7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
The Effect of Particle Deformation on the Rheology of Noncolloidal Suspensions
J. R. Clausen*, D. A. Reasor*, C. K. Aidun*
G. W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA
clausen@gatech.edu, daniel.reasor@gatech.edu, and cyrus.aidun@me.gatech.edu
Keywords: LatticeBoltzmann, suspension, deformable particle
Abstract
In this paper we describe our group's recent interest in simulating and modeling suspensions of deformable par
ticles. We simulated capsules, in which an elastic membrane surrounded a viscous internal fluid. These capsules
demonstrated a tanktreading dynamical behavior, in which the capsules maintained a steady orientation and the
membrane rotated about this ellipsoidal shape. In contrast to recent papers that concentrated on describing the rich
dynamics of vesicles and capsules, we concentrated on fully describing the impact of these capsules on the rheology
of capsule suspensions. Accordingly, we extensively probed the dilutelimit influence that a capsule had on the overall
suspension rheology, with a focus on accurately describing the normal stresses. We accurately simulated and modeled
the particle phase pressure, a quantity which must be known to accurately use a suspension balance model of particle
migration. Preliminary results of dense capsule suspensions were reported.
Introduction
Many industrial, commercial, and biological systems de
pend on the accurate understanding and modeling of sus
pension rheology. Often, these particles are treated as
idealized rigid shapes; however, in certain cases defor
mation of the particle phase becomes an important pa
rameter. Two practical examples of interest are fiber sus
pensions, in which the fibers deform appreciably from
straight slender bodies, and blood flow, in which de
formable red blood cells play a large role in the rheology.
The former example is important in the manufacture of
paper products, in which fibers composed of cellulose
have some amount of deformation. The latter example is
important in accurately modeling many biological situ
ations including predicting locations of disease, design
ing better artificial heart valves, and understanding the
process of clot formation during atherosclerosis.
This paper will focus on one of the simplest de
formable particles, the initially spherical microcapsule.
These capsules are composed of an elastic membrane
that surrounds an incompressible fluid. These particles
are large enough that Brownian effects are negligible,
yet small enough that inertial effects are not important.
As such, the governing equations for the motion of the
fluid are the Stokes flow equations,
Vp V2u (1)
where p is the fluid viscosity, p is the fluid pressure, and
u is the fluid velocity. The fluid phase is simulated using
the latticeBoltzmann (LB) method, which is a highly
scalable and efficient method for the simulation of par
ticle suspensions (Ladd 1994a,b; Aidun, Lu, and Ding
1998). The LB method serves as a NavierStokes solver,
and we perform the simulations in the lowReynolds
number limit.
Since both bending stiffness and inertia of the elas
tic membrane are minimal, dynamics are governed by
Cauchy's equation,
V, o 0,
where Vs is the surface divergence, or is the stress ten
sor, and Hooke's law is used to related stress and strain.
The surface divergence is defined as
Vs (I nn) V,
where I is the identity matrix, and n is the surface nor
mal. To simulate the elastic behavior of the membrane, a
linear finiteelement (FE) method is used. Owing to the
linearity of the method, rapid determination of capsule
dynamics can be recovered. The relevant nondimen
sional parameter governing the relative contribution of
viscous to elastic forces is the capillary number, defined
based on the Young's modulus, E, of the membrane as
G wEa
E'
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
where j is the shear rate, and a is the particle radius.
In this paper we study the dynamics of these elastic
capsules in the tank treading regime, in which the cap
sule deforms into a steady ellipsoid with a preferential
orientation, and the membrane rotates about this ellip
soidal shape. We make the distinction between capsules
and vesicles, in which the latter strictly conserve mem
brane surface area. Efforts are focused on relating the
microstructure of the capsule to the overall influence on
suspension rheology, and we discuss results presented
in Clausen and Aidun (2010). Comparisons are made
with a number of analytical theories and competing nu
merical methods. In addition to the typical theological
quantities of viscosity and first and second normal stress
differences, we evaluate the particle pressure, i.e., the
trace of the particle phase pressure. The inclusion of
this term requires several corrections to the LB method
owing to its pseudo compressibility. To provide some
analytical validation to the particlepressure calculation,
we extend the results of Roscoe (1967) to include the
pressure disturbance terms found in Jeffery (1922).
Methodology
The method developed to investigate suspensions of
deformable particles is a hybrid method that combines a
latticeBoltzmann (LB) fluid with a finiteelement (FE)
particle. The method will be briefly outlined below but
details can be found in MacMeccan, Clausen, Neitzel,
and Aidun (2009). This method allows the simulation of
capsules and elastic particles suspended in a Newtonian
fluid, and is capable of scaling on largescale supercom
puters (Clausen, Reasor, and Aidun 2010).
LatticeBoltzmann Method The LB method was
initially developed as an averaging procedure to im
prove the stability of lattice gas automata (Higuera and
Jimenez 1989); however, the method is formally a dis
cretization of the Boltzmann equation using a discrete
velocity set ej. The LB method relies on the collision
and propagation of the particle distribution function, f,,
where i denotes the lattice direction and i 1... Q.
The LB equation can be written as
fi(r + ei,t + 1)
fi(q) r,t)) (4)
where r is a position vector, fQ is the equilibrium dis
tribution, and T is the relaxation time of the collision
operation. The equilibrium distribution is a lowMach
number expansion of the MaxwellBoltzmann distribu
tion and is written as
f(eq)
Wip [1 + (e u) + i () 24 ( U)
U2 ] (5)
where w, are the lattice weight vectors, 1/3, 1/18, and
1/36 for the rest, nondiagonal, and diagonal directions,
respectively, and c, = 1/3 is the pseudo sound speed.
This relaxation towards the equilibrium distribution re
covers the diffusion of momentum in the longtime limit,
with the macroscopic viscosity recovered according to
v (2T 1)/6. Macroscopic properties are recov
ered via moments of the equilibrium distribution func
tion, shown as
fiej (r, t)
f 7(r, t)e
fie(r, t)eie,
i^M
p
pu
cplI + puu
Readers interested in a more thorough overview of
the LB method including its extensive capabilities are
directed to the review articles by Chen and Doolen
(1998) and Aidun and Clausen (2010).
FiniteElement Method The linear FE method cho
sen to simulate the elastic membrane provides an ef
ficient and flexible method of recovering capsule dy
namics. We use linear triangular shell elements, with
the meshing and modeling performed in the commer
cial software package ANSYS, and we import the re
sultant FE matrices into the LB/FE hybrid method. A
bodyfixed coordinate system is used to track the aver
age translation and angular rotation of the particle, thus
rendering the FE matrices invariant. MacMeccan et al.
(2009) describe the virtual work formulation and shape
functions used in detail.
The FE equation can be written as
MX + Cx + Kx= F, (7)
where M and K are the mass and stiffness obtained
using ANSYS. The damping matrix, C is calculated
using Rayleigh damping. We integrate (7) using
Newmark's method, which after some manipulation
and owing to the constant FE matrices, allows the time
evolution of the particle to be recast into a single matrix
multiplication (MacMeccan et al. 2009).
fi(r,t) 1(fi(r,t)
T
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Coupling and Corrections The LB method fluid is
coupled to the FE method solid through the standard
bounceback boundary condition. For arbitrary bound
aries this method has been shown to be firstorder accu
rate (Ginzbourg and Adler 1994). This method is predi
cated on the calculation of links that traverse the capsule
membrane while connecting neighboring lattice nodes.
The fluid distribution propagating towards the capsule
membrane in the direction i' are bounced in the opposite
direction (i) with a corresponding adjustment to the mo
mentum to account for the velocity of the boundary, ub.
The bounceback operation can be written as
fi(r, t + 1) fi(r,t+) + 6pwiUb ei, (8)
where t denotes the distribution postcollision, but pre
propagation. Likewise, there is a corresponding force on
the FE membrane, which is given by
F(b) 2ei [fi, (r, t+) + 3pwiUb ei], (9)
which must be interpolated to the nearest FE nodes (for
details, see MacMeccan et al. 2009). The bounceback
operation shown in (8) and (9) is done twice for each
link: once for the LB node external to the capsule, and
once for the LB node internal to the capsule.
Two corrections to the LB method are needed to accu
rately calculate the rheology of deformable particle sus
pensions. First, a Galilean error exists in the bounce
back method for determining the external surface forces
on a particle. As discussed in Clausen and Aidun (2009),
the nonlinear terms in the equilibrium distribution cre
ate an error in the bounceback force that scales as u2.
This error does not manifest in capsule dynamics, since
the internal and external bounceback operations create
offsetting errors; however, the use of just the external
surface force in rheology calculations can create large
errors in the flowdirection normal stresses. The cor
rection requires bouncing a virtual fluid node internal to
the particle, in which the distribution has been set to an
equilibrium distribution traveling with the velocity of the
particle boundary (Clausen and Aidun 2009).
Also, the pseudo compressibility of the LB method
must be accounted for when the particles deform. Ide
ally, the presence of an incompressible Stokes fluid (1)
internal to the particle would create an incompressible
particle; however, the LB method is weakly compress
ible with an equation of state p pcl. Consequently,
the pressure (density) internal to the particle increases
as the particle deforms to approximately maintain par
ticle incompressibility; however, the volume of the par
ticle does change slightly. Since the mass internal and
external to the capsule is conserved at short time scales,
when the capsule's volume decreases slightly, the pres
sure external to the particle decreases slightly, which can
create errors in the normal stresses reported. Further
more, as the particle translates, LB nodes are covered
and uncovered, which breaks mass conservation and al
lows the highdensity internal fluid to mix with the ex
ternal fluid, which causes the pressure to gradually re
lax, and the particle volume is no longer conserved. As
detailed in Clausen and Aidun (2010), the internal and
external densities (and hence pressures) are normalized
to a mean value of unity through an adjustment over all
fluid nodes. This correction does not preclude the for
mation of locally high pressures, but it does eliminate
the influence of particle deformation on the average fluid
pressure. A byproduct of this normalization is the lack
of an increased pressure internal to the particle; thus,
particle incompressibility must be enforced through an
artificial pressure term introduced to the boundary force
(9), shown as
F = 2e, [ (r, t ) + fA + 3pw Ub e] ,
(10)
where f is a static pressure adjustment defined as
W Wo 1P VO
po is the initial fluid density, Vo is the initial particle
volume, and Vp is the current particle volume (Clausen
and Aidun 2010).
Suspension rheology and microstructure
Analyzing the behavior of capsule suspensions re
quires investigating the suspension configuration or mi
crostructure, which occurs on the particle length scale,
and relating this microstructure to the largescale rheo
logical behavior of these flows. The microstructure of
these suspensions depends both on the singlebody dis
tribution function, i.e., the microstructure of an individ
ual particle, and the overall configurational microstruc
ture that defines the probabilistic relationship between
particle locations. This study focuses on fully describing
the effect of the singlebody distribution function, which
for an isolated capsule can be quantified by the orien
tation angle, 0, and the Taylor deformation parameter,
D,,, on the overall suspension rheology. In the tank
treading regime, the capsule deforms to an ellipsoidal
shape; the orientation angle is defined as the inclination
of the major axis of this ellipsoid with respect to the flow
direction, and D,, (L 1)/(L + 1), where L and I
are the major and minor axes in the xyplane. Figure 1
shows a snapshot of an isolated particle simulation with
the relevant microstructure parameters annotated, where
the flow direction is x and the sheargradient direction
is y. Details on the methods used to calculate the mi
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
045
04 
0
035
03
0 25
02 
015 
0 1 BarthesBlesel (1980) 
005 IFEA 2.0 
0 IrEA = 1.5
0 002 004 006 008 01 012
G
(a)
Figure 1: An FE capsule that has deformed into an el
lipsoidal shape with a steady orientation (0)
and shape (quantified by D1y).
crostructure can be found in Clausen and Aidun (2010).
Analytical analysis of capsule and vesicle dynam
ics and rheology is restricted to two cases: the near
sphere limit of small deformation (BarthesBiesel 1980;
Misbah 2006; Vlahovska and Gracia 2007), or the rhe
ological influence based on a prescribed particle mi
crostructure (Roscoe 1967; Keller and Skalak 1982).
Much numerical work exists describing capsule dynam
ics (Eggleton and Popel 1998; Pozrikidis 1993; Ramanu
jan and Pozrikidis 1998); however, a detailed analysis of
the influence of capsules on rheology is lacking, espe
cially for normal stresses, as discussed below. In Fig
ure 2(a and b) we show the microstructure as calcu
lated through the LB/FE simulation method (Clausen
and Aidun 2010) as compared with analytical small
deformation theory and the numerical results of Ra
manujan and Pozrikidis (1998). The simulations are per
formed in wallbounded shear, with a particle of radius
10, with a domain size of 160x 160x 160. The particle
Reynolds number, Rep, is 0.0375.
Good agreement is seen between the microstructure
reported using other numerical methods and the LB/FE
hybrid method. As the elasticity parameter G increases
in magnitude, the capsule's deformation increases, and it
progressively aligns with the flow direction. The pertur
bation results of BarthesBiesel (1980) are accurate for
the Taylor deformation parameter in the limit of small
deformation (G+0); however, prediction for the orien
tation angle is very poor. The vesicle results of Vla
hovska and Gracia (2007) do not represent a closed
form solution to capsule dynamics because Dy has
been prescribed based on the LB/FE simulation results.
The agreement in orientation angle is good, which sug
gests that the rheology results reported in Vlahovska and
0.26
Bar
024 Ramanujan
Vlahovsk
0.22 ........
0.2
0.2 o +....
0.18
0.16
0.16
0.14
0.1
0 0.02 0.04 0.06
G
0.08 0.1 0.12
(b)
Figure 2: Timeaveraged results for the (a) Taylor de
formation parameter and (b) orientation angle
for an isolated spherical capsule. Figure from
Clausen and Aidun (2010).
Gracia (2007) may serve as a useful comparison to our
simulation results. In Figure 2, two different levels of
discretization are reported: IFEA=1.5 and 1FEA=2.0,
where 1FEA is the average length for an FE. The re
sults suggest that the simple linear method used in the
LB/FE hybrid accurately recovers capsule dynamics in
the small to moderate deformation levels seen here. Fig
ure 3 shows model ellipses corresponding to several cap
illary numbers.
Analysis of the rheology relies on a twophase aver
aging approach detailed by Batchelor (1970), where the
suspension stress (I) is decoupled into fluid and particle
stresses, shown using f and p superscripts for the fluid
and particle phases as
(E) (IZ) +(), (11)
where angled brackets denote ensemble averages. Fol
lowing Batchelor (1970), ensemble averages over both
fluid and particle phases are approximated by volume in
L1
Dy L + l
y
t) x
r
r
r
r
Ifs
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
G 0.0
 = 0.007
 G 0.03
................. G = 0.1
1
0.9
0.8
0.7
y
L^
>x
Figure 3: Outlines of model ellipses matching D,, and
0 from simulation results. Note the progres
sively larger deformation and alignment with
the flow (x) direction.
tegrals, which is appropriate since a homogeneous distri
bution of stress exists in linear shear. By using the con
stitutive relationship for the stress in a Newtonian fluid,
the volumeaveraged stress in the fluid can be written as
(I) (pl) + 2/(E) + (EP), (12)
where I is the identity tensor, E is the rate of strain ten
sor, and
(SP) = f ({ nr + ro n)
p (un + nu)} dA. (13)
The surface integral term is called the stresslet (denoted
S), n is a surface normal vector, and V is the domain
volume. The summation is over all particles suspended
in the domain.
Dropping the angled brackets and assuming all quan
tities are averages, relevant theological parameters can
be calculated via (1113). For example, the relative vis
cosity of the suspension in simple shear is defined ac
cording to
/Peff E12
Pr = (14)
where peff is the effective or measured viscosity of the
suspension, and j is the shear rate. The first and second
normal stresses are defined as
N1 E 11 E22, (15)
and
N2 = 22 E33. (16)
The particle pressure is also of interest. Defined as
1
IP, 3tr(E), (17)
0
0 0.02 0.04 0.06 0.08
0.1 0.12
G
Figure 4: The the timeaveraged behavior of the shear
component of the particle stress for an iso
lated capsule in shear. Figure from Clausen
and Aidun (2010).
the particle pressure exists as the isotropic stress distur
bance due to the particle phase. Historically, the particle
pressure was neglected owing to the incompressibility of
the fluid and particle phases; however, the incompress
ibility of the entire suspension only implies that the sus
pension stress is divergence free, i.e.,
V I: 0,
which does not preclude the formation of gradients in
the stresses of the individual phases. Instead, a balance
is formed such that
V f + V. E = 0,
and these gradients create a movement of the two phases,
which is the driving force behind particle migration in
the suspension balance model (Nott and Brady 1994;
Morris and Boulay 1999). Accurate reconstruction of
IP requires considering the isotropic pressure distur
bance due to the particles, which our simulation method
is capable of resolving.
The dilutelimit shearthinning behavior of de
formable particle suspensions can be seen in Fig
ure 4. The particle stress has been normalized by
the shear component of the particle stress for a rigid
sphere (E1;=>mi), with normalized values denoted
by starred quantities. At large deformations (G=0.1)
the particle stresslet shows a nearly 20% decrease from
the rigidsphere case. Vesicle theory using microstruc
ture determined from our simulations displays exagger
ated shearthinning behavior. The results from Roscoe
(1967) are solutions to the volumeaveraged equations
assuming a prescribed microstructure and a homoge
neous deformation of the solid phase. Again, shear thin
ning is exaggerated, which is due to the homogenous de
Roscoe (1967) ...+ .
Ramanujan & Pozlkdis (1998) o
Vlahovska & Gracia (2007) o
IFEA = 2.0
.IFEA = 1.5 K
.... o
4
O
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
14
12
1
08
S06
04
02
02
01
0
01
02
03
04
05
0
0 05
01
015
02
0 25
03
035
.....+
.....+....
4.o
....Roscoe(1967) +.
Ramanujan & Pozrdis (1998) o
Vlahovska & Gracia (2007) o
SIFEA= 2.0 
IFEA 1.5 *
0 002 004 006 008 01
0 002 004 006 008 01
0 002 004 006 008 01
G
(C)
Figure 5: Longtime behavior of the capsule rhe
showing (a) N1, (a) N2, and (c) nI. Th
end is the same for (a) and (b). Figure
Clausen and Aidun (2010).
tributable to the underprediction of viscous dissipation
(Clausen and Aidun 2010). Both N* and N2 results
from Ramanujan and Pozrikidis (1998) show excellent
agreement to our simulation results.
The particle phase pressure has been neglected in past
analysis and numerical simulation of deformable par
ticle suspensions. In order to gain some analytical in
012 sight, we extend the results of Roscoe (1967) using the
pressure disturbance terms readily available in the solu
tion of rigid ellipsoidal bodies suspended in flow (Jef
fery 1922). By using the linearity of Stokes flow, we are
able to calculate the particle pressure for an the case of
Sa homogeneously deforming ellipsoidal body as
a=l
where g, and Aa are integrals and coefficients calcu
012 lated from Jeffery (1922). The definition of the coeffi
cients and derivation of (18) can be found in Clausen and
Aidun (2010).
Simulation results are plotted and compared with the
above analytical expression in Figure 5c. In contrast
to the normal stress differences, excellent agreement is
seen for HI because the errors in viscous dissipation
cancel.
Dense suspensions
012 The analysis of more dense suspension rheology of de
formable capsules is in the formative stages. For rigid
spherical suspensions, the viscosity of dense suspen
sions is often modeled by semiempirical methods like
ology the KriegerDaugherty equation, shown as
eleg
from
formation representing a minimum in energy dissipation
(Clausen and Aidun 2010).
Normal stresses are also normalized by E1", as
shown in Figure 5(ac). As the particles become more
aligned in the flow direction, E11 dominates and the first
normal stress difference is positive. At the most de
formable case, the first normal stress difference is com
parable to the viscous stresses in the particle phase (Fig
ure 5a). Vesicle theory appears fairly accurate at describ
ing N1; however, the results of Roscoe (1967) slightly
overpredict the magnitude. The second normal stress
difference shows poorer agreement with analytical mod
els, with both Vlahovska and Gracia (2007) and Roscoe
(1967) overpredicting the magnitude of N2. For the re
sults of Roscoe (1967), this overprediction is directly at
Pr = (1
2 P1w
4m)
where p is the volume fraction, 4m is the maxi
mum packing fraction (0.630.64 for spheres), and [p]
is the intrinsic, or dilute limit viscosity (Krieger and
Dougherty 1959). Figure 6 shows the numerical re
sults for the relative viscosity at 40% volume frac
tion as a function of the elasticity parameter, where
Geff=rbPla/E. Also shown are the ideal predictions of
(19) and a crude prediction model which is based upon
scaling the intrinsic viscosity by the dilutelimit capsule
results shown in the previous section, i.e.
[/1]= [P]rigid(yP ). (20)
Although this prediction does show shearthinning be
havior, it does not predict the rapidity of the thinning
in the nearrigid limit. Instead, the shear thinning be
haves as the dilute case in Figure 4, in which the shear
. ...... ..
.. .... ........ .
+
Clausen & Aldun (2010) +.....
IFEA = 2.0 x
7IFEA=1.5 I
..........

6
5.5
5
4.5
4
3.5
3
2.5
2
0.01
simulation x
KriegerDaugherty (1959) o
prediction *
o0
0 0.01 0.02 0.03 0.04
Geff
Figure 6: Shearthinning behavior in a suspension of
capsules at 40% volume fraciton showing the
relative viscosity as a function of the elasticity
parameter, G. Also plotted are the semiempir
ical formula of Krieger and Dougherty (1959)
and a prediction based on adjusting the intrin
sic viscosity in the KriegerDaugherty equa
tion.
thinning is less pronounced at small elasticities. We can
only speculate that particleparticle interactions become
an important factor in describing this behavior. Unfor
tunately, the full rheology and microstructure of dense
capsule suspension results are not available at this time
and remain the focus of future work.
Conclusions
We have presented a thorough investigation into the mi
crostructure and dynamics of isolated initially spherical
capsules in the tanktreading regime, compared these re
sults with leading analytical and numerical results, and
have thoroughly described the impact of these capsules
on the overall rheology. We fully describe the normal
stresses in dilute rheology as discussed in Clausen and
Aidun (2010), and we anticipate that accurately describ
ing the particle pressure will be critical in applying the
suspension balance model (Nott and Brady 1994) to sus
pensions of deformable particles.
We have also presented preliminary results show
ing the shearthinning nature of dense suspensions of
deformable particles, and we are accurately working
on describing the complete rheology, including normal
stresses, of more dense suspensions. We also will at
tempt to describe the microstructure of dense suspen
sions, with consideration to describing the relative ef
fects of the singlebody microstructure, i.e., the Taylor
deformation parameter and capsule orientation, with the
suspension or configurational microstructure, i.e., the
relative position of particles with respect to one another.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
It is our hope that accurately describing these suspen
sions will allow inroads into modeling capsule migration
effects using the suspension balance model. Such mod
eling is critical to applying these results to industrial or
biologicalscale systems, in which the number of parti
cles may be well in excess of a million.
Acknowledgements
Simulations results used resources of TeraGrid, specif
ically the cluster Steele at Purdue. J.C. is funded by
the Institute of Paper Science and Technology; D.R. is
funded by ASEE through the SMART fellowship pro
gram.
References
C. K. Aidun and J. R. Clausen. The latticeBoltzmann
method for complex flows. Annu. Rev. Fluid Mech., 42
(1):43972, 2010.
C. K. Aidun, Y. Lu, and E. J. Ding. Direct analy
sis of particulate suspensions with inertia using the dis
crete Boltzmann equation. J. Fluid Mech., 373:287311,
1998.
D. BarthesBiesel. Motion of a spherical microcapsule
freely suspended in a linear shear flow. J. Fluid Mech.,
100(04):83153, 1980.
G. K. Batchelor. The stress system in a suspension of
forcefree particles. J. Fluid Mech., 41:54570, 1970.
S. Chen and G. D. Doolen. Lattice Boltzmann method
for fluid flows. Annu. Rev. Fluid Mech., 30(1):32964,
1998.
J. R. Clausen and C. K. Aidun. Galilean invariance in
the latticeBoltzmann method and its effect on the cal
culation of theological properties in suspensions. Int. J.
Multiphas. Flow, 35:30711, 2009.
J. R. Clausen and C. K. Aidun. Capsule dynamics and
rheology in shear flow: Particle pressure and normal
stress. submitted to Phys. Fluids, 2010.
J. R. Clausen, D. A. Reasor, and C. K. Aidun. Par
allel performance of a latticeBoltzmann/finite element
cellular blood flow solver on the IBM Blue Gene/P ar
chitecture. in press, Comput. Phys. Commun., DOI:
10.1016/j.cpc.2010.02.005, 2010.
C. D. Eggleton and A. S. Popel. Large deformation of
red blood cell ghosts in a simple shear flow. Phys. Flu
ids, 10(8):18341845, 1998.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
I. Ginzbourg and P.M. Adler. Boundary flow condition P. M. Vlahovska and R. S. Gracia. Dynamics of a vis
analysis for the 3dimensional lattice Boltzmann model. cous vesicle in linear flows. Phys. Rev. E, 75(1): 16313,
J. Phys. II, 4(2):191214, 1994. 2007.
F. J. Higuera and J. Jimenez. Boltzmann approach to
lattice gas simulations. Europhys. Lett., 9(7):66368,
1989.
G. B. Jeffery. The motion of ellipsoidal particles im
mersed in a viscous fluid. Proc. R. Soc. London, Ser. A,
102(715):16179, 1922.
S. R. Keller and R. Skalak. Motion of a tanktreading
ellipsoidal particle in a shear flow. J. Fluid Mech., 120:
2747, 1982.
I. M. Krieger and T. J. Dougherty. A mechanism for
nonNewtonian flow in suspensions of rigid spheres. J.
Rheol., 3(1):13752, 1959.
A. J. C. Ladd. Numerical simulations of particulate sus
pensions via a discretized Boltzmann equation. Part 1.
Theoretical foundation. J. Fluid Mech., 271:285309,
1994a.
A. J. C. Ladd. Numerical simulations of particulate sus
pensions via a discretized Boltzmann equation. Part 2.
Numerical results. J. Fluid Mech., 271:31139, 1994b.
R. M. MacMeccan, J. R. Clausen, G. P. Neitzel, and
C. K. Aidun. Simulating deformable particle sus
pensions using a coupled latticeBoltzmann and finite
element method. J. Fluid Mech., 618:1339, 2009.
C. Misbah. Vacillating breathing and tumbling of vesi
cles under shear flow. Phys. Rev. Lett., 96(2):028104,
2006.
J. F. Morris and F. Boulay. Curvilinear flows of non
colloidal suspensions: The role of normal stresses. J.
Rheol., 43(5):121337, 1999.
P. R. Nott and J. F. Brady. Pressuredriven suspension
flow: Simulation and theory. J. Fluid Mech., 275:157
99, 1994.
C. Pozrikidis. On the transient motion of ordered sus
pensions of liquid drops. J. Fluid Mech., 246:301320,
1993.
S. Ramanujan and C. Pozrikidis. Deformation of liquid
capsules enclosed by elastic membranes in simple shear
flow: large deformations and the effect of fluid viscosi
ties. J. Fluid Mech., 361:11743, 1998.
R. Roscoe. On the rheology of a suspension of viscoelas
tic spheres in a viscous liquid. J. Fluid Mech., 28(02):
27393, 1967.
