
is reached. This is done to eliminate the effect of wall-induced migration. The
99
0 10 20 30 40 500
0.1
0.2
0.3
0.4
t∗ − t0∗
<(y
− y
0)2 >/a
2
Polymer (β = 0.994, Wi = 20)Newtonian
Ca = 0.08Ca = 0.142Ca = 0.30Ca = 0.60
(a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0
0.001
0.002
0.003
0.004
0.005
Ca
Ds/(
γa2 )
NewtonianPolymer (β = 0.994, Wi = 20)
⋅
(b)
Figure 6.19: (a) Mean squared displacement of suspensions of capsules in the wall-normal direction at steady state in Newtonian and polymer ( β = 0.994,Wi = 20)solutions as a function as a function of time t∗ and the corresponding short-time dif-fusivities (b) in the wall-normal direction as a function of Ca. The standard deviation(error bar) is based on results from different initial configurations.
100
0 1 2 3 4 5 0
0.001
0.002
0.003
0.004
y/a
Ds(y
)/(γ
a2 )
Polymer (β = 0.997, Wi = 20)
Newtonian
Ca = 0.08Ca = 0.142Ca = 0.30Ca = 0.60
⋅
Figure 6.20: Short-time diffusivities in the wall-normal direction as a function of y inNewtonian (solid line) and polymer(dashed line, β = 0.994,Wi = 20) solutions.
subscript ‘ss’ denotes the value at steady state. Fig. 6.19(a) shows the mean squared
displacement of the suspensions of capsules at different Ca in Newtonian (solid lines)
and polymer (dashed line) solutions. The figure shows that the polymers suppress
the diffusive motions of the capsules. This is also reflected in Fig. 6.19(b) which
shows the short time diffusivity Ds in wall-normal direction of the suspensions of
capsules in Newtonian (solid lines) and polymer (dashed line) solutions as a function
of Ca. The short time diffusivity Ds is calculated by averaging the diffusive motions
of all the capsules after steady state for a time period of t − tss = 50. The effect
of polymers is to suppress the diffusive motions of the capsules resulting in lower
diffusivities than in the Newtonian case. This is in agreement to the simulation
results of pair collisions of elastic capsules in polymer solutions by Pranay et al.74.
They showed that polymers suppress the net final displacements of the capsules after
101
the collisions. Since in suspensions the random motions due the collisions drive the
diffusive motion of the capsules, the effect of polymer will be to suppress this diffusive
motion reducing the mean squared displacement and the corresponding diffusivities.
The dependance of short time diffusivity on y for capsules at different Ca is shown
in Fig. 6.3.4. As observed from the figure the short time diffusivity is an increasing
function of the local volume fraction of the capsules – showing a maxima near the
center where the capsules are accumulated and having a zero value near the walls
since there are no capsules there. Fig. 6.19(b) also shows that the diffusivity is a
weak function of Ca for the Newtonian case. The weak dependance of diffusivity on
Ca has also been observed for the case of suspensions of deformable drops45 which
is analogous to deformable capsules. Interestingly, the figure shows a non-monotonic
behavior of the diffusivity with Ca around Ca = 0.10. This is in excellent agreement
with the numerical simulation results of pair collision of elastic capsules by Pranay
et al.74. They showed that at low Ca, where the deformations are small, the net
final displacement of the capsules after the collision decreases as Ca decreases. This
is because at low Ca, as Ca decreases, the capsules approach the rigid particle limit
and the net displacement of rigid particles is zero due to the reversibility of Stokes
flow. At high Ca, where the deformations of the capsules are significant, each capsule
becomes more elongated in the flow direction but less so in wall-normal direction as
Ca increases and accordingly the net displacement due to collision decreases. The
net final displacement of the capsules after collision shows a non-monotonic behavior
with Ca around Ca = 0.10.
102
6.4 Comparison with the theoretical model
In section 6.1 we presented a prediction about the dependance of the capsule-free layer
thickness on two competing mechanisms – wall-induced migration and shear-induced
diffusion (eq. 6.10) as
δ ∼ K (N1 −N2)
fsφbηγ. (6.14)
Eq. 6.14 shows a scaling predcition δ ∼ φ−1. Our numerical simulation results
(Fig. 6.14) on suspensions of elastic capsules at two different volume fraction (φ =
0.10 and φ = 0.20) show that the the thickness of the capsule depleted layer is a
decreasing function of the volume fraction of the capsules. Eq. 6.14 shows δ ∼(
(N1−N2)ηγ
)(
1fs
)
≈ umig
Ds(eqs. 6.3 and 6.4) – indicating that the wall-induced migration
favors formation of thicker capsule free layer where as shear-induced diffusion opposes
migration leading to a thinner capsule-free layer. Numerical simulation results on
suspensions of capsules (Fig. 6.19(b)) show that diffusivity is a weak function of Ca
where as results on single capsule migration show that in the intermediate range of
Ca (between 0.1 and 1.0) the normal stress difference scales (Fig. 6.4(b)) as (N1 −
N2)/(ηγ) ∼ Ca0.60. This leads to a prediction δ ∼ (Ca)0.60. In suspensions, the
effect of wall-induced migration gets weak due to finite concentration and finite size
of capsules, so it would be expected that the exponent of the scaling to be less than
0.60. Our simulation results on suspensions of capsules show δ ∼ (Ca)0.40. This
is in a good agreement to the experimental7 and numerical34 observations. These
observations indicate that the capsule-depleted layer thickness can be determined
primarily by the mechanism of wall-induced migration.
Numerical simulation results on the effect of polymer on the single capsule migra-
103
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.2
0.4
0.6
0.8
1
Ca
δ p/a
SimulationTheory
Figure 6.21: Comparison of the thickness of the capsule free layer for suspensions(φ = 0.10) of capsules in polymer ( β = 0.994,Wi = 20) solution obtained fromsimulations and predicted from theory (eq. 6.16) as a function of Ca.
tion show that polymer suppresses the migration of capsules. In suspension this would
be expected to lead to lower thickness of the capsule-free layer. Numerical simulation
results on shear-induced diffusion of suspensions of capsules show that polymer sup-
presses the diffusive motions of the capsules which would be expected to lead larger
thickness of the capsule-free layer. But since the influence of migration is dominant
in the dynamics of suspensions, the net effect of the polymer on the suspensions of
capsules would be expected to suppress the collective motion of capsules towards the
center of the channel and to decrease the thickness of the capsule-free layer. In fact
we can use the theoretical model to show that the balance of wall-induced migration
and shear-induced diffusion leads to a smaller capsule-free layer thickness in the poly-
mer case. The thickness of the capsule-free layer for the suspensions of capsules in
104
polymer solutions from eq. 6.14 can be expressed as
(
δPδN
)
=
(
KP
KN
)(
(N1 −N2)P(N1 −N2)N
)(
fs,Nfs,P
)
, (6.15)
where subscripts “N” and “P” denote results for Newtonian and polymer (β =
0.994,Wi = 20) solutions respectively. Numerical simulation results on single capsule
migration have shown thatKP/KP ∼ 0.42 (Fig. 6.8(b)) and (N1 −N2)P / (N1 −N2)N ∼
0.95 (Fig. 6.8(a)). Numerical simulation results on the diffusivity of capsules in sus-
pensions show that fs,N/fs,P = Ds,N/Ds,P ∼ 1.8 (Fig. 6.19(b)). These values yields a
prediction(
δPδN
)
= 0.72. (6.16)
Indeed our numerical simulation results on the dynamics of suspensions show that
polymer leads to a redistribution of capsules at steady state with a reduction in the
thickness of the capsule-free layer. Fig. 6.21 shows the comparison of δP obtained
from simulations and predicted from theory as a function of Ca. As observed from
the figure the theoretical predictions are reasonably accurate considering the fact the
the prediction was derived from the basic knowledge of the wall-induced migration
and shear-induced diffusion of capsules. Our results show that the thickness of the
capsule-free layer under the influence of polymer also scales with the Ca as δ ∼ (Ca)n.
The exponent of this scaling is less than that of the Newtonian case indicating that
the effect of polymer becomes substantial at high Ca. This is in a good qualitative
agreement to the experimental observations7.
105
6.5 Discussion
In the present work, we have studied the suspensions of elastic capsules in Couette
flow in Newtonian and polymer solutions. We have presented a theory which suggests
that the competition between the effect of wall-induced migration and shear-induced
diffusion leads to a steady-state distribution of capsules with a capsule-depleted layer
near the walls. The thickness of the capsule-free layer is derived analytically and is
found to be dependent on the capillary number. Numerical simulations is performed
to study the motion of elastic capsules in polymer solutions. Capsule membranes
are modeled using a neo-Hookean (NH) constitutive model and polymer molecules
are modeled as bead-spring chains with finitely extensible non-linearly elastic (FENE)
springs; parameters were chosen to loosely approximate 4000 kD poly(ethylene oxide).
Simulations are performed with a novel Stokes flow formulation of the Immersed
Boundary Method (IBM) for the capsules, combined with Brownian dynamics for the
polymer molecules. The polymer chains interact hydrodynamically with one another
and with the capsules. An approximate treatment of the Brownian motion of the
polymer beads is used, allowing substantial simplification in methodology. Results
for an isolated capsule indicate that the wall-induced migration depends strongly on
the capillary number. Numerical simulation of suspensions of capsules in Newtonian
fluid illustrates the inhomogeneous distribution of capsules at steady-state with the
formation of capsule-depleted layer near the walls. The thickness of this layer is found
to be strongly dependent on the capillary number. The shear-induced diffusivity,
on the other hand, show a weak dependence on capillary number. These results
indicate that the mechanism of wall-induced migration is the primary source for
determining the capsule-depleted layer thickness of capsules in suspensions. These
106
results are in qualitative agreement to the theoretical model. Numerical simulation
results on the effect of polymer show that both the wall-induced migration and the
shear-induced diffusive motion of the capsules are suppressed under the influence of
polymer. Since the influence of migration is dominant in the behavior of suspensions
of capsules, the net effect of the polymer on the suspensions of capsules is to suppress
the collective motion of capsules towards the center of the channel leading to a lower
thickness of the capsule-free layer and a redistribution of the capsules at steady state.
These observations are representative of the fact that the presence of small amount of
polymers can significantly influence the motion of deformable particles in suspensions.
107
Chapter 7
Conclusion
Blood flow in the microcirculation is strongly affected by the presence of drag reducing
polymer additives, for reasons that are not understood. The present work represents
an initial step toward understanding these observations, and more broadly, shedding
light on the dynamics of complex multiphase fluids. Simulations of simple particle
motions and pair collisions of nominally spherical elastic capsules in shear flow are
performed using a novel immersed boundary method for Stokes flow; the suspended
polymer molecules are simulated as bead-spring trimers with parameters chosen to
model poly(ethylene oxide). The polymer chains interact hydrodynamically with one
another and with the capsules. An approximate treatment of the Brownian motion
of the polymer beads is used, allowing substantial simplification in methodology and
savings in computation time. Because the simulations are performed at high Peclet
number, this approximation does not introduce substantial error into the results. The
dynamics of isolated capsules in shear are found to be almost completely unaffected
by the polymer additives at the very dilute concentrations considered here. This
108
result is a reflection that the flow field in this case is shear-dominated, and thus
does not lead to substantial polymer stretching. Results for pair collisions are more
complex. At high Ca, for neo-Hookean capsules or for Skalak capsules with a small
penalty for area change, there is almost no effect of polymer additives on the collision
trajectories, while for Skalak capsules with a substantial energy penalty for area
change, there is a substantial effect – in this case the net displacement of the capsules
in the wall-normal (gradient) direction is substantially decreased from the Newtonian
case. This effect strongly depends on the proximity of the capsules at the time of
collision, concentration of polymer and the area incompressibility of SK capsules and
originates in the increased rate of deformation in the gap between departing capsules
that is observed with membranes that strongly resist area change. At low Ca, where
the deformations are small, the effect of polymer on the collision dynamics of NH and
SK capsules are substantial.
A simple theoretical model is presented to describe the cross-stream migration
of deformable capsules in suspensions which comes from a balance of two competing
mechanisms – wall-induced migration and shear-induced diffusion. The wall-induced
migration favors continuous migration of capsules away from the vessel wall where
as shear-induced diffusion opposes migration. The balance between these two effects
leads to an inhomogeneous distribution at steady state with higher concentration of
capsules near the center and a capsule-depletion layer near the walls. The thick-
ness of this layer depends on both mechanisms. Results for an isolated capsule near
a wall indicate that the wall-induced migration depends strongly on the capillary
number. Numerical simulation of suspensions of capsules in Newtonian fluid illus-
trates the inhomogeneous distribution of capsules at steady-state with the formation
109
of capsule-depleted layer near the walls. The thickness of this layer is found to be
strongly dependent on the capillary number. The shear-induced diffusivity, on the
other hand, show a weak dependence on capillary number. These results indicate
that the mechanism of wall-induced migration is the primary source for determining
the capsule-depleted layer thickness of capsules in suspensions. Numerical simulation
results show that both the wall-induced migration and the shear-induced diffusive
motion of the capsules are suppressed under the influence of polymer. Results on sus-
pensions of capsules illustrate that the net effect of polymers is to reduce the thickness
of the capsule-depleted layer resulting in a redistribution of capsules at steady state.
The results are in qualitative agreement to the experimental observations. These
observations are concrete indications that the presence of small amounts of polymer
additives can substantially change the flow of suspensions of model cells, and that the
membrane properties of the cells play an important role in the observed changes.
110
Chapter 8
Current work: Suspensions of
capsules in a pressure-driven flow
8.1 Overview
We consider the motion of spherical NH capsules of nominal radius a, suspended in
in Newtonian fluid, in a pressure-driven flow as shown in Fig. 8.1. The simulation
box is periodic in the flow (x) and vorticity (z) directions. The wall-normal direction
is y. The walls are at y = 0 and y = By. The undisturbed flow field in x -direction
is a fully developed Poiseuille (parabolic) flow given by
u(y) = 4U
[
(
y
By
)
−(
y
By
)2]
, (8.1)
where U is the channel centerline velocity. The capillary number of the capsule is
defined as Ca = ηsγwa/G, where γw is the shear rate at the wall given as: γw = 4U/By.
Time is non-dimensionalized with By/U as t∗ = tU/By. The dimensionless time step
111
y
z
2a
y = By
xy = 0
Figure 8.1: Schematic of suspensions of fluid-filled elastic capsules in a pressure-drivenflow.
used in our simulations is 5 × 10−3. In all our simulations we use the size of the
simulation domain as a slit of height 10a, 16a and 40a. Considering the nominal
radius of the capsule to be 4µm, the size of the slit corresponds to lengths 40µm,
64µm and 160µm respectively. To avoid wrinkling instabilities56,6, the capsules are
initially preinflated such that the radius of the prestressed capsule a is related to the
radius of the unstressed capsule a0 as a = 1.05a0. We will consider two cases: the
dynamics of an isolated capsule place near a wall and suspensions of capsules in a
pressure-driven flow in a slit geometry as shown in Fig. 8.1. The numerical details of
these two situations are outlined below.
112
8.2 Migration of an isolated capsule in a pressure-
driven Flow
In this section we report the results of wall-induced migration of an isolated NH
capsule in a pressure-driven flow in a Newtonian fluid. For all the simulations of single
capsule migration, unless otherwise specified, we use the simulation domain to be a
cubic box of size 10a. The capsule is initially placed at a distance of 2a (y/By = 0.20)
from the bottom wall (y = 0) and is then subjected to the shear flow. For all the
simulations of single capsule migration in pressure-driven flow the capsule surface
is discretized into 1280 triangular elements corresponding to 642 beads. Fig. 8.2(a)
shows the trajectory of the center of mass of the capsule in the wall-normal direction as
a function of time at different values of Ca. The figure shows that the capsule migrates
continuously away from the wall towards the center of the channel with the rate of
migration increasing with an increase in Ca. As shown in the figure for a time period
of 300 time units, the capsule with Ca = 0.60 almost reaches the center of the channel
y = 5a. As Ca increases, the capsule becomes more deformable – the capsule gets more
stretched in the x−direction than in the y−direction. The shape of the capsule can be
characterized by the Taylor deformation parameter, D = (Lmax − Lmin) / (Lmax + Lmin)
where Lmin and Lmax are the smallest and largest dimensions of the capsule in the shear
(x−y) plane. Fig. 8.2(b) shows plots of Taylor deformation parameter as a function of
channel height for NH capsule at various values of Ca. As the flow starts, D attains
its steady state value within a short time interval (t∗ ≤ 5) implying that capsules
attain their equilibrium shape quickly before moving significantly in the flow. It
should be noted that since the flow is pressure-driven, the shear rate is not constant
113
0 50 100 150 200 250 3000
1
2
3
4
5
t∗
y/a
Ca = 0.10Ca = 0.142Ca = 0.30Ca = 0.60
(a)
2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
y/a
D
Ca = 0.10Ca = 0.142Ca = 0.30Ca = 0.60
(b)
Figure 8.2: Migration of NH capsules in a Newtonian fluid in a pressure-driven flow.(a) Trajectory of the center of mass of a capsule y as a function of time t∗ in the wall-normal direction. The walls are at y = 0 and y = By = 10a. (b) Capsule deformationD as a function of center of mass of a capsule y.
114
2 2.5 3 3.5 4 4.5 50
0.01
0.02
0.03
0.04
0.05
0.06
y/a
(N1−
N2)/
(ηγ w
)
Ca = 0.10Ca = 0.142Ca = 0.30Ca = 0.60
(a)
0 50 100 150 200 250 300
0.2
0.25
0.3
0.35
0.4
0.45
0.5
t∗
y/B
y
2a/By = 0.20
2a/By = 0.05
NH, Ca = 0.142SK, Ca = 0.60
(b)
Figure 8.3: (a) Difference between first (N1) and second (N2) normal stress differencesas a function of y. (b) The trajectory of an isolated capsule in a Newtonian fluid fordifferent channel heights. Solid lines are simulation results for channel of heightBy = 10a and dashed are for By = 40a.
115
along the wall-normal direction. The shear is zero at the center and maximum (in
magnitude) at the walls. The figure shows that as the capsule move towards the
center in the low shear region, the deformation parameter decreases and approaches
to zero (as seen for Ca = 0.60) at the center. Fig. 8.3(a) shows the variation of
the normal stress difference difference N1 −N2 of an individual capsule as a function
of channel height. The the first and second normal stress differences are defined as
N1 = τxx − τyy; N2 = τyy − τzz. Here τ is the extra stress due to the presence of
the capsule. As observed from the figure the more deformable a capsule (higher Ca),
the larger (N1 − N2) it can exhibit and higher is its tendency to migrate away from
the wall. The figure also shows that normal stress difference decreases as the capsule
move towards the lower shear rate region.
It should be noted that migration of an isolated capsule near a single wall is
qualitatively different for pressure-driven flows than for Couette flow. The shear rate
is constant in the case of Couette flow where as the shear rate varies across the channel
height for the case of pressure-driven flows. Theoretical predictions38,87 of migration
of a deformable drop (analogous to a deformable capsule) suggest that in the case
of linear unidirectional shear flow (Couette flow) the wall-induced migration has the
dominant effect (the effect is O(ε4) where ε = 2a/By) . However, in the case of
quadratic unidirectional shear flow (Poiseuille flow) the dominant contribution comes
from the curvature (the effect is O(ε3)). In fact the migration of a deformable particle
is faster in the pressure-driven flow than in the Couette flow having same shear rates
at the walls. The effect of confinement (ε = 2a/By) on migration is shown in Fig.
8.3(b). The figure shows that the effect of channel height on the migration capsule
is significant. Increasing the channel size (decreasing ε ) decreases the the migration
116
rate.
8.3 Suspensions of capsules in a pressure-driven
Flow
We now consider the dynamics of dilute suspensions of NH capsules in a pressure-
driven flow in a Newtonian fluid as shown in Fig. 8.1. Simulation box of three
different sizes are considered – a cubic box of size 10a, a cubic box of size 16a and a
slit of size 15a,40a and 6.5 in x, y and z directions respectively. The volume fraction
of capsules considered is φ = 0.10 which corresponds to simulations of 25, 100 and 100
capsules in the corresponding simulation domains. For all simulations of suspensions
of capsules, we use 320 triangular elements to discretize the capsule surface which
corresponds to 162 nodes per capsule. These parameters of capsules were chosen to
keep the computational cost moderate while still capturing the suspension dynamics
with a reasonable accuracy (see chapter 6). The capsules are initially distributed
randomly in the computational domain and are then subjected to a the Poiseuille
flow for a time period of 300 time units. The net motion of the capsules towards the
center of the channel is quantified by examining the time evolution of the average
distance of the capsule’s center from the center of the channel < |y − ycenter| >. Fig.
8.4(a) shows < |y − ycenter| > for the capsules as a function of time in a Newtonian
fluid at different Ca. The value of < |y−ycenter| > initially decreases and then reach a
steady state indicating initial migration of capsules towards the center of the channel
leading to an accumulation of the capsules in the core of the channel at steady state.
We consider the time at which the steady state is reached as the time after when the
117
0 50 100 150 200 250 3001
1.5
2
2.5
t∗
<|y
−y ce
nter|>
/a
NH, Ca = 0.08NH, Ca = 0.142NH, Ca = 0.30NH, Ca = 0.60
(a)
0 50 100 150 200 250 3000.5
0.6
0.7
0.8
0.9
1
t∗
<u>
/U
−−− <U>/U
NH, Ca = 0.08NH, Ca = 0.142NH, Ca = 0.30NH, Ca = 0.60
(b)
Figure 8.4: (a) Average distance from the centerline < |y − ycenter| > of suspensions(φ = 0.10) of NH capsules in a Newtonian fluid in a a cubic box of size 10a as afunction of time t∗. (b) Average velocity of the center of mass of capsules in flow(x)direction as a function of time t∗. The dashed line represents the average velocityof the undisturbed flow (〈U〉/U = 2/3).
118
change in average distance from the center or average center of mass of the capsules,
is less than 5%. For all simulations of the suspensions of capsules, we take this time
as t∗ ≥ 200 as observed from the figure. The figure also shows that the extent to
which the capsules move towards the center of the channel increases with an increase
in Ca indicating that as the deformability of the capsules increases, the capsules
have a greater tendency to migrate away from the walls and move closer toward the
center of the channel. Comparison of the present results to the results of suspensions
in Couette flow (chapter 6, Fig. 6.12(a)) indicate that the migration of capsules
towards the center of the channel in pressure-driven flow is faster than that of the
Couette flow (both cases have same shear rate at the walls) indicating that curvature
enhances the net migration of capsules.
Migration of capsules towards the center and accumulation at the core of the
channel can also be quantified by examining the average velocity of the capsule’s
center. Fig. 8.4(b) shows the average velocity of the center of mass of capsules in
flow (x)direction as a function of time. The dashed line represents the average velocity
of the undisturbed flow given as 〈U〉 = 2/3U . It should be noted that the average
velocity of the capsules in the flow direction is proportional to the flow rate of the
capsules. The figure shows that as the flow starts, the capsules migrates towards the
center and hence their average velocity increases (in pressure-drive flow the velocity of
the fluid is maximum at the center). Interestingly, the average velocity of the capsules
is more than the average velocity of the undisturbed flow indicating that capsules get
accumulated away from the walls of the channel and start flowing with the faster
moving fluid near the center. The figure also shows the effect of deformability on the
average velocity of the capsules. More deformable capsules (high Ca) migrate more
119
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
y/a
u/U
NH, Ca = 0.08NH, Ca = 0.142NH, Ca = 0.30NH, Ca = 0.60
(a)
0 2 4 6 8 100
0.01
0.02
0.03
0.04
0.05
0.06
y/a
frac
tion
NH, Ca = 0.08NH, Ca = 0.142NH, Ca = 0.30NH, Ca = 0.60
(b)
Figure 8.5: (a) Steady-state distribution of the velocity of capsule’s center (symbols)in flow direction (x) as a function of y in a Newtonian fluid in a cubic box of size10a. The dashed lines are the quadratic fits to the symbols. The solid line (black)represents parabolic profile of the undisturbed velocity. (b) Steady state distributionof capsules as a function of y. The walls are at y = 0 and y = 10a.
120
towards the center and hence have higher average velocity in the flow direction (∝
flow rate).
The steady-state distribution of the velocity of capsule’s center (symbols) in flow
direction (x) as a function of y is shown in Fig. 8.5(a). Symbols are the simulation
data for the velocity of capsule’s center at various Ca and the dashed lines are the
corresponding quadratic fits. For reference the parabolic profile of the undisturbed
flow is also shown (solid black line). The primary observation is that suspensions of
capsules leads to the blunting of the velocity profiles. As noted in chapter 3.2, the
capsule membranes are required to move with the fluid velocity in order to satisfy
no-slip boundary condition. Hence the velocity of the center of capsules is directly
related to the velocity of the fluid elements. The figure shows that at steady-state the
capsule accumulate near the center of channel and move slowly than the corresponding
fluid element in the undisturbed flow. This leads to the blunting of the velocity
profiles. The blunting of the velocity profiles has also been observed in suspensions of
rigid particles43, emulsions of deformable liquid droplets89 and suspensions of elastic
capsules94,70. The shear-induced particle drift in this leads to the blunting of the
velocity. The deformability of the capsule paly an important role in the blunting of
the velocity profiles. More deformable capsules cause less blunting of the velocity and
vice versa.
Fig. 8.5(b) shows the steady state distribution of capsules, represented as the
fraction of capsules, at various values of Ca as a function of channel height. The
fraction is calculated as an areal fraction of membrane surface as described in chap-
ter ??. Fig. 8.5(b) shows that at steady state the distribution of the capsules is
inhomogeneous – capsules have higher concentration near the center of the channel
121
0 50 100 150 200 250 3000.12
0.14
0.16
0.18
0.2
0.22
0.24
t∗
<|y
−y ce
nter|>
/By
2a/By = 0.20
2a/By = 0.125
2a/By = 0.05
Figure 8.6: Average distance from the centerline < |y − ycenter| > of suspensions(φ = 0.10) of NH (Ca = 0.142) capsules in a Newtonian fluid for different channelheights.
and a capsule-depleted region near the walls. The thickness of the depleted layer as
well as the concentration near the center of the channel increases with an increase
in Ca indicating that more deformable capsules move farther away from the walls
and accumulate closer towards the channel center. The figure also shows the layer
formation of capsules near the walls, as shown by the off-center peaks in the distri-
bution. The layer formation has also been observed by Li and Pozrikidis89 in their
simulations of suspensions of liquid drops in wall bounded flows. They observed that
the layer formation was a result of using confined channel size. We have studied the
effect of box size on the steady state distribution of capsules in Couette flow (chapter
??). Results show formation of layers diminishes in large box indicating that layer
122
formation is an artifact of using small size of the channel.
The effect of confinement (2a/By) on suspensions of capsules is shown in Fig. 8.6.
As observed from the figure, increasing the channel height decreases the migration
rate of the capsules towards the center i.e. the capsules in a bigger channel take longer
time to reach a steady state than in the case of smaller one. The capsules in a channel
of height 40a is far from reaching the steady state even at a time period of 300 time
units where as the capsules in a channel of height 10a reach a steady state around
150 time units. If we assume migration to be a dominant phenomenon in suspension
dynamics then the time it takes to reach a steady state should scale as the time it
takes a capsule to migrate to the capsule-free layer. Theoretical predictions38,87 of
migration of a deformable drop in a pressure-driven flow show that migration velocity
scales as O(ε3) where ε = 2a/By. Hence, the time to reach a steady state tss should
scale as tss ∼ (By/2a)3. For example, in the figure, the time it takes to reach a
steady state for capsules in a slit geometry of height 40a should be 64 times the time
the capsules would take to reach a steady state in a slit of height 10a (for the same
volume fraction). Similar scaling was obtained for the suspensions of rigid particles
in pressure-driven flow43.
8.4 Discussion and future work
Numerical simulation results show that migration of an isolated capsule in a pressure-
driven flow in a Newtonian fluid depends strongly on the Ca. The effect of curvature
enhances the migration rate of the capsules. The results of suspensions of elastic
capsules in pressure-driven flow show that capsules collectively migrates towards the
123
center of the channel and reach a steady state showing an inhomogeneous distribution
– having higher concentration at the core of the channel and a depletion layer near
the walls. The migration of capsules leads to the blunting of the velocity profiles. The
Ca has a strong influence on the suspension dynamics – more deformable capsules
migrates more towards the center leading to a thicker depletion layer near the walls
and more concentration of capsules near the center.
As a part of the future work, we would like to investigate the effect of long chain
drag reducing additives on the dynamics of suspensions of capsules in pressure-driven
flows. Since most of the experiments of DRAs on blood flow pertains to the case of
pressure-driven flow, it would be of high interest to perform simulations to observe
the effects, at least qualitatively, on suspensions of capsules in pressure-driven flow.
The main focus of the study would be to gain a mechanistic understanding on the
effect of polymers on concentration distribution, flow resistance, capsule-depletion
layer and velocity profiles of the capsules at steady state.
124
Chapter 9
Future work
9.1 Suspensions of Red Blood Cells
With the current numerical method and model, a variety of interesting issues can
be addressed. Until now the focus of the study have been on the dynamics of elastic
capsules which were nominally spherical in shape. The spherical capsules were chosen
for the study because of the simplicity of shape. The short term goal now is to
incorporate the model to study the dynamics of suspensions of fluid-filled elastic
capsules which have the same nominal shape as that of the red blood cell i.e. capsules
having biconcave discoid shapes. As a starting point, we will use a discoid shape as
the equilibrium shape of the capsules. The analytical form of the discoid shape is
given as95,96,70
x = R0x′, y =
1
2R0
√
(1− r2)(
C0 + C2r2 + C4r
4)
, z = R0z′, (9.1)
125
Figure 9.1: A schematic showing suspensions of RBCs in the microcirculation.
where x, y, z are the coordinates of the cell surface, x′, y′, z′ are the coordinates of
a unit sphere, R0 is a constant to preserve the volume, r2 = x′2 + z′2, C0 = 0.32,
C2 = 2.003 and C4 = −1.123. Fig. 9.1 illustrates the basic situation of interest. As
noted in our methodology (chapter 3.2), we use two different models for the capsule
membrane – neo-Hookean (NH) model and Skalak (SK) model. The SK model is more
accurate representation of RBCs than NH model as it can be parametrized to yield
a strong resistance to area change relative to its resistance for shear deformation
as is the case for RBCs60. The Skalak model contains a shear modulus G and an
additional parameter C associated with the energy penalty for area change. Typically,
C � 1 indicating approximate area-incompressibility. For modelling the suspensions
of RBCs, SK model will be used.
126
In our model capsule membranes are assumed to have no bending. It is because the
membrane is assumed to have zero thickness that displays no resistance to bending.
In reality, the RBCs have membranes that have a finite thickness (∼ 4nm) and hence
have small resistance to bending. For example, for an RBC, R ∼ 1µm and G ∼
10−6J/m32. The bending modulus kb of the membrane is kb ∼ 10−20J97. The relative
contribution of shear deformation over bending can be estimated as R2G/kb ≈ 10−2−
10−3. Thus for modelliing spherical capsules, where there are no sharp curvatures,
the contribution of bending to the total membrane force can neglected. However, an
RBC can display sharp curvature depending on the flow geometry. For example, for
bending energy to have the same order as the elastic energy, κ/κ0 ∼ 103, where κ is
the local mean curvature and κ0 is the curvature in the undeformed state. In most
physical conditions, for example in suspensions of RBCs in confined geometry, the
value of κ/κ0 can be of that order indicating the need to incorporate bending in the
existing model. Here we outline the implementation of bending in our model.
For a closed membrane Γ, bending energy called Canham-Helfrich energy98,99 is
given by:
E =KB
2
∫
Γ
(2κH)2dS +KB
∫
Γ
κGdS, (9.2)
where κH , κG are the mean and gaussian curvature of the surface, and EB and EB are
bending moduli. Gauss-Bonnet theorem ensures that the second term is a constant
when no topological changes are involved. Force density due to bending is given by
first variation of the free energy Eq. 9.2:
fbi = KB
(
2∆sκH + 2κH(κ2H − κG)
)
n, (9.3)
127
where ∆sH is the Laplace-Beltrami operator over the surface. The resultant mem-
brane force density is the combination of bending and mechanical tension, fci = fei +fbi ,
where fei is the force density due to elasticity of membrane as discussed in chapter
3.2. The details of the methodology is provided by Boedec et al.100 in their work on
vesicle dynamics simulation.
9.2 Drug delivery
In recent years, colloid, polymer and lipid chemistry have been harnessed to create
a great diversity of particles based on silica, polymer gels, polymeric micelles or
liposomes101,102,103,104,105,106,107. Furthermore, a great variety of targeting moities (
antibodies, aptamers, vitamins, targeting peptides) have been conjugated to these
particles. Some interesting drug delivery particles are shown in Figure 9.2.
Recent years have seen increase in shape of these particles other than spherical.
Evidence shows that wormlike “filomicelles” developed by Discher and coworker106
stay longer in blood than spherical counterpart. Mitragotri and coworkers104 synthe-
sized nonspherical particles and observed similar results. DeSimone and coworkers108
developed a variant of imprint lithography to generate particles of wide variety of
shapes. With advances in chemistry and nanotechnology for particle synthesis, there
is a growing realization that shape of particle is a important determinant of their life
in blood stream and their specificity. Numerical complexity will augment in studying
behavior of these complex shape particles in an already complex blood circulation.
Principles which govern the distribution of such particles in the circulation are not
yet known and will be investigated as a part of the future work.
128
Figure 9.2: (a) Spheres. (b) Rectangular disks. (c) Rods. (d) Worms. (e) Oblateellipses. (f) Elliptical disks. (g) UFOs. (h) Circular disks. (Scale bars: 2 µm.)104
We can use the same theoretical approach developed in chapter ?? to make a
prediction of the evolution of a drug delivery particle in suspensions of deformable
particles (capsules/RBCs). If we simply take the evolution of the concentration c of
the drug delivery particle to obey the same Fokker-Planck equation (eq. 6.6) given
above for a dilute suspensions of capsules, then the balance of shear-induced diffusion
and hydrodynamic migration of the particle can be expressed as
∂c
∂t= −∂jy
∂y= − ∂
∂y
(
vmig,pc−∂ (Ds,p(y)c)
∂y
)
. (9.4)
Here vmig,p is cross-stream drift and Ds,p(y) is the short time diffusivity of the drug
delivery particle. The above equation is equivalent to writing the evolution equation
for the probability of finding a particle at a given y−position in the dilute limit; with
the assumption that the motion of particle is a continuous Markov process85,86 and is a
good approximation for time scales longer than about γ−1. In suspensions of capsules
with drug delivery particles, the short time diffusivity of the particle will be dominated
by the shear-induced diffusion because of the presence of capsules. Here Ds,p takes
129
rigid particles −marginated
critically marginatedparticles
highly deformable particles−not marginated
Figure 9.3: A schematic showing hypothesized distributions of drug delivery particlesin the microcirculation.
the same functional form as Dself for capsules in eq. 6.4. Consider the limiting case of
a rigid particle, which is a good approximation for a leukocyte, platelet or for many
proposed drug delivery particles. For rigid particles, the wall-induced hydrodynamic
migration effect vanishes and there are no other obvious mechanisms for drift (at low
Reynolds number), so at steady state, the flux balance simply reduces to
0 = − ∂
∂y
(
∂ (Ds,p(y)c)
∂y
)
, (9.5)
which gives the solution as
c = Kp/Ds,p(y). (9.6)
Here Kp is a constant, and Ds,p(y) is an increasing function of capsule concentra-
130
tion, so near the wall, where the capsule concentration is lowest, this simple equation
predicts that the particle concentration is highest, which is indeed the case. This
idea can be generalized to make predictions for finding the ideal location of the drug
delivery particles in suspensions of capsules/RBCs based on the physical properties
of the particle such as Ca, size, aspect ratio. For example, a relatively rigid parti-
cle in a flowing suspension of capsules/RBCs, as determined by its size, shape and
mechanical properties through capillary number, will most likely be found in the cap-
sule depletion layer. In contrast, a highly flexible particle will have larger migration
velocity than an capsule and so it will probably not be found in the depletion layer
but rather migrate into the concentrated layer of capsules where it will then diffuse
by shear-induced diffusion. A schematic of the hypothesized prediction is shown in
Fig. 9.3. Therefore as a part of future work, we want to extend the current model
to incorporate flow of mixtures of particles having different physical properties and
then to characterize the mechanical properties of drug delivery particles in flowing
suspensions of capsules/RBCs.
9.3 Leucocyte margination under the influence of
polymers
Finally, we would want to extend the our current model and methodology to facilitate
the study of mixture of particles (capsules, RBCs) with different physical properties
along with the effect of polymers. As noted above (chapter 2.1) the migration of RBCs
towards the center of a vessel is assumed to increase the probability of less deformable
leucocytes, such as white blood cells and platelets, to be found near the blood vessel
131
walls, a phenomenon known as margination. Margination is found to play a key role
in the process of inflammation35,36. On the other hand, microchannel experiments7
have shown that the addition of small amount of these polymer additives to the
suspensions of RBCs resulted in a redistribution of RBCs with a significant reduction
in the thickness of the cell-free layer. A recent experiment by Zhao et al.47 showed
that the addition of small amount of drag reducing polymers (10ppm of 4×106D PEO)
to blood can not only decrease the thickness of the cell-free layer layer but also reduce
the concentrations of leucocytes in these layers. Therefore as a part of future work
we would want to use our methodology and computational approach to examine the
dynamics of suspensions of elastic capsules (having discoid shapes representing RBCs)
with small concentration of leucocytes, modelled as capsules with low Ca, under the
presence of small amount of DRAs. Our simulation results have shown that DRAs
can influence the behavior of suspensions of capsules leading to the decrease in the
thickness of capsule-depleted layer. We now want to extend our study to incorporate
the presence of small concentrations of rigid particles (low Ca capsules) to find out if
we can reproduce the results of Zhao et al.47 qualitatively.
132
Appendix A
Single Polymer Dynamics
The fluctuation-dissipation theorem implies a nontrivial coupling between the Brow-
nian motion and the configurations of the polymer molecules and capsules. This
coupling is expensive to compute78. In dilute solutions and at high Wi, the effect of
Brownian motion on polymer dynamics gets relatively weak (Weissenberg number is
proportional to particle Peclet number). The Brownian motion of polymer under such
conditions can be approximated by using a free draining approach in the Brownian
term in an unbounded domain – where the contribution of hydrodynamic interaction
to the Brownian motion of polymer is neglected. In this section, we validate our
approximation of neglecting the hydrodynamic interaction in the Brownian term for
the polymers in the limit of high Wi and infinite dilution.
As a test case, we place a single polymer molecule (3-bead chain) in an unbounded
domain under shear flow with shear rate γ. We then study its evolution under the
influence of flow with and without the effect of hydrodynamic interactions in the
Brownian term respectively. The equation of motion of each bead of the chain is
133
governed by the balance of forces:
0 = fhi + f si + f ei + f bi . (A.1)
Here the superscripts h, s, e and b denote the hydrodynamic drag, spring, external
and Brownian forces respectively. This force balance leads to the following evolution
equation:
dxp = [u∞ +M · fp] dt+B · dW, (A.2)
where fp = f e + f s, M is the mobility tensor expressed as
Mi,j =1
ζδδij + (1− δij)G
R,ξ∞ (xi,xj) , (A.3)
B ·BT = (2kBT )M, (A.4)
u∞ (= γyex) is the externally imposed fluid velocity field, dW is a vector of indepen-
dent random variables, each chosen from a Gaussian distribution with zero mean and
variance dt and GR,ξ∞ (xi,xj) is the free space regularized Green’s function78 given by
GR,ξ∞ (xi,xj) = 1
8πηs
[
δ + xxr2
]
(
erf(ξr)r
)
+ 18πηs
[
δ − xxr2
]
(
2ξπ1/2 e
−ξ2r2)
, (A.5)
x = xi − xj, r = |x|.
Here ξ is the regularization parameter; its reciprocal represents the length scale over
which the force is spread. ξ scales as the size of the bead radius i.e. ξ ∼ a−1p . Eq. A.5
reduces to the Oseen-Burgers tensor as ξ → ∞. Further, the mobility tensor will be
134
positive definite for ξ−1 ≥ 3ap/√π. We take this value here. The term B · dW in the
evolution equation (eq. A.2) represents the Brownian contribution to the motions of
polymer molecules. The coupling between the Browinian motion and configuration of
polymer molecules is represented in eq. A.4. Note that computation of B is expensive
(scales as O (N3) for N particle system). Neglecting the hydrodynamic interaction in
the Brownian term simplifies eq. A.4 as
B =
√
2kBT
ζδ, (A.6)
and reduces the evolution equation (eq. A.2) to
dxp = [u∞ +M · fp] dt+√
2kBT
ζδ · dW. (A.7)
The above equation is identical to the evolution equation (eq. 3.25) used for polymer
molecules in bounded domain where the hydrodynamic interaction in the Brownian
term is neglected. Note that computation of B in eq. A.6 now scales as O (N).
The results on the dynamics of single polymer molecule in an unbounded shear
flow as a function of Wi are shown in Fig. A.1. We denote “HI” (solid lines) as the
simulation results including the hydrodynamic interactions in the Brownian term i.e.
using eq. A.2 and eq. A.4; and denote “FD” (dashed lines) as a free-draining model
representing simulations neglecting the hydrodynamic interactions in the Brownian
term i.e. using eq. A.7. Each data point the figures corresponds to averaging over 25
different initial configurations of polymer molecule. The mean properties of the poly-
mer dynamics in the figure are calculated after the polymer molecules are equilibrated
for a time period of 50 units. Fig .A.1(a) shows the steady state RMS end-to-end
135
0 5 10 15 20
0.01
0.1
1
Wi
<R
0>/L
c
HIFD
(a)
0 5 10 15 200.001
0.01
0.1
1
10
Wi
< τ
p>/(
η γ
n)
HIFD
τxx
τyy
τzz⋅
(b)
Figure A.1: (a) Average steady state RMS end-to-end distance < R0 > and (b) av-erage steady state polymer stress < τ p > as a function of Wi for a single polymermolecule in an unbounded shear flow. x, y and z represents flow, gradient and neutraldirections respectively. “HI” denote simulations including hydrodynamic interactionsin the Brownian term. “FD” represents simulations neglecting hydrodynamic inter-actions in the Browninan term.
distance < R0 > as a function of Wi. The figure shows that the difference in < R0 >
between HI and FD case decreases as Wi increases. Fig. A.1(b) shows the normal
components of steady state polymer stress < τ p > as a function of Wi for HI and FD
cases respectively. Here the expressions for the stress tensor for a bead spring chain,
as given by Kramers-Kirkwood method77, is
τ p = n
beads∑
i=1
(xi − xc) fpi , (A.8)
where xc is the center of mass of a polymer molecule and n is the number density. Here
also the figure shows that the results for HI and FD case converge at high Wi (≥ 5)
indicating that the effect of hydrodynamic interaction in the Brownian motion gets
weak as Wi increases. This shows that the approximation of neglecting hydrodynamic
interaction in the Brownian term holds good at high Wi and low concentration of
136
polymer.
137
Appendix B
Comparison of GGEM/IBM with
other methods
B.1 Overview of Conventional IBM Method
The immersed boundary method (IBM) has been used to study fluid-structure inter-
actions, especially in the field of biological fluid dynamics. This section deals with a
brief review on the conventional IBM described by Peskin75. The IBM , which is both
a mathematical formulation and a numerical scheme, involves a mixture of Eulerian
and Lagrangian variables. The coupling between these two types of variables is done
by the Dirac delta function. The Eulerian variables are defined on a fixed Cartesian
mesh, and the Lagrangian variables are defined on a curvilinear mesh that moves
freely through the fixed Cartesian mesh without being constrained to adapt to it in
any way at all. The interaction equations of the numerical scheme involve a smoothed
approximation to the Dirac delta function75,80.
138
The problem under consideration is motion of liquid capsules and polymer molecules,
immersed in a surrounding fluid. The Eulerian description applies to the fluid and
the Lagrangian description applies to the body (capsules and polymers in our case).
At zero Reynolds number, the fluid obeys Stokes equations. Let x be the Eulerian
coordinate, u be the Eulerian velocity and ρ be the Eulerian force density that the
immersed body applies to the fluid. Let η be the fluid viscosity and P be the pressure.
In the IBM, the Stokes equations hold in all of space, including that occupied by the
body. These equations are :
−∇P (x) + η∇2u(x) = −ρ(x),
∇ · u(x) = 0,(B.1)
with the boundary condition as
u(xW ) = u|W , (B.2)
where xW represents the solid boundary and u|W is velocity at the boundary. All
quantities in eqs. B.1 and B.2 are function of Eulerian variable x.
The Lagrangian variables define the configuration of body immersed in the fluid.
Let xc denote the coordinate of the surface of body and uc denote its velocity. Let
f c be the Lagrangian force applied by the immersed body. A capsule, for example,
comprises a nominally spherical elastic membrane of radius, enclosing a Newtonian
liquid with density equal to that of the outside fluid. The viscosities of the liquids
inside and outside the drop are considered to be identical and equal to ηs. The
membrane is represented as a collection of points (or elements). The forces resisting
deformation are calculated by computing displacements of the vertices of the deformed
139
element with respect to the un-deformed element. The equation of motion is :
dxci
dt= uc(xc
i), (B.3)
where subscript i denote a point on the capsule membrane.
It is now important to understand how the Eulerian variables interact with the
Lagrangian variables. The interactions can be expressed by using the property of
Dirac delta function as
ρ(x) =∑
δξIBM(|x− xc
i |) f ci , (B.4)
dxci
dt= uc =
∑
δξIBM(|xc
i − x|)u (x) , (B.5)
where δξIBMis a smoothed approximation of delta function and ξ−1
IBMdenotes its
length scale for smoothing. Eq. B.4 shows the conversion of Lagrangian to Eulerian
variables. It states that the Eulerian force density and the Lagrangian force density
are, at corresponding points, equal as densities if integrated over a volume. Eq. B.5
shows the conversion of Eulerian to Lagrangian variables. It states that the Eulerian
velocity, u, and the Lagrangian velocity are equal at corresponding points. This also
comes from the fact that the fluid obeys no-slip condition on the surface of the body.
Let hc be the characteristic element size of the membrane and let hg be the size
of the Eulerian mesh. To avoid leaks, the restriction needed is
hc < hg/2, or hg ∼ hc. (B.6)
140
In the IBM method by Peskin75, force distributions at moving interfaces or mem-
branes are discretized as distributions of regularized point forces, where the length
scale for smoothing the delta function scales as the Langrangian grid spacing used in
the simulation, i.e.
ξIBM
−1 ∼ hc. (B.7)
This ensures that the force density associated with each node is uniformly spread over
the length scale of the associated elements, thereby preventing fluid from penetrating
the membrane surface, and also preventing the un-physically large fluid (and thus
node) velocity that would be present at the node if ξIBM is made too large (in which
case δ approaches a true delta function). Moreover, this scaling is a natural one so
that δξIBM→ δ as hc → 0.
Eqs. B.6 and B.7 suggest that for the accuracy of the method all the length scales
should be equal i.e. all the length scales should resolve the membrane mesh size
(Lagrangian mesh), hc. A schematic of the length scales are shown in Fig. B.1(a).
Therefore, a conventional IBM requires h3c mesh points (if a uniform mesh is used).
An important remark is that the equations of motion that are derived for a viscous
incompressible fluid containing an immersed elastic boundary are mathematically
equivalent to the conventional equations that one would write down involving the
jump in the fluid stress across that boundary.
B.2 IBM and GGEM
With the same definition of Eulerian and Lagrangian variables defined in previous
section, the approach taken in our present work is to use an O (N) (where N is
141
hg ∼ ξ−1
IBM
hc ∼ ξ−1
IBM
(a) length scales associated with conventional IBM method
hc
hg ∼ α−1
∼ ξ−1GGEM
(b) length scales associated with IBM/GGEM
Figure B.1: (a) Schematic of different length scales associated with the conventionalimmersed boundary method. (b) Schematic of different length scales associated withIBM/GGEM.
142
the number of nodes to represent the immersed body) particle-particle/particle-mesh
method that we have recently developed for Stokes flow problems in nonperiodic
geometries, called GGEM (General geometry Ewald-Like Method)78,81. GGEM starts
with splitting the interaction equations for force density into two parts,
ρ(x) = ρl(x) + ρg(x),
ρl(x) =∑
i
[δξGGEM(|x− xc
i |)− gα (|x− xci |)] f ci ,
ρg(x) =∑
i
gα (|x− xci |) f ci ,
(B.8)
where δξGGEM(r) is a regularized delta function and the screening function, gα (r)
is used to split the force density into a “local” and “global” part denoted by the
subscripts l and g. The regularized delta function and the screening functions are
given by
δξGGEM(r) =
ξGGEM
3
π3/2e(−ξGGEM
2r2)[
5
2− ξGGEM
2r2]
(B.9)
gα(r) =α3
π3/2e(−α2r2)
[
5
2− α2r2
]
, (B.10)
where α is the screening parameter. The screening function takes the same mathe-
matical form as the regularizing delta function. The flow field is a superposition of
flow driven by ρl(x) and ρg(x):
u(x) = ul(x) + ug(x), (B.11)
143
or equivalently in terms of Lagrangian velocities:
uc(xci) = uc
l (xci) + uc
g(xci). (B.12)
The equation of motion is :
dxci
dt= uc(xc
i) = ucl (x
ci) + uc
g(xci). (B.13)
The general idea in GGEM is to separately compute the velocity fields due to
the two force densities. The total velocity is then simply the sum of the individual
contributions.
The local velocity ul(x), which results from the force density ρl(x). This is com-
puted analytically using solutions to Stokes’ equations in an unbounded domain - the
error incurred by assuming an unbounded domain will be canceled out in the global
solution described below. One can show that the modified force distribution, eq. B.8
yields a velocity field given by
ul(x) =∑
i
GRl (x− xc
i) · f ci , (B.14)
where the subscript l indicates the “local” contribution to the velocity. The modified
Green’s function, GRl is given by,
GRl (x) =
1
8πηs
[
δ +xx
r2
]
[
erf (ξr)
r− erf (αr)
r
]
− 1
8πηs
[
δ − xx
r2
]
[
2α
π1/2e−α2r2 − 2ξ
π1/2e−ξ2r2
]
. (B.15)
144
Because of the presence of the screening function in the local force density, this
function decays rapidly to zero over a distance proportional to α−1. The calculation of
the local velocity field begins by first identifying point forces located within a sphere
of fixed radius 4/α. Any force outside this cut-off radius is ignored. The value of α is
empirically determined. For a Lagrangian point xcj on the immersed body, the local
Lagrangian velocity can be written as
ucl
(
xcj
)
=
n∑
i=1,i 6=j
GR,cl
(
xf − xci
)
· f ci + limx→0
GR,cl (x) · f cj . (B.16)
Here n is the number of Lagrangian points which lie within the cut-off sphere with
center at(
xcj
)
. In eq. B.16, the velocity at Lagrangian point includes a contribution,
a “self-term”, due to the force exerted by the same point on the fluid. For a point
force, this velocity is infinite. However, the Green’s function is regularized, and hence
has a finite limit as the separation vector vanishes.
This gives the local velocity at any point (both Eulerian and Langrangian) in the
system.
The global contribution to the velocity, ug (x), is due to the global part of the force
distribution, ρg (x) in eq. B.8. The solution to the Stokes equations with a forcing
function ρg (x) is calculated numerically, requiring that the total velocity u (x) satisfy
appropriate boundary conditions. The Stokes equation for the global field are:
−∇P (x) + η∇2ug(x) = −ρg(x),
∇ · ug(x) = 0.(B.17)
All quantities in eq. B.17 are function of Eulerian variable x. At points xW on the
145
bounding walls, boundary conditions has to be satisfied. Therefore, the corresponding
boundary condition for the global velocity field is given by
ug|W = −ul|W + u|W , (B.18)
where u|W is the actual boundary condition and ul|W is the known local component
of the velocity field. Note that by imposition of this boundary condition for ug (x),
we cancel out the error introduced by using unbounded-geometry solutions for the
local problem.
The interaction equation for global force density from Lagrangian to Eulerian
quantities is shown in eq. C.10. The global calculations are performed on Eulerian
mesh having characteristic length as hg. With the same argument as in the previous
section, the constraint on the mesh size is :
hg ∼ α−1. (B.19)
This is the key difference between conventional IBM and GGEM. It is due to the
fact that the interaction equations of global force density, ρg(x), between Lagrangian
and Eulerian quantities are linked by the screening function, gα(x), having a char-
acteristic length scale as α−1. Hence the global calculations can be performed on a
coarse grid depending on α. However, the the interaction equations of local force
density, ρl(x), between Lagrangian and Eulerian quantities are linked by the delta
function, δξ(x), having a characteristic length scale as ξ−1. So to maintain numerical
accuracy in the local calculations, ξ−1 scales as the Lagrangian grid spacing used in
146
the simulation, i.e.
ξ−1 ∼ hc. (B.20)
Fig. B.1(b) shows the schematic of the length scales associated with GGEM. Fig.
B.1 shows these differences between the two methods.
Now what remains is the interactions from Eulerian variables to Lagrangian vari-
ables. The global velocity in Lagrangian coordinate, ucg(x
ci), can be obtained from
the interpolation of the global velocity in Eulerian coordinate, ug(x), as :
ucg(x
ci) = L(xc
i )ug(x), (B.21)
where L(x) is a interpolation function in three dimensions. In our work we use second
order Lagrange interpolation.
B.3 BIM and GGEM
We turn now to the relationship between IBM/GGEM and boundary integral equation
methods. A Stokes flow u(x) driven by a force density ρ(x) is given by:
u(x)− u∞(x) =
∫
V
G (x,x′) · ρ(x′) dx′, (B.22)
where G (x,x′) is the Green’s function for the geometry and boundary conditions of
interest and u∞(x) is the velocity in the absence of the forcing. If the force density
is localized on an interface S within the fluid (e.g. a capsule membrane), the force
density can be written as:
147
ρ(x′) = −∫
S
∆t(x′′)δ(x′ − x′′) dS(x′′), (B.23)
where −∆t(x′′) dS(x′′) is the total force exerted by the surface on the fluid at point
x′′. Inserting eq. B.23 into eq. B.22 yields:
u(x)− u∞(x) = −∫
V
∫
S
G (x,x′) ·∆t(x′′)δ(x′ − x′′) dS(x′′) dx′, (B.24)
which simplifies to:
u(x)− u∞(x) = −∫
S
G (x,x′′) ·∆t(x′′) dS(x′′). (B.25)
This is the standard boundary integral equation for an interface immersed in a single
fluid.
In IBM/GGEM, eq. B.23 is discretized by (1) approximating δ(x′−x′′) by δc(x′−
x′′); (2) approximating −∆t(x′′) dS(x′′) by f c(x′′); and (3) approximating the integral
as a sum, where the x′′ are nodal positions xi. This yields the IBM/GGEM force
density
ρ(x′) =
nodes∑
i
δc(x′ − xi)fci . (B.26)
This is identical to eq. 3.26 in the absence of polymer molecules. Inserting this into
eq. B.22 leads to a discretization of the boundary integral equation eq. B.25:
u(x)− u∞(x) =
nodes∑
i=1
GR,c (x,xci) f
ci , (B.27)
148
where GR,c (x,xci) is the regularized Green’s function that corresponds to δc(r). This
velocity field is the solution to eq. C.1 with the relevant boundary conditions and
force density eq. 3.26 (again with no polymers), and is precisely what is computed
by GGEM. These results establish the relationship between IBM/GGEM and the
boundary integral method. An identical analysis, replacing δc(r) by δcIBM
shows the
same result for IBM in the Stokes flow case.
149
Appendix C
Implementation Issues-Slit
The problem under consideration is a Stokes flow in a slit geometry of fluid with
velocity u, pressure P , viscosity η. To treat the bounded domains, the present method
starts with restatement of the force-density expression: ρ(x) = ρl(x) + ρg(x). Here
ρl(x) is the “local” density which drives the local contribution of the velocity field,
ul(x), and ρg(x) is the “global” density which drives the global contribution of the
velocity field, ug(x). By linearity u(x) = ul(x) + ug(x). The local velocity ul(x) is
calculated analytically assuming an unbounded domain. The global velocity ug(x) is
calculated numerically on a coarse grid with appropriate boundary conditions. The
Stokes equation for the global field are:
−∇P (x) + η∇2ug(x) = −ρg(x),
∇ · ug(x) = 0,(C.1)
or
− ∂P∂xi
+ η∂2ug,i
∂x2j
= −ρg,i,
∂ug,j
∂xj= 0,
(C.2)
150
where x1, x2 and x3 are the flow, wall-normal and vorticity directions respectively
and ug,i, are the corresponding velocity components for the global field. We also use
interchangeably x,y,z for xi and ug, vg, wg for ug,i. The box is periodic in x− and z−
directions and bounded by walls in y− direction. At points xwall on the bounding
walls, boundary conditions has to be satisfied. Therefore, the corresponding boundary
condition for the global velocity field is given by
ug|wall = −ul|wall + u|wall, (C.3)
where u|wall is the actual boundary condition and ul|wall is the local component of
the velocity field.
C.1 Different flow profiles:
Different flow behavior can be solved by using different expressions for pressure and
boundary conditions. For example the pressure P can be restated as
P = −Gx+ p, (C.4)
where G is the mean pressure gradient in the flow direction and p is the pressure driven
by the forces that particles exert on the fluid. The following cases show different
expressions for G and boundary conditions:
For Couette flow in a slit geometry between two parallel plates the unperturbed
151
velocity profile is linear and is given by
u(y) = γy, v = 0, w = 0, (C.5)
where y = ±By/2 represents wall, By is the height of the channel and γ is strain rate.
The expression for G and boundary conditions are given as
G = 0,
ug|wall = −ul|wall ± γBy/2,
vg|wall = −vl|wall,
wg|wall = −wl|wall.
(C.6)
Here, G is zero and the boundary conditions are no-slip at the walls.
For pressure-driven channel flow the unperturbed velocity profile is parabolic and
is given by
u(y) = Ucl
[
1−(
y
By/2
)2]
, v = 0, w = 0, (C.7)
where y = ±By/2 represents wall, By is the height of the channel and Ucl is the
channel centerline velocity. The expression for G and boundary conditions are given
as
G = 8ηUcl
B2y,
ug|wall = −ul|wall,
vg|wall = −vl|wall,
wg|wall = −wl|wall.
(C.8)
Here G is constant and the boundary conditions are no-slip at the walls.
152
C.2 Assignment: putting ρg on the mesh - collo-
cation
The global calculations are performed on mesh consisting of Mx ×My ×Mz discrete
points. The global density, ρg(x), which drives the global contribution of velocity
field ug(x), is calculated as
ρg(x) =∑
i
g(|x− xi|) fi(xi), (C.9)
where g(r) is the screening function and fi(xi) is the force exerted by the particle i
at position xi. The mathematical form of the screening function is given by
g(r) =α3
π32
e(−α2r2)[
5
2− α2r2
]
, (C.10)
where α is the screening parameter. The screening function, g(r), satisfies
∫
V
g(r) dV =
∫ ∞
0
4πr2g(r) dr = 1, (C.11)
where V is the infinite space. One of the properties of the screening function is that
it decays rapidly to zero over a distance proportional to α−1. We take this domain
to be a sphere of radius 4/α, eq. C.11 then becomes
∫ 4/α
0
4πr2g(r) dr ≈ 1. (C.12)
153
The calculation of global density, ρg(x), can then be simplified by identifying all the
particle positions within this domain. Contribution of any particle outside this cut-off
radius is ignored. For any point xM on the mesh, the calculation of the global density
is given by
ρg(xM) =
n∑
i
g(|xM − xi|) fi(xi). (C.13)
Here n is the number of particles which lie within the cut-off sphere with center at
xM .
C.3 Solution approach - FFT/finite differences
For problems involving periodic boundary conditions, Fourier techniques is adopted
to guarantee the periodicity of global velocity ug(x). Using Fast Fourier Transforms
(FFT) eqs. C.2 and C.3 can be written as
ikxηp− G
ηδ(k=0) +
(
k2 − ∂2
∂y2
)
ug =1ηρg,x,
1η∂p∂y
+(
k2 − ∂2
∂y2
)
vg =1ηρg,y,
ikzηp+
(
k2 − ∂2
∂y2
)
wg =1ηρg,z,
ikxu+ ∂v∂y
+ ikzw = 0,
(C.14)
with the boundary conditions as:
ug|y=wall = −ul|y=wall + u|y=wall,
vg|y=wall = −vl|y=wall + v|y=wall,
wg|y=wall = −wl|y=wall + w|y=wall,
(C.15)
where k =√
k2x + k2
z , and kx and kz denote the wave numbers in the x- and z-
154
directions respectively and superscript ∧ refers to the Fourier transforms as:
ug,i(y) =∑
x
∑
z
ug,ie−ikxxe−ikzz, (C.16)
p(y) =∑
x
∑
z
pe−ikxxe−ikzz. (C.17)
C.3.1 Treatment of periodic boundary condition:
GGEM was originally conceived as an Ewald-like approach to solving Stokes flow
problems in non-periodic domains. In this situation, we imagine that the total velocity
u = ul + ug satisfies appropriate boundary conditions, such as no-slip. In this case
the solution to the global problem satisfies the boundary condition
ug|wall = −ul|wall + u|wall. (C.18)
In the case where one or more of the directions for the flow is periodic, there is an
important subtlety to the implementation of GGEM that needs to be understood. For
simplicity let us just consider the case where there is one periodic direction, x (e.g.
the geometry is a tube). In this case the boundary condition on the total velocity is
simply
u(0, y, z) = u(Bx, y, z), (C.19)
where Bx is the periodic box length and thus that
ul(0, y, z) + ug(0, y, z) = ul(Bx, y, z) + ug(Bx, y, z). (C.20)
155
Rearranging we have then that
ug(0, y, z)− ug(Bx, y, z) = ul(Bx, y, z)− ul(0, y, z). (C.21)
This result is exact.
Now let’s turn to the exact form of the local solution in the case of the periodic
domain. The exact local velocity field (again treating only the x-direction as periodic)
is
ul(x) =
∞∑
j=−∞
N∑
i=1
Gl(x− (xi + jBxex)) · fi. (C.22)
This velocity field satisfies the condition
ul(Bx, y, z)− ul(0, y, z) = 0, (C.23)
allowing the use of the periodic boundary condition for the global velocity (which we
use):
ug(Bx, y, z)− ug(0, y, z) = 0, (C.24)
In practice, however, the local velocity is treated as:
ul(x) =N∑
i=1
Gl(x− x∗i ) · fi, (C.25)
where x∗i is the image of particle i which is closest to x (minimum image convention).
Eq. C.25 does not exactly satisfy eq. C.23 which in turn leads to a small error in
the global solution if we impose eq. C.24 as the boundary condition for the global
problem.
156
C.3.2 Proper treatment of k = 0:
For k = 0 (kx = 0 and kz = 0) eq. C.14 becomes:
−Gη+(
− ∂2
∂y2
)
ug(k = 0) = 1ηρx(k = 0),
1η∂p(k=0)
∂y+(
− ∂2
∂y2
)
vg(k = 0) = 1ηρy(k = 0),
(
− ∂2
∂y2
)
wg(k = 0) = 1ηρz(k = 0),
∂v(k=0)∂y
= 0,
(C.26)
with
ug(k = 0)|y=wall = −ul(k = 0)|y=wall + u(k = 0)|y=wall,
vg(k = 0)|y=wall = −vl(k = 0)|y=wall + v(k = 0)|y=wall,
wg(k = 0)|y=wall = −wl(k = 0)|y=wall + w(k = 0)|y=wall,
(C.27)
as the boundary condition.
Divergence-free condition of total velocity gives:
∫
V
∇ · u dv = 0, (C.28)
or∫
V
∇ · ul dv +
∫
V
∇ · ug dv = 0, (C.29)
where V is the total volume of the simulation domain. The local velocity, by definition,
is also divergence free:∫
V
∇ · ul dv = 0. (C.30)
157
Eqs. C.29 and C.30 result in making the global velocity divergence free. Therefore,
∫
V
∇ · ug dv = 0. (C.31)
Using Guass-Divergence theorem on equations C.29, C.30 and C.31 gives:
∫
S
n · u dS = 0, (C.32)∫
S
n · ul dS = 0, (C.33)∫
S
n · ug dS = 0, (C.34)
where n is a unit vector normal to the surface, S. Since the domain under considera-
tion is periodic in x− and z− directions, the surface integral of u and w components
of velocity is zero. Eqs. C.32, C.33 and C.34 reduces to
∫
top
v dS −∫
bottom
v dS = 0, (C.35)
∫
top
vl dS −∫
bottom
vl dS = 0, (C.36)
∫
top
vg dS −∫
bottom
vg dS = 0, (C.37)
(C.38)
where top refers to top wall (y = +By/2) and bottom refers to bottom wall (y =
−By/2). Since the walls are assumed to be impenetrable, there is no net flux across
the walls. This condition give rise to
∫
top
v dS =
∫
bottom
v dS = 0, (C.39)
158
or,∫
top
vl dS +
∫
top
vg dS =
∫
bottom
vl dS +
∫
bottom
vg dS = 0. (C.40)
The local force density, ρl (x), decays rapidly to zero over a distance proportional
to α−1, where α is the screening parameter. The calculation of local velocity field
begins by identifying interactions located within a sphere of fixed radius 4/α. Any
interactions outside this cut-off distance is ignored. The cut off distance for the local
calculation is always set such that no point has interactions that intercepts both walls
i.e.
4/α < By/2. (C.41)
Therefore, it can be assumed that the local velocity, vl, due to a point force is negligible
at at least on of the walls. For example consider a point near the top wall. Its
contribution to the bottom wall will be negligible. Therefore
∫
bottom
vl dS = 0. (C.42)
Comparing eqs. C.36 and C.42:
∫
top
vl dS = 0. (C.43)
Substituting eqs. C.42 and C.43 in eq. C.40 gives
∫
top
vg dS =
∫
bottom
vg dS = 0. (C.44)
The same analogy can be applied for a point close to the bottom wall to obtain eq.
159
C.44 and as such eq. C.44 can be obtained for any point inside the domain. Eq. C.44
can be represented in Fourier space as
vg(k = 0)|y=wall = 0. (C.45)
Taking a closer look at equation(C.26)gives
∂v(k = 0)
∂y= 0. (C.46)
This equation, when coupled with the boundary conditions for vg obtained at the
walls (eq. C.45) give rise to
vg(k = 0) = 0. (C.47)
Using the above result, eq. C.26 is then reduced to a system of decoupled equations:
−Gη+(
− ∂2
∂y2
)
ug(k = 0) = 1ηρx(k = 0),
1η∂p(k=0)
∂y= 1
ηρy(k = 0),
(
− ∂2
∂y2
)
wg(k = 0) = 1ηρz(k = 0),
vg = 0,
(C.48)
with
ug(k = 0)|y=wall = −ul(k = 0)|y=wall + u(k = 0)|y=wall,
wg(k = 0)|y=wall = −wl(k = 0)|y=wall + w(k = 0)|y=wall,(C.49)
The above system of decoupled equations can be solved individually to get the cor-
responding values at kx = 0 and kz = 0 mode.
The velocities and pressure in physical space can be evaluated by taking the inverse
160
Fourier transforms as:
ug,i(x) =∑
kx
∑
kz
ug ,i(y)eikx xe ikz z , (C.50)
p(x) =∑
kx
∑
kz
p(y)e ikx xe ikz z . (C.51)
C.3.3 Finite Difference:
Finite difference scheme can be used to solve for the wall-normal, y−, direction.
Staggered grid can be used to solve for the above problem. On a staggered grid
the scalar variables (pressure, force density) are stored in the cell centers of the
control volumes, whereas the velocity variables are located at the cell faces. This is
different from a collocated grid arrangement, where all variables are stored in the same
positions. For every kx and kz mode, there are My velocity variables and (My − 1)
pressure variables. HereMy refers to the mesh in y− direction. Using finite difference,
eqs. C.14 and C.15 are reduced to
(
−1∆2
y
)
ug(yi−1) +(
k2 + 2∆2
y
)
ug(yi) +(
−1∆2
y
)
ug(yi+1)
+(
ikx2η
)
p(yi−1) +(
ikx2η
)
p(yi) =1ηρg,x(yi) · · · ∀ i = 2 . . .My − 1,
(
−1∆2
y
)
vg(yi−1) +(
k2 + 2∆2
y
)
vg(yi) +(
−1η∆2
y
)
vg(yi+1)
+(
−1η∆y
)
p(yi−1) +(
1∆y
)
p(yi) =1ηρg,y(yi) · · · ∀ i = 2 . . .My − 1,
(
−1∆2
y
)
wg(yi−1) +(
k2 + 2∆2
y
)
wg(yi) +(
−1∆2
y
)
wg(yi+1)
+(
ikz2η
)
p(yi−1) +(
ikz2η
)
p(yi) =1ηρg,z(yi) · · · ∀ i = 2 . . .My − 1,
(
ikx2
)
ug(yi) +(
ikx2
)
ug(yi+1) +(
−1∆y
)
vg(yi) +(
1∆y
)
vg(yi+1)
+(
ikz2
)
wg(yi) +(
ikz2
)
wg(yi+1) = 0 · · · ∀ i = 1 . . .My − 1,
(C.52)
161
with
ug(y1) = −ul|y=bottomwall + u|y=bottomwall, ug(yMy) = −ul|y=top wall + u|y=top wall,
vg(y1) = −vl|y=bottomwall + v|y=bottomwall, vg(yMy) = −vl|y=top wall + v|y=top wall,
wg(y1) = −wl|y=bottomwall + w|y=bottomwall, wg(yMy) = −wl|y=top wall + w|y=top wall,
(C.53)
as the boundary conditions. Here ∆y is the mesh resolution in y−direction. The
above equations can then be solved as a system of linear equations:
A ·U = Q, (C.54)
162
where A is a coefficient matrix of size (3My − 1)× (3My − 1), U,and Q are vectors
of size (3My − 1). A, U and Q are defined as
A(i, i− 1) = A(i, i+ 1) = −1∆2
y
A(i, i) = k2 + 2∆2
y
A(i, 3My + i− 1) = A(i, 3My + i) = ikx2η
Q(i) = ρxη
A(My + i,My + i− 1) = A(My + i,My + i+ 1) = −1∆2
y
A(My + i,My + i) = k2 + 2∆2
y
−A(My + i, 3My + i− 1) = A(2My + i, 3My + i) = 1∆yη
Q(My + i) = ρyη
A(2My + i, 2My + i− 1) = A(2My + i, 2My + i+ 1) = −1∆2
y
A(2My + i, 2My + i) = k2 + 2∆2
y
A(2My + i, 3My + i− 1) = A(2My + i, 3My + i) = ikz2η
Q(2My + i) = ρzη
∀ i = 2 . . .My − 1
A(3My + i, i) = A(3My + i, i+ 1) = ikx2
−A(3My + i,My + i) = A(3My + i,My + i+ 1) = 1∆y
A(3My + i, 2My + i) = A(3My + i, 2My + i+ 1) = ikz2
∀ i = 1 . . .My − 1
163
A(1, 1) = A(My,My) = 1
Q(1) = ug|y=bottomwall, Q(My) = ug|y=top wall
A(My + 1,My + 1) = A(2My, 2My) = 1
Q(My + 1) = vg|y=bottomwall, Q(2My) = vg|y=top wall
A(2My + i, 2My + 1) = A(3My, 3My) = 1
Q(2My + 1) = wg|y=bottomwall, Q(3My) = wg|y=top wall
Boundary Conditions
U =
ug(y1)
...
ug(yMy)
vg(y1)
...
vg(yMy)
wg(y1)
...
wg(yMy)
p(y1)
...
p(yMy−1)
where yi are the mesh points in y− direction. The above system of linear eqs. C.54
can be solved using any linear algebra package for all kx and kz modes except for
k = 0 (kx = 0 and kz = 0) mode.
164
Using finite difference for k = 0 modes eqs. C.48 and C.49 can be represented as
(
−1∆2
y
)
ug(yi−1, k = 0) +(
2∆2
y
)
ug(yi, k = 0) +(
−1∆2
y
)
ug(yi+1, k = 0)
= 1ηρg,x(yi, k = 0) + G
η· · · ∀ i = 2 . . .My − 1,
(
−1η∆y
)
p(yi−1, k = 0) +(
1∆y
)
p(yi, k = 0)
= 1ηρg,y(yi, k = 0) · · · ∀ i = 2 . . .My − 1,
(
−1∆2
y
)
wg(yi−1, k = 0) +(
2∆2
y
)
wg(yi, k = 0) +(
−1∆2
y
)
wg(yi+1, k = 0)
= 1ηρg,z(yi, k = 0) · · · ∀ i = 2 . . .My − 1,
(C.55)
with
ug(y1, k = 0) = −ul(k = 0)|y=bottomwall + u|y=bottomwall,
ug(yMy,k=0) = −ul(k = 0)|y=top wall + u(k = 0)|y=top wall,
wg(y1, k = 0) = −wl(k = 0)|y=bottomwall + w(k = 0)|y=bottomwall,
wg(yMy,k=0) = −wl(k = 0)|y=topwall + w(k = 0)|y=topwall,
(C.56)
as the boundary conditions. The above system of equations can then be solved for
the velocity fields individually as systems of linear equations:
A0 ·Ux = Qx,
A0 ·Uz = Qz,(C.57)
where A0 is a coefficient matrix of size (My)× (My), Ux, Uz,Qx and Qz are vectors
165
of size (My). They are defined as follows:
A0(i, i− 1) = A0(i, i+ 1) = −1∆2
y
A0(i, i) =2∆2
y
Qx(i) =ρxη+ G
η
Qz(i) =ρzη
∀ i = 2 . . .My − 1
A0(1, 1) = A0(My,My) = 1
Qx(1) = ug(k = 0)|y=bottomwall, Qx(My) = ug(k = 0)|y=top wall
Qz(1) = wg(k = 0)|y=bottomwall, Qz(My) = wg(k = 0)|y=topwall
Boundary Conditions
Ux =
ug(y1, k = 0)
...
ug(yMy , k = 0)
, Uz =
wg(y1, k = 0)
...
wg(yMy , k = 0)
.
C.4 Interpolation: getting ug at the particle posi-
tions
Many interpolation techniques can be used to get the value of global velocity at any
desired lactation. The interpolation used in this method is quadratic interpolation
using 3 points.
166
C.4.1 Lagrange interpolation of degree 2 :
Given a set of 3 data points
(x1, f(x1)), (x2, f(x2)), (x3, f(x3)) (C.58)
where no two xi are the same, the Lagrange form of interpolating polynomial of degree
2 is a linear combination
L2(x) = l1(x)f(x1) + l2(x)f(x2) + l3(x)f(x3) (C.59)
of Lagrange polynomial, li, of degree 2
li(x) =
3∏
j=1,j 6=i
(x− xj)
(xi − xj). (C.60)
li is associated with the interpolating point xi in the sense:
li(xi) = 1 ∀ i 6= j,
= 0 ∀ i = j. (C.61)
The Lagrange form of the interpolation polynomial shows the linear character of
polynomial interpolation and the uniqueness of the interpolation polynomial.
C.4.2 Interpolation of global velocity:
Let x = {x, y, z} be the point where the value of global velocity, ug(x, y, z) =
{ug(x, y, z), vg(x, y, z), wg(x, y, z)}, needs to be interpolated. Let xi,j,k = {xi, yj, zk}
167
be the nearest mesh point and ug(xi, yj, zk) = {ug(xi, yj, zk), vg(xi, yj, zk), wg(xi, yj, zk)}
be the corresponding global velocity at that mesh point. The value of ug(x, y, z) are
obtained by doing one-dimensional interpolation sequentially in all the three direc-
tions using eq. C.59. As the interpolation in each direction requires 3 points, a total
of 3× 3× 3 = 27 mesh points is required to interpolate at one point in three dimen-
sion. The order of interpolation in any direction does not change the result. The
interpolation of ug(xi, yj, zk) in y− direction gives intermediate velocities as :
ug(xi, y, zk) = ly,1(y)ug(xi, yj−1, zk) + ly,2(y)ug(xi, yj, zk) + ly,3(y)ug(xi, yj+1, zk),
(C.62)
where ly,i is a lagrange polynomial for y− direction interpolation
ly,1 =y−yj
yj−1−yj
y−yj+1
yj−1−yj+1,
ly,2 =y−yj−1
yj−yj−1
y−yj+1
yj−yj+1,
ly,3 =y−yj−1
yj+1−yj−1
y−yjyj+1−yj
,
(C.63)
and ug(xi, y, zk) is the interpolation of velocity in y−direction. The velocity, ug(xi, y, zk),
is then interpolated in z−direction to obtain
ug(xi, y, z) = lz,1(z)ug(xi, y, zk−1) + lz,2(z)ug(xi, y, zk) + lz,3(z)ug(xi, y, zk+1), (C.64)
where lz,i is a lagrange polynomial for z− direction interpolation
lz,1 =z−zk
zk−1−zk
z−zk+1
zk−1−zk+1,
lz,2 =z−zk−1
zk−zk−1
z−zk+1
zk−zk+1,
lz,3 =z−zk−1
zk+1−zk−1
z−zkzk+1−zk
,
(C.65)
168
and ug(xi, y, z) is the interpolation of velocity in z−direction. The velocity, ug(xi, y, z),
is then interpolated in x−direction to obtain velocity at particle position, x, as
ug(x, y, z) = lx,1(x)ug(xi−1, y, z) + lx,2(x)ug(xi, y, z) + lx,3(x)ug(xi+1, y, z), (C.66)
where lx,i is a lagrange polynomial for x− direction interpolation
lx,1 =x−xi
xi−1−xi
x−xi+1
xi−1−xi+1,
lx,2 =x−xi−1
xi−xi−1
x−xi+1
xi−xi+1,
lx,3 =x−xi−1
xi+1−xi−1
x−xi
xi+1−xi.
(C.67)
Eqs. C.62, C.64 and C.66 can be written in a compact form as
ug(x, y, z) =3
∑
p=1
3∑
q=1
3∑
r=1
lx,p ly,q lz,r ug(xi+p−2, yj+q−2, zk+r−2). (C.68)
169
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