lattice boltzmann simulation of three-dimensional capsule

Scientia Iranica B (2015) 22(5), 1877{1890

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

Lattice Boltzmann simulation of three-dimensionalcapsule deformation in a shear ow with di�erentmembrane constitutive laws

Z. Hashemia, M. Rahnamaa;� and S. Jafarib

a. Department of Mechanical Engineering, Faculty of Engineering, Shahid Bahonar University of Kerman, Kerman, P.O. Box76169-13, Iran.

b. Department of Petroleum Engineering, Faculty of Engineering, Shahid Bahonar University of Kerman, Kerman, P.O. Box76169-13, Iran.

Received 27 October 2014; received in revised form 14 December 2014; accepted 3 February 2015

KEYWORDSLattice Boltzmannmethod;Immersed boundarymethod;Capsule deformation;Shear ow;Finite elementmethod.

Abstract. In this paper, the deformation of an elastic spherical capsule suspendedin a shear ow is studied in detail using Lattice Boltzmann method for uid owsimulation, immersed boundary method for uid-membrane interaction and �nite elementmethod for membrane force analysis. While Lattice Boltzmann method is capable ofimplementing inertia e�ects, computations were carried out for small Reynolds numberin which inertia e�ects are negligible. E�ects of three membrane constitutive equationson capsule deformation, including Neo-Hookean, zero-thickness shell approximation andSkalak's law with di�erent area-dilation modulus, are studied in detail. Results presentedin the form of Taylor deformation parameter, inclination angle and period of tank-treadingmotion of capsule, show close agreement between those obtained from Neo-Hookeanand zero-thickness shell approximation with previous published ones. Such agreementis partially observed for Skalak's law implementing di�erent area-dilation modulus. Ingeneral, behaviors of all three constitutive laws are similar for nondimensional shear ratesof less than 0.05 while some di�erences were observed for its values of 0.1 and 0.2. As ane�cient computational framework, it is shown that combined Lattice Boltzmann, ImmersedBoundary and Finite element method is a promising method for such ow con�guration,implementing di�erent membrane constitutive laws.© 2015 Sharif University of Technology. All rights reserved.

1. Introduction

Deformation of a spherical capsule under shear owhas been a subject of research for more than threedecades due to its physical complexity as well as itsimportance in many applications such as motion ofred blood cell in blood ow and arti�cial capsules infood industries. Consisting of a thin elastic membranewith an incompressible uid inside, a capsule starts

*. Corresponding author. Tel.: +98 343 2111763;Fax: +98 343 2120964E-mail address: [email protected] (M. Rahnama)

to deform under shear ow with its �nal con�gura-tion depending on Reynolds number and shear rate.The complexity of such ow con�guration stems fromsimultaneous implementation of Eulerian descriptionfor uid ow, Lagrangian description of membranedeformation and their interconnection through a uid-membrane interaction model.

Among the pioneers in investigating deformationof a capsule freely suspended in a shear ow, Barth�es-Biesel and Rallison [1] derived the equations governingtime-dependent deformation of membrane resultingfrom viscous stresses more than three decades ago.Their study was limited to a spherical capsule with

1878 Z. Hashemi et al./Scientia Iranica, Transactions B: Mechanical Engineering 22 (2015) 1877{1890

small departure from sphericity. Substantial progressin computational methods and computer hardware, inthe last few decades, created an opportunity to studythree dimensional time-dependent capsule deformationin shear ow extensively [2-5]. A comprehensive reviewof these progresses was reported by Barth�es-Biesel [6],in 2011, in which di�erent modeling strategies insimulation of a spherical capsule suspended in a shear ow were presented along with their limitations andadvantages. Progress in implementation of di�erentviscosities for uids inside and outside of the capsulemembrane [4,7] and membrane constitutive laws andmembrane bending resistance [8,9] are also reviewed inthis paper.

Analytical methods based on perturbation solu-tion for deformation of an initially spherical capsulein weak simple shear ow were presented by Barth�es-Biesel and Rallison [1] and Barth�es-Biesel [10]. Thesemethods are limited to small deformations at Stokes ow regimes. On the other hand, computationalmethods are capable of representing details of owand capsule deformation in three dimensions for largedeformation, considering the e�ect of changing param-eters such as di�erent membrane constitutive modelsand uid viscosities. Boundary Element Method(BEM) [2-5], Finite Di�erence Method (FDM) [7,11,12]and Lattice Boltzmann Method (LBM) [13-17] areamong frequently used computational methods in sim-ulation of capsule deformation in shear ow. BEMwas the �rst successful method in revealing detailedphysics of capsule deformation since two decades ago.Pozrikidis [2,9,18], Kraus et al. [19], Ramanujan andPozrikidis [4], Navot and Sackler [20], Diaz et al. [21],Barth�es-Biesel et al. [22] and Lac et al. [5] usedBEM in their simulation, and their results were inclose agreement with experimental data of Chang andOlbricht [23] and Walter et al. [24,25]. In their in-vestigations, Ramanujan and Pozrikidis [4] studied thee�ect of large capsule deformation and uid viscosity indetail for the �rst time. In spite of their great success,BEM simulation of capsule deformation su�ered froma limitation regarding ow regime: BEM applicationis limited to Stokes ow where the e�ect of inertia isneglected, and also the full dynamics of capsules withcomplex geometries, such as biconcave capsules, cannotbe handled due to instabilities in computations.

There are some applications for which uid inertiae�ect has to be considered, such as arti�cial capsules infood and polymer processes. Moreover, the background uid might be viscoelastic as is the case in biological ows which might not be simulated using BEM. Theselimitations motivated researchers to employ alterna-tive methods among which there are �nite di�erenceof Navier-Stokes equations [7,11,12], and LBM withImmersed Boundary Method (IBM) [26,27] for uid-particle interaction. Meanwhile, IBM has been used

in simulation of many problems, such as deformationof elastic membranes. Eggleton and Popel [28] studiedlarge deformation of capsules in shear ow using IBM;however, due to computational costs, simulations werecarried out on a coarse mesh for a short period of time.Li and Sarkar [12] employed a front tracking methodto simulate deformation of a spherical capsule in shear ow, and compared and discussed the performance ofthe method with the results obtained from BEM. Theirresults showed that, in spite of low-order surface stresscalculations, the front tracking method is comparedvery well with high-order BEMs in predicting capsuledeformation, orientation, and tank-treading motions.

Recently, LBM has emerged as a useful methodfor simulation of uid-particle interaction problems dueto some of its intrinsic features, such as �xed Euleriangrid and computational e�ciency [29], as experiencedby Hashemi et al. in particle ow simulation [30-32]. Acombination of IBM and LBM was used by Feng andMichaelides [33,34] to study particulate ow. Basedon progresses in combined IB-LBM method, Sui etal. [13-16] proposed a hybrid method consisting of IB-LBM and Finite Element Method (FEM) to study thedeformation of a capsule in simple shear ow. Intheir earlier simulations, Sui et al. [13] used multi-block strategy to re�ne the mesh near the capsule toincrease the accuracy and e�ciency of computation.They used a �nite element method to obtain forcesacting on the membrane nodes of the three-dimensionalcapsule which was discretized into at triangular ele-ments. Compared with other published results in theliterature, it was shown that this method is capableof predicting dynamic behavior of deformable capsulesaccurately. Sui et al. [13] studied deformation ofinitially spherical and oblate-spheroidal capsules withvarious membrane constitutive laws under shear owwhere it was observed that there were good agreementsbetween previous theory and numerical results. It wasreported in their paper that the inertia e�ects werefound to have signi�cant impact on capsule deforma-tion. Moreover, the unsteady tank-treading motion,in which the capsule undergoes periodic shape defor-mation and inclination oscillation while its membraneis rotating around the liquid inside, was revealed forthe �rst time in simulations using IB-LBM. Later on,Sui et al. [16] coupled front-tracking method with theLBM to investigate the e�ect of di�erent viscosities for uids inside and outside of the membrane. These two uids were treated as one uid with varying physicalproperties in LBM simulation. Their computationof Taylor deformation parameter (de�ned in Section3), capsule inclination angle and the tank-treadingfrequency showed good correspondence with previouslypublished numerical results.

More recently, Kruger et al. [17] studied capsuledeformation in shear ow using IB-LBM with FEM

Z. Hashemi et al./Scientia Iranica, Transactions B: Mechanical Engineering 22 (2015) 1877{1890 1879

used for obtaining membrane forces from its deforma-tion in the limiting case of small deformation wherethe analytical expressions were reported. Their mainobjective was to show the in uence of mesh tessellationmethod, the spatial resolution, and the discrete deltafunction of the immersed boundary method on thenumerical results. They came to the conclusion thatdetails of the membrane mesh, as tessellation methodand resolution, play only a minor role while thehydrodynamic resolution, i.e. the width of the discretedelta function, can signi�cantly in uence the accuracyof the simulations. Their study also showed thatLBM is capable of reducing the computational timefor simulation of deformable object immersed in a uidwhile maintaining high accuracy.

The present study was performed using combinedIBM, LBM and FEM for capsule deformation under ashear ow. While physics of such ow con�guration hasbeen matured in recent years as mentioned in the aboveliterature review, there are still some issues whichhave not been addressed clearly in this computationalframework such as the e�ect of di�erent membrane con-stitutive equations on capsule deformation parameterssuch as length of major axis and area changes of capsuleduring its deformation as well as computational time.The �rst aim of the present paper is to compare thecomputational e�ciency and accuracy of the combinedmethod with other available techniques. Then thedynamic response of three di�erent membrane con-stitutive laws namely zero-thickness shell formulation,Neo-Hookean law and Skalak's law at di�erent shearrates are studied. Since the biological cells tend todeform at constant area, the area change during thedeformation is an important factor. In the present workthe resistance of di�erent constitutive equations toarea change under di�erent conditions is also veri�ed.Following the Skalak's law, the area dilation moduluscontrols the changes in area during the deformation.When this modulus approaches in�nity, the area willbe constant. The e�ects of di�erent area dilationmodulus are studied here to �nd a value which isappropriate for dynamic behavior of biological cells.The paper starts with a description of LBM for uid ow simulation, IBM for uid-capsule interaction mod-eling and FEM which is used for obtaining membraneforces from its deformation. In the results part ofthe paper, authors express their �ndings regardingthe computational strategies, comparison of di�erentmembrane constitutive laws and major axes lengthsand capsule area variations in more details comparedto the previous published papers.

2. Numerical method

In order to study deformation of a spherical capsule ina shear ow, a combined computational frame, capable

of simulating uid motion, capsule membrane deforma-tion and uid-membrane interaction is required. In thepresent computations, a combination of three meth-ods is used for such simulation composed of LatticeBoltzmann method for uid ow simulation, immersedboundary method which models uid-membrane inter-actions, and �nite element method for computation ofmembrane forces from its deformation. In the followingsubsections, each of the above mentioned methods isdescribed in more details.

2.1. Lattice Boltzmann methodLattice Boltzmann method is a particle-based kineticmethod which uses a mesoscopic approach for uid owcomputations. It is based on the numerical solution ofBoltzmann equation from which velocity distributionfunction is obtained. The discretized form of thisequation, using single relaxation time approximation(BGK model) [35], is expressed as:

f�(x+ e��t; t+ �t)� f�(x; t)

= �1�

[f�(x; t)� feq� (x; t)] + �tF�; (1)

where f�(x; t) is velocity distribution function in �thdiscrete velocity direction; x is the spatial positionvector; t is time; � is the relaxation time; feq� (x; t)is equilibrium velocity distribution function; and F�is the external force. Di�erent discretizations ofthree dimensional velocity vector have been proposed,however, the most widely used one is nineteen-velocitymodel which is called D3Q19, as shown in Figure 1.Discrete velocity sets fe�g can be de�ned in this modelas:e0 = (0; 0; 0);

e� = (�1; 0; 0)c; (0;�1; 0)c; (0; 0;�1)c

for � = 1 � 6;

Figure 1. Velocity discretization in three-dimensional LBmethod: D3Q19 model.


e� = (�1;�1; 0)c; (0;�1;�1)c; (�1; 0;�1)c

for � = 7 � 18; (2)

where c = �x=�t is the lattice speed. The equilibriumdistribution function can be expressed as:

feq� = !��

"1 +

3~e�:~uc2

+9 (~e�:~u)2

2c4� 3 (~u:~u)

2c2

#: (3)

The coe�cients !� are weighting factors that dependson the discrete velocity set in three spatial dimensions.In D3Q19 model, !0 = 1=3 and !� = 1=18 for � = 1 to6, and !� = 1=36 for � = 7 to 18.

Macroscopic variables such as density and velocitycan be obtained from velocity distribution function asfollows:

� =X

f�; (4)

�~u =X

~e�f�: (5)

Fluid viscosity is related to the relaxation time in LBMas � = (� � 0:5)c2s�t, where cs is the lattice speed ofsound and is equals to c=

p3. External force term,

which is introduced in the present LBM, is based onthe method of Luo [36], which is expressed as:

F� = �3!��e�:f=c2: (6)

Here f is external force density at Eulerian nodes whichis more discussed in Sections 2.2 and 2.3.

2.2. Immersed boundary methodImmersed boundary method was initially developedby Peskin [26] in 1970s to model blood ow in thehuman heart. In this method, the e�ect of an objecton uid ow is speci�ed by adding a body forceto momentum equations. It can be explained usinga set of �xed uid Eulerian nodes, x, and movingLagrangian points s with position vectors X(s; t), formembrane, see Figure 2. At each time step, a forceF (s; t) is computed for each membrane node due toits deformations. This Lagrangian force which iscomputed using a procedure explained in Section 2.3,is distributed on the Eulerian points by the followingDirac delta function interpolation:

f(x; t) =Xs

F (s; t)�(x�X(s; t)): (7)

For three-dimensional cases the discrete delta functioncan be written as:

�(x�X(s; t))

= �(x�X(s; t))�(y � Y (s; t))�(z � Z(s; t)); (8)

Figure 2. Schematic diagram showing uid Eulerian andmembrane Lagrangian points.

where for example,

�(x�X(s; t))

=

(1� jx�X(s; t)j jx�X(s; t)j � 10 otherwise

(9)

The new velocities of membrane Lagrangian nodes canbe obtained from the local uid velocities, u(x; t), bythe same interpolation function used in Eq. (7):

U(s; t) =Xx

u(x; t)�(x�X(s; t)): (10)

At the surface of the particle, the no-slip boundarycondition is satis�ed by equating boundary-node ve-locities with velocity obtained from its adjacent uidnodes expressed as:

@X(s; t)@t

= u(X(s; t); t): (11)

2.3. Membrane model: �nite element methodIn order to compute elastic forces on membrane nodes,an appropriate constitutive equation, which describesthe membrane properties, is required. Among di�erentconstitutive equations, the Neo-Hookean (NH) law isthe simplest one with the strain energy function in theform of:

WNH =Es6

��2

1 + �22 � 3 +

1�2

1�22

�; (12)

in which Es is the surface shear elasticity modulus and�1 and �2 are the principle stretch ratios. The NHconstitutive equation models membrane as a volume-incompressible isotropic material which means anyincrease in the area of membrane due to its deformationaccompany with thinning of membrane thickness to


preserve membrane volume. Another version of NHlaw is Zero-Thickness (ZT) shell formulation which wasused by Ramanujan and Pozrikidis [4]. Here, a strain-energy function is expressed as:

WZT =Es6

��2

1 + �22 � 2� log(�2

1�22)

+12

log2(�21�

22)�: (13)

Skalak et al. [37] proposed another constitutive equa-tion, denoted by SK, to describe elastic behavior of redblood cells. A red blood cell tends to deform easilyat constant area. In the SK law, the resistance of themembrane to area changes is represented by coe�cientC in its equation which is written as follow:

W SK =Es12�(�4

1 + �42 � 2�2

1 � 2�22 + 2)

+C(�21�

22 � 1)2� : (14)

The �rst term on the right-hand side of the aboveequation describes the shear e�ects, and the secondterm represents the area dilation. A membrane withzero change in area is obtained if C approaches in�n-ity.

The current study was done using �nite elementmethod developed by Charrier et al. [38] and Shri-vastava and Tang [39] from which elastic forces ofmembrane nodes are obtained from their deforma-tion, taking into account the selected constitutive law.Membrane surface is discretized using at triangularelements which remain at after deformation. Theprocedure of generating unstructured mesh on thesurface of the membrane is presented in Section 3. Thisidealization allows us to employ in-plane-membranetheory for each element. To obtain element deforma-tion, each individual undeformed triangular element istransferred to the plane of the deformed element (seeFigure 3). Such pure displacement does not a�ectnodal forces of the element. Assuming constant dis-placement gradients within the element, displacement

of any point on the surface of element can be expressedas:

u =3Xi=1

Niui;

in which Ni is the linear shape function de�ned asNi = �ix + �iy + i for each node i. The unknowncoe�cients in this function can be obtained by thefact that Ni = 1 at node i and Ni = 0 at two othernodes. By virtue of this assumption, the displace-ment gradients and corresponding principal stretchratios can be immediately determined [38]. Using theprinciple of virtual work, one can derive the relationbetween the nodal displacements and nodal forces.Since the membrane material is assumed to be volume-incompressible and initially isotropic, the strain energyfunction is a symmetric function of the principal stretchratios and is only a function of in-plane stretch ratios.Therefore one can �nd the nodal forces in the localsystem as:�

FLx

= Ve�@W@�1

@�1

@uL+@W@�2

@�2

@uL

�; (15)

�FLy

= Ve�@W@�1

@�1

@vL+@W@�2

@�2

@vL

�; (16)

where Ve is the original area of the element, u andv are the x- and y-component of the displacements.These nodal forces in local system are transferred backto the global system and a summation is made over allelements that meet at one node. The opposite of theresulting force is the elastic force that is implementedon uid nodes by membrane. More details of thisprocedure can be found in [38,39].

3. Results and discussion

The present results are related to the ow geometryof deformation of an initially spherical capsule locatedin the middle of an unbounded shear ow (illustratedin Figure 4) with velocity ~Vs = ( _ y; 0; 0) in which_ is shear rate and y is distance from center of the

Figure 3. Undeformed (a) and deformed (b) triangular elements in space along with their superposition (c) shown in thedeformed plane.


Figure 4. Schematic illustration of an initially sphericalcapsule in an unbounded shear ow.

computational domain. The computational domain isa cubic box whose top and bottom walls move withthe equal velocity in opposite directions to producea simple shear ow and follow the no-slip boundarycondition. In order to simulate an unbounded shear ow, the periodic boundary condition is applied at thefour other boundaries of the domain to reach repeating ow conditions. Fluids inside and outside the capsuleare considered as Newtonian with the same densityand viscosity. No bending resistance is considered forthe capsule membrane. Two dimensionless parametersa�ecting such ow con�guration are Reynolds number,Re = � _ R2=�, and dimensionless shear rate, G =� _ R=Es which express the ratio of viscous force toelastic force. Here, � and � are the density and dynamicviscosities of surrounding uid, respectively, R is cap-sule radius, and Es is the membrane shear elasticity.The present study was performed at Re = 0:025 whichcorresponds to Stokes ow regime. Starting from itsspherical shape, membrane starts to deform gradually,changing to an ellipsoidal shape with its major axishaving an angel of � with respect to the ow direction(x). Capsule deformation can be represented by Taylordeformation parameter which is de�ned as:

Dxy =L1 � L2

L1 + L2: (17)

Here, L1 and L2 are the largest and the smallest semi-axes of the ellipsoid with the same inertia tensor as thecapsule. Using the inertia tensor in the form of [12]:

Id =ZV

[r2I � xx]d3r =15

Z@V

[r2Ix� xxx]:nd2r;(18)

one can �nd the principal directions and the principalmoments of the tensor (Idi ) and compute the major

Figure 5. E�ect of the size of computational domain ontransient evolution of Taylor deformation parameter atG = 0:1 and Re = 0:025 for a sphere with 1280 elements(642 nodes).

axes of the ellipsoid (Li) as:

Id1 =15�V (L2

2 + L23); Id2 =

15�V (L2

1 + L23);

Id3 =15�V (L2

1 + L22): (19)

The �rst concern in numerical simulation of shear owover a spherical capsule is selecting an appropriateminimum size for the computational domain. Thelength of the computational domain in x-, y- and z-directions is generally expressed based on the radius ofthe sphere. In order to investigate the e�ect of thedistance between shear-producing plates, H, on thepresent computation, Taylor deformation parameter isplotted for H equals to 5R, 10R and 12R in Figure 5.In this �gure t� is the dimensionless time de�ned as_ t. As is observed from this �gure, computed valuesof Taylor deformation parameter for H = 10R and12R correspond to each other compared to a slightlysmaller values for H = 5R. Despite small di�erencebetween results obtained from H = 5R and H = 10R,the latter one was used in the present computationsto better compare the results with previous publishedworks with H = 10R. It should be mentioned thatsphere radius is selected as 6 lattice units (lu) in thepresent calculations which results in a computationaldomain of 60� 60� 60.

The number of elements on membrane surface isanother important factor which a�ects the accuracy ofthe results and computational time considerably. Increating membrane mesh, homogeneity and isotropyare two important factors which need to be considered.As implemented by many researches [16,17], a uniformand symmetric mesh can be generated starting from a


Figure 6. Di�erent steps of mesh subdivision on thesurface of a sphere for generating triangular elements: (a)A regular icosahedron; (b) split each face of icosahedron;(c) project each new vertex on the surface of the sphere;(d) a sphere with 162 nodes and 320 elements; (e) a spherewith 462 nodes and 1280 elements; and (f) a sphere with2562 nodes and 5120 elements.

regular icosahedron (Figure 6(a)), which is a polyhe-dron with 20 identical equilateral triangular faces, 30edges and 12 vertices. The process starts with splittingeach edge of the triangular faces of the icosahedroninto two equal parts (Figure 6(b)). The next stepis to project the new vertices on the surface of thesphere (Figure 6(c)). This procedure is repeated untila desired resolution is obtained (Figure 6(d)-(f)).

In order to show the e�ects of di�erent meshresolutions of sphere on the accuracy of results, vari-ation of Taylor deformation parameter is representedin Figure 7 for three sets of membrane elements. Therequired time for each iteration of di�erent resolutionsatH = 10R is also tabulated in Table 1. Based on theseresults, it is concluded that using a sphere with 1280elements and 642 nodes is appropriate for obtainingaccurate and e�cient modeling. Moreover, Table 1shows that increasing the number of nodes on spherehas much less e�ect on iteration time as compared toincreasing the domain size. It should be mentionedthat all of the present computations were done using aCore-i7/2.4 GHz computer.

It should be noted that the required number ofmembrane nodes and domain size used in the presentcomputations are considerably smaller than those used

Figure 7. E�ect of di�erent sphere grid resolution ontransient evolution of Taylor deformation parameter atG = 0:1 and H = 10R.

in previous works, such as Li and Sarkar [12], whichused a computational grid of 96 � 96 � 48 nodes with5120 elements and 10242 nodes for a membrane, Doody& Boghchi [7] with a computational grid of 120�120�120 nodes and 1280 elements on a membrane, and Suiet al. [13] who used multi-block technique with 8192elements and 4098 nodes on a membrane.

As a main criterion for evaluating the results,Figure 8 shows the evolution of Taylor deformationparameter, Dxy, and inclination angle, �=�, withdimensionless time for a spherical capsule with mem-brane constitutive equation of zero-thickness shell, inshear ow at di�erent dimensionless shear rates, G.The well-known behavior of initial increase followedby a steady value for Taylor deformation parameter,and initial decrease followed by a steady value ofinclination angle with increasing dimensionless shearrate are clearly observed in this �gure. Results ofRamanujan and Pozrikidis [4], which were obtainedusing boundary element method, are shown in this�gure too. Close agreement between the present resultsand those of Ramanujan and Pozrikidis reveals theaccuracy of the computational method along with itsrequired resolution.

When a spherical capsule is exposed to the shear ow, it starts to elongate until an equilibrium stateis reached. At this stage, the capsule is no longerdeformed and starts to rotate around the interior uid. The transient shapes of the deformable spherical

Table 1. The required time for one complete iteration in di�erent test cases.

Case

255 elements,162 nodes



H = 10R H = 5R H = 10R H = 12R H = 10RCPU time 0.63 0.064 0.67 1.14 0.76


Figure 8. Comparison of transient evolution of (a)deformation parameter, and (b) inclination angle for acapsule with zero-thickness model in shear ow atdi�erent dimensionless shear rates at Re = 0:025.

capsule at di�erent stages and G = 0:1 are shownin Figure 9. The �rst three �gures show transientdeformation and alignment of the capsule, while thelast three ones represent steady deformed shapes thatalign with a �xed angle with respect to the x-axis.A material point is shown in these �gures whichrepresents tank trading motion of the membrane in thelast three �gures; a continuous rotation of a membranearound its steady shape.

To investigate the e�ect of constitutive equationson the results, three most important models, namelyZero-Thickness shell formulation (ZT), Neo-Hookeanlaw (NH) and Skalak's law (SK) were implemented andcompared in the present computations. Figure 10(a)and (b), respectively, show steady deformation param-

Figure 9. Deformation of an initially spherical capsuleexposed to simple shear ow at di�erent times; G = 0:1and Re = 0:025..

Figure 10. E�ect of di�erent constitutive equations on(a) a steady deformation parameter, and (b) steadyinclination angle at di�erent G.


eter and steady inclination angle for di�erent values ofG. The results of Ramanujan and Pozrikidis [4], Suiet al. [13] and small deformation theory of Barth�es-Biesel and Rallison [1] are also included in these�gures for comparison. Excellent agreement is observedbetween present computations and those obtained fromsmall deformation theory for G = 0:025. It is worthmentioning that small deformation theory is not validat relatively large deformations, e.g. for G � 0:05.Among these three membrane laws, the ZT modelpredicts the closest values to those of Ramanujan andPozrikidis [4] who used the same model. As observedfrom Figure 10, the aforementioned three membranelaws produce nearly equal values of Dxy (and �)for G � 0:05 which reveals their accuracy at smalldeformations. This trend is not observed for relativelylarger deformations. In accordance with the results ofRamanujan and Pozrikidis [4], the predicted results byZT shell law is 1.7% and 2.4% less than those obtainedby NH law for G = 0:1 and 0.2, respectively. It shouldalso be noted that SK law underestimates the values ofDxy and �, slightly, compared to the ones obtained byRamanujan and Pozrikidis [4]. In other words, usingSK membrane law results in less deformation comparedto ZT or NH membrane law for the same shear rate.Such behavior for SK law is reasonable due to itsstrain hardening behavior. In fact, resistance to areacompression in SK law prevents further deformationof the capsule which, in turn, a�ects the alignment ofthe membrane and increases the inclination angle withrespect to the ow direction (x-axis).

One important issue in the present computa-tions is the e�ect of grid uniformity on the resultsas compared to the published results obtained fromnonuniform grid distribution. Sui et al. [13] whosecomputations were based on IBM/LBM/FEM, usedmulti-block technique for increasing concentration ofgrid points near the membrane. They divided thecomputation domain into two blocks: the interior onewhich is a cubic box with length of 4R around themembrane and has �ne mesh with grid resolution of�xf = �yf = �zf = R=12, and the exterior onewith the coarse mesh and grid resolution �xc = �yc =�zc = R=6. The present computations were done witha uniform grid with resolution R=6 in all directions forthe whole computational domain. As observed fromFigure 10(a) and (b), results of present computationsare in good agreement with those of Sui et al. [13] usingmulti-block method. It is concluded that the conditionof local grid re�nement can be relaxed without losingconsiderable accuracy.

As mentioned before, when a capsule passesthrough the transient stage and reaches the steadystate behavior, the material points on the membranestart to rotate along a �xed trajectory on the steadydeformed shape (see Figure 9). At this level, there is

no normal velocity for the membrane nodes and onlytangential velocity is experienced by those nodes. Thiswell-known periodic motion is called \tank-treading-motion". To reveal the characteristics of this motion,the position of a material point, which is initially on thetop of the sphere, is plotted versus time in Figure 11for di�erent values of G with ZT shell formulation forthe membrane. The periodic motion of this materialpoint is evident in these �gures. With increasingdimensionless shear rate, G, the capsule elongates morewhich results in a larger path of rotation of the materialpoint and its subsequent longer periodic time.

Following the position of a material point for along time, one can �nd the tank-treading period _ T ofthe membrane. The tank-treading period is plotted inFigure 12 as a function of G for di�erent constitutivelaws along with the results of the earlier works. In thepresent study, the tank-treading period is computed byfollowing a marker point and measuring the time periodfor one complete cycle. Due to larger deformation andlonger rotation path at higher G, the tank-treadingperiod increases with increasing G. Results obtainedfrom the present computations are in a reasonableagreement with those of Ramanujan and Pozrikidis [4]and Lac et al. [5] for G < 0:1. However, closeragreement is observed between present computationsand those of Ramanujan and Pozrikidis [4] for G = 0:2using ZT and NH laws. At small deformations thethree constitutive equations show similar behavior andpredict approximately the same value for the tank-treading period. However, at large deformation, the SKlaw predicts the smallest value among them. The strainhardening behavior of the SK law leads to smallerdeformation and shorter rotating path, compared withother laws, which results in an underestimation. AtG = 0:2, the predicted result by ZT law is 3% less thanthat obtained using NH law.

The steady shape of the deformable capsule is rep-resented in Figure 13 for di�erent values of G. Viscousforces become relatively stronger than elastic forceswith increasing G which results in more deformationand less inclination angle with respect to the x-axis.

Surface area of a deformable capsule in shear ow might not remain constant. Investigation of thise�ect is of prime importance in RBC deformation. Toreveal the e�ect of shear rate on surface area andlengths of the major and minor axes of capsule, aseries of computations were carried out for di�erentdimensionless shear rates, and their results are illus-trated in Figure 14(a), (b) and (c). In these �gures,time variation of these parameters is expressed innondimensional forms; lengths and area are divided bytheir corresponding values for the undeformed initialsphere. Figure 14(a) shows variation of major axiselongation with time. Opposite behavior is observed forthe length of the minor axis of the capsule; it reduces


Figure 11. Time evolution of the position of a material point on a ZT membrane surface.

Figure 12. Comparison of tank-treading period fordi�erent G and di�erent constitutive equations withearlier works.

with both increasing time and dimensionless shear rate.At equilibrium state, both lengths reach some constantvalues. As is observed from Figure 14(c), surfacearea of capsule increases with time, reaching a steadyvalue which increases with increasing G. According to

Figure 13. Steady deformed shape of the capsule in theplane of shear at di�erent G and Re = 0.025.

Figure 14(a)-(c), results obtained by ZT, NH and SKlaws correspond to each other for small deformation(G � 0:05). For G = 0:2, the steady area of SK lawis about 3.1% and 4.6% less than that for ZT and NHcapsules. Unlike the SK law, at the same G, the NH


Figure 14. E�ect of di�erent constitutive equations on a)the lengths of major axes, b) lengths of minor axes, and c)area ratio at di�erent G.

law makes the biggest changes in the capsule area whichshows its strain-softening behavior.

At the end of this section the e�ect of di�erentvalues of the coe�cient C in SK law is investigated.The variations of deformation parameter, inclination

Figure 15. E�ect of di�erent values of coe�cient C on a)the Taylor deformation parameter, b) an inclination angle,and c) area ratio in SK law at di�erent C.

angle and area ratio with dimensionless shear rate fordi�erent values of C are plotted in Figure 15. Resultsof Ramanujan and Pozrikidis [4] which were obtainedfor a membrane with ZT shell formulation is also addedto Figure 15(a) and (b). The SK constitutive equation


was proposed to describe the elastic behavior of redblood cell membranes. A red blood cell tends to deformeasily at a constant area. In the SK law, the resistanceof the membrane to area changes is represented bycoe�cient C. If this coe�cient approaches in�nity, amembrane with zero change in the area is obtained.However, the behavior of SK law with small value of Cis close to the behavior of a ZT or NH membrane. Amembrane obeying the SK law with larger values of Chas a larger area dilation modulus and becomes morearea incompressible and less deformed at a constantshear rate. According to Figure 15(c) with C = 10, the�nal area change for G = 0:025, 0.05, 0.1 and 0.2 isabout 0.002, 0.015, 0.03 and 0.054, respectively. Withincreasing the value of C, the inclination angle, withrespect to x-axis, increases.

4. Conclusions

A three-dimensional numerical simulation is performedto study the dynamic behavior of an initially spher-ical capsule in simple shear ow. Fluid ow, uid-membrane interaction and membrane analysis werestudied using Lattice Boltzmann method, immersedboundary method and �nite element method, respec-tively. Membrane analysis was done using three di�er-ent constitutive equations consisting of Neo-Hookianlaw, zero-thickness shell approximation and Skalak'slaw with di�erent area-dilation modulus. Comparisonof results, in the form of Taylor deformation param-eter and capsule inclination angle, showed negligiblevariation using di�erent constitutive laws for G < 0:05while more discrepancies were observed for G = 0:1and 0.2. Results of membrane area variation duringcapsule deformation showed that area conservation isa good assumption for small G, namely G < 0:05,no matter which constitutive model is used for themembrane. Such behavior was not observed for G =0:1 and 0.2, and more discrepancies appeared usingdi�erent constitutive laws. Using di�erent values ofthe parameter, C, in SK model (see Eq. (14)) incomputation of Taylor deformation parameter, incli-nation angle and area ratio showed that SK modelis sensitive to such variations; good correspondencewith previous published results of Ramanujan andPozrikidis [4] could be obtained for C = 0:01 forsmall G values. For C = 10, the maximum areachange is about 5%, so since higher values of thisparameter may lead to numerical instability, it seemsthat C = 10 is large enough that one can ignore thearea change during the deformation. In comparisonwith previous techniques available in the literaturethe combined method can predict accurate results atlow computational cost, and according to the intrinsicfeatures of lattice Boltzmann method in paralleliza-tion processes, this method has a great capability in

simulating biological problems with large number ofdeformable cells.

Nomenclature

c Lattice speedC A coe�cient in Skalak's lawcs Speed of sounde� Discrete velocity vectorDxy Taylor deformation parameterEs Shear elasticity modulusf� Density distribution functionfeq� Equilibrium density distribution

functionF� External forcef Eulerian external force densityF Lagrangian forceG Dimensionless shear rateH Domain height

Id Inertia tensorL1; L2 Semi-axes of ellipsoidN Shape functionR Capsule radiuss Membrane nodet Timet� Dimensionless time~u Fluid velocity vectorU Velocity of Lagrangian nodeu; v DisplacementV VolumeVe Original area of elementW Strain energy functionx; y; z Eulerian position coordinatesX;Y; Z Lagrangian position coordinates

Greek symbols

_ Shear rate�(x) Dirac delta function� Inclination angel� Principal stretch ratio� Fluid dynamic viscosity� Fluid kinematic viscosity� Density� Dimensionless relaxation time!� Weighting factor

Subscripts/superscripts

� Lattice direction


eq Equilibrium stateL Local system! A vector

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Biographies

Zahra Hashemi received her MSc degree in Mechan-ical Engineering from Shiraz University, Iran. She iscurrently a PhD student at Mechanical EngineeringDepartment of Shahid-Bahonar University of Kerman,Iran. Her main research interest includes numericalsimulation of Biological systems such as deformationand motion of red blood cell in uid ow using thelattice Boltzmann and �nite element methods.

Mohammad Rahnama is Professor at MechanicalEngineering Department of Shahid Bahonar Universityof Kerman, Iran. He graduated from Shiraz Universitywith a PhD degree in 1997. Since then he has beeninvolved in uid ow simulations. His interest in Lat-tice Boltzmann Method (LBM), as a strong simulationmethod in computational uid dynamics, started a fewyears ago with LBM simulation of a microchannel ow.He recently focused on investigation of solid particlesmotion immersed in a uid ow using LBM which wasa starting point for simulation of deformable particles.Now, he is continuing his research on motion anddeformation of red blood cells in blood ow using LBM,FEM and immersed boundary method.

Saeed Jafari received his BS degree from Shahid Ba-honar University of Kerman, MS degree from IsfahanUniversity of Technology and PhD degree from ShahidBahonar University of Krman, Iran, in 2004, 2006and 2010, respectively, all in Mechanical Engineering(Energy Conversion). He is currently Assistant Pro-fessor of Mechanical Engineering at Shahid BahonarUniversity of Kerman. His research interests include:mathematical and numerical modeling, computational uid dynamics, particle science and technology, andsimulation of multiphase ows and lattice Boltzmannmethod.

lattice boltzmann simulation of three-dimensional capsule

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