Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 4.7.2 - The Effect of Particle Deformation on the Rheology of Noncolloidal Suspensions
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 Material Information
Title: 4.7.2 - The Effect of Particle Deformation on the Rheology of Noncolloidal Suspensions Colloidal and Suspension Dynamics
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Clausen, J.R.
Reasor, D.A.
Aidun, C.K.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: Lattice-Boltzmann
suspension
deformable particle
 Notes
Abstract: In this paper we describe our group’s recent interest in simulating and modeling suspensions of deformable particles. We simulated capsules, in which an elastic membrane surrounded a viscous internal fluid. These capsules demonstrated a tank-treading 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 dilute-limit 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.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Volume ID: VID00116
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Resource Identifier: 472-Clausen-ICMF2010.pdf

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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: Lattice-Boltzmann, 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 tank-treading 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 dilute-limit 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 lattice-Boltzmann (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 Navier-Stokes solver,
and we perform the simulations in the low-Reynolds-
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 finite-element (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 particle-pressure 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
lattice-Boltzmann (LB) fluid with a finite-element (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 large-scale supercom-
puters (Clausen, Reasor, and Aidun 2010).

Lattice-Boltzmann 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 low-Mach-
number expansion of the Maxwell-Boltzmann 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, non-diagonal, 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 long-time 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).

Finite-Element 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
body-fixed 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
bounce-back boundary condition. For arbitrary bound-
aries this method has been shown to be first-order 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 bounce-back operation can be written as

fi(r, t + 1) fi(r,t+) + 6pwiUb ei, (8)

where t denotes the distribution post-collision, 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 bounce-back
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 bounce-back force that scales as u2.
This error does not manifest in capsule dynamics, since
the internal and external bounce-back operations create
offsetting errors; however, the use of just the external
surface force in rheology calculations can create large
errors in the flow-direction 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 high-density 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 large-scale rheo-
logical behavior of these flows. The microstructure of
these suspensions depends both on the single-body 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 single-body 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 xy-plane. Figure 1
shows a snapshot of an isolated particle simulation with
the relevant microstructure parameters annotated, where
the flow direction is x and the shear-gradient 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 Barthes-Blesel (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 (Barthes-Biesel 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 wall-bounded 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 Barthes-Biesel (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: Time-averaged 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 two-phase 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) -(I-Z) +(), (11)

where angled brackets denote ensemble averages. Fol-
lowing Batchelor (1970), ensemble averages over both
fluid and particle phases are approximated by volume in-


L-1
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 volume-averaged 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 (11-13). 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 time-averaged 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 dilute-limit shear-thinning 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 rigid-sphere case. Vesicle theory using microstruc-
ture determined from our simulations displays exagger-
ated shear-thinning behavior. The results from Roscoe
(1967) are solutions to the volume-averaged 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: Long-time 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 Krieger-Daugherty 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(a-c). 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 P1-w
4m)


where p is the volume fraction, 4m is the maxi-
mum packing fraction (0.63-0.64 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 dilute-limit capsule
results shown in the previous section, i.e.

[/1]= [P]rigid(yP ). (20)

Although this prediction does show shear-thinning be-
havior, it does not predict the rapidity of the thinning
in the near-rigid 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-
Krieger-Daugherty (1959) o
prediction *
o0








0 0.01 0.02 0.03 0.04


Geff

Figure 6: Shear-thinning 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 Krieger-Daugherty equa-
tion.


thinning is less pronounced at small elasticities. We can
only speculate that particle-particle 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 tank-treading 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 shear-thinning 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 single-body 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
biological-scale 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.


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