Applied Mathematical Modelling 36 (2012) 2890–2899

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

A lattice Boltzmann model for blood flows

Yanhong LiuCollege of Mathematics, Jilin University, Changchun 130012, PR China

a r t i c l e i n f o

Article history:Received 26 August 2010Received in revised form 18 August 2011Accepted 23 September 2011Available online 2 October 2011

Keywords:Lattice Boltzmann methodLattice Boltzmann Bi-viscosity modelBlood flow

0307-904X/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.apm.2011.09.076

E-mail address: yanhongliu6765@gmail.com

a b s t r a c t

A lattice Boltzmann model for blood flows is proposed. The lattice Boltzmann Bi-viscosityconstitutive relations and control dynamics equations of blood flow are presented. A non-equilibrium phase is added to the equilibrium distribution function in order to adjust theviscosity coefficient. By comparison with the rheology models, we find that the latticeBoltzmann Bi-viscosity model is more suitable to study blood flow problems. To demon-strate the potential of this approach and its suitability for the application, based on thisvalidate model, as examples, the blood flow inside the stenotic artery is investigated.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

In recent years, the lattice Boltzmann method (LBM) has developed into an alternative and promising numerical schemefor simulating fluid flow and modeling physics in fluids [1–4]. This method originated from the classical statistical physics, inwhich the fluid is modeled as a collection of pseudo-particles, and such particles propagate and collide over a discrete latticedomain [5]. Unlike traditional numerical methods which solve equations for macroscopic variables, the LBM is based on themesoscopic kinetic equation for the particle distribution function. The kinetic nature of the LBM brings certain advantagesthat distinguish it from other numerical methods, such as its suitability for parallel algorithms, simple structure, easy hand-ing of complex boundary conditions, and efficient numerical simulations. Due to these advantages, the LBM has been suc-cessfully used to simulate many complex problems [6–16], including compressible flows [6], multiphase and multi-component fluids [7,8], particulate suspensions [9], reaction–diffusion system [10], and flow through porous media[11,12]. Additionally, this method can be used in several equations of mathematical physics, including wave motion equa-tions [13], Burgers equations [14], KdV equation [15], and the nonlinear Schrödinger equations [16].

Recently, the application of the LBM to blood flow has attracted growing interest [17–28]. The blood flow through amechanical heart valve [17,18] and the blood flow in the stented aneurysm [19–22] are given. Some LBM results for thered blood cells (RBCs) are presented [23–25]. The blood flow in the microvascular bifurcations is simulated [26,27]. An im-mersed boundary method to simulate moving solid surfaces is used [28]. The blood flow and the red blood cells informationare also studied by other methods [29–38]. The thrombus formation in blood flow is predicted [29,30]. Multiphase flowdynamics to study hemodynamic computation is used [31]. The gas bubble motion in a blood vessel is studied [32]. Bloodflow inside arterial branches is computed [33]. Blood cells shape and interactions in flow are given [34–37]. The rheology ofRBCs rouleaux in microchannels is simulated [38].

In this paper, a lattice Boltzmann model for the blood flow is proposed. The lattice Boltzmann Bi-viscosity constitutive rela-tions and the control dynamics equations of blood flow are presented. A non-equilibrium phase is added to the equilibriumdistribution function in order to adjust viscosity. By comparison with the rheology models, the lattice Boltzmann Bi-viscosity

. All rights reserved.

Y. Liu / Applied Mathematical Modelling 36 (2012) 2890–2899 2891

model is more suitable to study blood flow problems. To demonstrate the potential of this approach and its suitability for theapplication, based on this validate model, as examples, the blood flow inside the stenotic artery is investigated.

The paper is organized as follows. In Section 2, a mathematical description of these problems is provided. In Section 3, thenumerical method employed to solve these problems is presented. The results and discussion are given in Section 4. The finalsection deals with the major conclusions of the present study.

2. Mathematical formulations

The rheology model for blood viscosity is a key factor in the simulation of hemodynamic flows. In this paper, blood isassumed to be a shear-thinning fluid, and its constitutive relation is described by the Bi-viscosity model. The constitutiveequation suggested by Nkayama and Sawada [39–42] are

rij ¼2ðl0 þ r0=

ffiffiffiffiffiffiffi2Pp

Þsij; P P Pc;

2ðl0 þ r0=ffiffiffiffiffiffiffiffiffi2Pcp

Þsij; P < Pc;

(ð1Þ

where rij is shear stress, r0 is yield stress, l0 is a constant viscosity, P = sijsij is the (i, j) component of deformation rate, and

the strain rate tensor is sij ¼ 12

@ui@xjþ @uj

@xi

� �. In the Newtonian part, the fluid can be considered as a Newtonian fluid of high vis-

cosity whose constitutive relation is described by l0 þ r0=ffiffiffiffiffiffiffiffiffi2Pcp

; usually, for ordinary Newtonian fluid, r0 = 0. The Bi-viscosity model can be expressed

l ¼ 2ðl0 þ r0=ffiffiffiffiffiffiffi2Pp

Þ; P P Pc;

2ðl0 þ r0=ffiffiffiffiffiffiffiffiffi2Pcp

Þ; P < Pc;

(ð2Þ

Blood is assumed to be an impressible fluid, and the governing equation of the blood flow may be expressed as

@qui

@tþ @quiuj

@xj¼ � @P0

@xiþ @

@xjl @qui

@xjþ @quj

@xi

� �� �; ð3Þ

In this paper, we propose a lattice Boltzmann model for Eq. (3).

3. Lattice Boltzmann method

The present study is based on the lattice Boltzmann method. The key steps in the numerical algorithm are listed below.

(1) Given the lattice Boltzmann method theory on square grids.(2) Given the lattice Boltzmann Bi-viscosity model.(3) Given the equilibrium distribution functions.

These steps are briefly explained in the following.

3.1. Series of partial differential equations in different time scales

We consider a two-dimensional square lattice (see Fig. 1) and choose the vectors associated with the moving directions ofthe particles as follows:

ea ¼ð0;0Þ a ¼ 0

ðcos ha; sin haÞ ha ¼ ða� 1Þp=2 a ¼ 1;2;3;4ffiffiffi2pðcos ha; sin haÞ ha ¼ ða� 5Þp=2þ p=4 a ¼ 5;6;7;8

8><>: : ð4Þ

Fig. 1. Schematic diagram of 9-speed (D2Q9) lattice Boltzmann model.

2892 Y. Liu / Applied Mathematical Modelling 36 (2012) 2890–2899

On every lattice node x, at time t, the distribution function fa(x, t) is stored, which is defined as one-particle distributionfunction with velocity ea, here, a ¼ 1; . . . ;8 denotes moving particles, a = 0 represents the rest particle. The macroscopic massand momentum are defined as

q �X

afaðx; tÞ; ð5Þ

qui �X

afaeai; ð6Þ

We assume that fa(x, t) possesses the equilibrium distribution function f eqa ðx; tÞ; and it meets the conservation conditions

X

af eqa ¼ q; ð7Þ

Xa

f eqa eai ¼ qui: ð8Þ

Combining Eqs. (5)–(8), we have

Xa

f ðkÞa ¼ 0;X

af ðkÞa eai ¼ 0; k P 1: ð9Þ

The distribution function fa(x, t) satisfies the lattice Boltzmann equation,

faðxþ ea; t þ 1Þ � faðx; tÞ ¼ �1s

faðx; tÞ � f eqa ðx; tÞ

� �; ð10Þ

where s is the single-relaxation time factor.Let us take � ¼ d=c the small parameter controlling the limit to continuum, as the time step, here, dx is the spatial step,

and c is the speed of particle. In this case, the lattice Boltzmann equation can be written as

faðxþ eea; t þ eÞ � faðx; tÞ ¼ �1s

faðx; tÞ � f eqa ðx; tÞ

� �: ð11Þ

By performing a Taylor expansion of Eq. (11) up to O(e3), we have

faðxþ eeaÞ � faðx; tÞ ¼X3

n¼1

en

n!

@

@tþ ea

@

@x

� �n

faðx; tÞ þ Oðe4Þ: ð12Þ

The Chapman–Enskog expansion [43] is applied to fa(x, t) under the assumption of small parameter e; it is

fa ¼X3

n¼0

enf iðnÞa þ Oðe4Þ: ð13Þ

Introducing t0, t1, t2 as different scale times, we define them as

tn ¼ ent; n ¼ 0;1; . . . ;3; ð14Þ

and

@

@t¼X3

n¼0

en @

@tnþ Oðe4Þ: ð15Þ

We can obtain a series of partial differential equations as the following:

C1Df ð0Þ;ia ¼ � 1sf

f ð1Þ;ia ; ð16Þ

@

@t1f ð0Þ;ia þ C2D

2f ð0Þ;ia ¼ � 1sf

f ð2Þ;ia ; ð17Þ

@

@t2f ð0Þ;ia þ 2C2D

@

@t1f ð0Þ;ia þ C3D

3f ð0Þ;ia ¼ � 1sf

f ð3Þ;ia ; ð18Þ

where f ð0Þa � f ðeqÞa ; and the partial differential operator D � o/ot0 + ea(o/ox).

Eqs. (16)–(18) are the so-called series of partial differential equations in different times scales. It is suitable for one-dimensional, two-dimensional, and three-dimensional cases. There are three polynomials of the relaxation time factor sin Eqs. (16)–(18); they are the following:

Y. Liu / Applied Mathematical Modelling 36 (2012) 2890–2899 2893

C1 ¼ 1; ð19Þ

C2 ¼12� s; ð20Þ

C3 ¼ s2 � sþ 16¼ C2

2 �1

12; ð21Þ

3.2. Moments of the equilibrium distribution function

Other moments of the equilibrium distribution function are denoted as the following:

Xa

f eqa eaieaj ¼ pð0Þij ; ð22Þ

Xa

f eqa eaieajeak ¼ Pð0Þijk ; ð23Þ

Selecting the moments

pð0Þij ¼ P0dij þ quiuj �k2

@qui

@xjþ @quj

@xi

� �þ k

2@quk

@xkdij; ð24Þ

Pð0Þijk ¼13ðqujdik þ quidjk þ qukdijÞ: ð25Þ

summing Eqs. (16) and (17) over a and making (16) + (17)�e, we obtain

@q@tþ @qui

@xi¼ Oðe2Þ: ð26Þ

Summing Eqs. (16) and (17) over a and making ((16) + (17)�e)eaj, we obtain

@quj

@tþ@pð0Þij

@xjþ e

12� s

� �@2pð0Þijk

@xj@t0þ e

12� s

� �@2Pð0Þijk

@xj@xk¼ Oðe2Þ: ð27Þ

Combining Eq. (24), we have

@2pð0Þij

@xj@t0¼ @

@xj

@P0

@q@q@t0

dij þ@quiuj

@t0� k

2@

@t0

@qui

@xjþ @quj

@xi

� �þ @qui

@xidij

� �: ð28Þ

Combining Eqs. (9) and (17), Eq. (28) becomes

@2pð0Þij

@xj@t0¼ @

@xj

@P0

@q� @quk

@xk

� �dij þ ui �

@pð0Þjk

@xk

!þ quj

@

@t0

qui

q

� �� k

2@

@xj� @p

ð0Þik

@xk

!þ @

@xi�@pð0Þjk

@xk

!� @

@xi� @p

ð0Þik

@xk

!dij

" #( )

¼ @

@xj

@P0

@q� @quk

@xk

� �dij � ui

@pð0Þjk

@xk� uj

@pð0Þik

@xkþ uiuj

@quk

@xk� k

2@

@xj� @p

ð0Þik

@xk

!" #:

ð29Þ

According to Eq. (24), we get

@pð0Þjk

@xk¼ @P0

@xkdjk þ

@qujuk

@xk� k

2@

@xk

@quj

@xkþ @quk

@xj

� �þ @

@xk

@qum

@xm

� �djk

; ð30Þ

@pð0Þik

@xk¼ @P0

@xkdik þ

@quiuk

@xk� k

2@

@xk

@qui

@xkþ @quk

@xi

� �þ @

@xk

@qum

@xm

� �dik

: ð31Þ

Combining Eqs. (30) and (31), and omitting higher order quantity, Eq. (29) becomes

@2pð0Þij

@xj@t0¼ @P0

@q@

@xj� @quj

@xj

� �dij � ui

@q@xj� uj

@q@xiþ k

2@2q@xi@xj

" #( ): ð32Þ

If we assume the density q is uniform, for the incompressible flow, Eq. (32) becomes

@2pð0Þij

@xj@t0¼ @P0

@q@

@xj� @quj

@xj

� �dij

� �: ð33Þ

2894 Y. Liu / Applied Mathematical Modelling 36 (2012) 2890–2899

Combining Eq. (25), we have

@2Pð0Þijk

@xj@xk¼ 1

3@

@xj

@qui

@xjþ @quj

@xi

� �þ 1

3@

@xi

@quj

@xj

� �: ð34Þ

According to Eqs. (30), (33), and (34), then Eq. (27) becomes

@quj

@tþ @quiuj

@xj¼ � @P0

@xiþ @

@xjkðtÞ @qui

@xjþ @quj

@xi

� � þ @

@xikðqÞ @quj

@xj

� � þ Oðe2Þ: ð35Þ

where

kðtÞ ¼ tþ k2; ð36Þ

t ¼ eð2s� 1Þ=6; ð37Þ

kðqÞ ¼ �3t@P0

@qþ t� k

2; ð38Þ

here @P=@q ¼ c2s . As the fluid is assumed to be impressible, oquj/oxj = 0, then Eq. (35) becomes

@quj

@tþ @quiuj

@xj¼ � @P0

@xiþ @

@xjkðtÞ @qui

@xjþ @quj

@xi

� � þ Oðe2Þ: ð39Þ

By comparing Eq. (39) with Eq. (3), if we assume that

kðtÞ ¼ l; ð40Þ

then Eq. (39) is agree with Eq. (3).

3.3. Lattice Boltzmann Bi-viscosity model

The LBM is based on the mesoscopic kinetic equation for the particle distribution function. Thus, we shall now inject themacroscopic rheology model (Eq. (1)) into the lattice Boltzmann scheme. According to Refs. [22,44], the stress tensor

pð1Þij ¼X

aðfa � f ð0Þa Þeaieaj ð41Þ

is related to the strain rate tensor sij, and sij ¼ �pð1Þ

ij

2qec2s s

(see Refs. [22,44]), thus,ffiffiffiffiffiPp¼ r

2qec2s s

, here r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipð1Þij pð1Þij

qis directly

computed from the distribution functions, and as P < Pc, r = r0. According to Eq. (40), we have

eð2s� 1Þ6

þ k2¼ 2l0 þ

ffiffiffi2p

r0ffiffiffiffiffiPp : ð42Þ

If we assume that eð2s�1Þ6 ¼ 2l0; and k

2 ¼r0ffiffiffiffiffi2Pp ; then we get

s ¼ 6l0=eþ 1=2; ð43Þ

k ¼ 4ffiffiffi2p

hqec2s s; ð44Þ

where h = r0/r, as h < 1, P P Pc; h = 1, P < Pc. Eqs. (43) and (44) are called the lattice Boltzmann Bi-viscosity model. Thecoefficient k appears in the equilibrium distribution function f eq

a ðx; tÞ: In order to adjust the viscosity coefficient, a non-equi-librium phase is added to the equilibrium distribution function. Fig. 2(a) gives the resulting value of k as a function of thepossible values of h, and (b) gives the resulting value of the viscosity l as a function of the solution of k. From these figureswe may see the changes of the value k and the viscosity l.

3.4. The equilibrium distribution functions

Let us consider a two-dimensional square lattice (D2Q9) model as shown Fig. 1. A non-equilibrium phase is added to theequilibrium distribution function in order to adjust the viscosity. Assuming the equilibrium distribution function f eq

a ðx; tÞ hasthe form

f eqa ¼ xaq 1þ eai � ui

c2sþ eaieajuiuj

2c4s� u2

2c2s

� �þ Gijeaieaj; ð45Þ

where c2s is the speed of sound, and the weight coefficient xa is expressed as follows

Fig. 2. (a) The value of k; (b) the value of viscosity l. The parameters are: e = 0.001, c2s ¼ 1=3, q = 1, l0 = 0.5.

Y. Liu / Applied Mathematical Modelling 36 (2012) 2890–2899 2895

xa ¼4=9 a ¼ 01=9 a ¼ 1;2;3;41=36 a ¼ 5;6;7;8

8><>: ; ð46Þ

and

Gij ¼k

10c4

@qui

@xjþ @quj

@xi

� �; ði – jÞ ð47Þ

Gxx ¼ �Grr ¼ �k

10c4

@qux

@x� @qur

@r

� �: ð48Þ

4. Numerical results and discussion

4.1. Case of a channel flow

Inlet boundary, at x = 0. At the inlet, we use the velocity in Ref. [22]:

@uð0; rÞ@r

¼ r2l0

dpdx� r0

l0þ 2

l0

ffiffiffiffiffiffiffiffiffiffiffiffi� rr0

2

rdpdx; ð49Þ

where r is the distance from the channel center, r0 is 10�6, and dP/dx is pressure gradient. Here, P0 � PL determines thehydrodynamic force acting on this pipe, P0 is the pressure at the inlet, PL is the pressure at the outlet, and P0 = 0.33,PL = 0.29 [45]. The pressure gradient dP/dx at each segment of the physical boundary is obtained by linear extrapolation

@P=@x ¼ xðP0 � PlÞ=L; ð50Þ

here, L is the tube length.Outlet boundary, x = L. Since the flow occurs in a long vessel, for computational purpose, here, the outlet boundary is set

at x = L = 6R.An outflow boundary condition at x = L is prescribed as follows:

@uðL; r; tÞ@x

¼ 0: ð51Þ

when x = L, u(L,r, t) = u(L � dx,r, t).

Fig. 3. Schematic diagram of bounce-back boundary conditions.

2896 Y. Liu / Applied Mathematical Modelling 36 (2012) 2890–2899

Bounce-back boundary, r = 0, r = R. For the walls, we have implemented a simple bounce-back scheme in which particleshitting the walls simply reverse their direction toward the fluid as shown in Fig. 3. The bounce back rule seems good enoughfor this benchmark and is suitable for complex arterial systems, since it is adaptive, simple and fast.

We compare our results for the velocity profile of the lattice Boltzmann Bi-viscosity model with that of the latticeBoltzmann Casson’s rheology model [22,32]. The Casson’s viscosity model l is given

l ¼ð ffiffiffiffiryp þ

ffiffiffiffiffiffigj _cjp

Þ2

j _cj ; for r > ry;

1; otherwise;

8<: ð52Þ

where g is a constant viscosity, r is shear stress, ry is the so-called yield stress, and _c is the shear strain rate. In the literature

[22], j _cj ¼ rslc2

s qdt, where sl is the relaxation time, and r ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPð1Þij Pð1Þij

q, where Pð1Þij ¼

Pamaeaieajðfa � f ð0Þa Þ, ma are weights asso-

ciated with the lattice directions and m0 = 1 by definition. In our simulation, the artery width D is set to 40 lattice sites, andthe yield stress r0 is 10�6, which is the same as the literature [22]. Fig. 4 shows that the velocity profile ux of the lattice Boltz-mann Bi-viscosity model fits very well with that of the lattice Boltzmann Casson’s model, and in the literature [22], thenumerical solver of the lattice Boltzmann Casson’s model fits with the expected analytical profile. However, in the latticeBoltzmann Casson’s model, only when r > ry, the model is valid, otherwise, the viscosity is infinite. The author introduceda ceiling viscosity and cutoff viscosity to produce a relaxation time. In this paper, a non-equilibrium phase is added to theequilibrium distribution function in order to adjust the viscosity coefficient. It is easy to verify that the velocity and the pres-sure gradients are continuous at the interface of the Newtonian and non-Newtonian parts [41]. This rheology model avoidsthe error caused by selecting the cutoff viscosity.

Fig. 5 shows the velocity profiles and the wall shear stresses at Re = 30, Re = 200 and Re = 1000. The vessel diameter is 40lattice nodes and the length is 120 lattice nodes. As these figures show, when the Reynolds number Re = 30, the flows exhibitthe well-known parabolic velocity profile, the corresponding wall shear stress curve is shown as Fig. 5(b) at Re = 30. Thevelocity profiles show the non-Newtonian behavior with the Reynolds number increasing, as shown in Fig. 5(a). In Ref.[46], it gives the difference of the velocity profile between the Newton fluids and the shear-thinning fluids. SeeingFig. 5(b), the corresponding wall shear stresses tend to zero because of the smoothness of the velocity profiles. Accordingto the function of shear stress rij = loui/oxj, here, l = uD/Re, the wall shear stress will decrease with the increasing of Rey-nolds number , and these changes agree with the changes of the wall shear stress in the Ref. [39].

4.2. Case of a stenotic channel flow

In the Ref. [39], the author gives the flow difference between blood fluids and the Newtonian fluids. As shown Fig. 6, wehave carried out a number of lattice Boltzmann simulations for steady flow in the stenotic artery at Re = 30, Re = 200 andRe = 1000. The streamlines fields of the blood fluids and the Newtonian fluids obtained by LBM are shown for three Reynoldsnumbers. For the Newtonian fluids, the viscosity m = 1.0, and the relaxation time s ¼ 1=2þ m=c2

s dt; here the speed of sound c2s

is 1/3. In this simulation, the vessel diameter is 40 lattice nodes and the length is 120 lattice nodes. A bounce-back boundarycondition is imposed at the walls, corresponding to a non-slip wall boundary condition. The flow velocities at the inlets areimposed according to Eq. (49), and the outflow velocities are computed according to Eq. (51). The pressure at any locationcan be measured with Eq. (50). After a certain time in computing, the flow evolves to a steady state. As shown in these fig-ures, the typical semi-circumferential flow features are all captured, including the higher-pressure regions upstream, thelower-pressure regions downstream, vortexes near the bottom of the stenotic part, and detached flow at the end of the ste-notic part. We may observe the streamlines difference between the blood fluids and the Newtonian fluids. As the Reynoldsnumber Re = 30, the streamlines of two kinds of fluids are almost identical, and with the increasing of Reynolds number , thestreamlines show a difference. For the blood fluids the vortex intensity is relatively large behind the narrow area. The flow

Fig. 4. Comparison of the Bi-viscosity model profiles with the Casson’s model profiles. Re = 30.

Fig. 5. (a) Comparison of velocity profiles at different Reynolds numbers. (b) Comparison of the wall shear stress at different Reynolds numbers. Theparameters are: lattice size 120 � 40, dx = 1/120, dt = dx/c, t = 2000dt, c = 2, e = dt.

Fig. 6. The blood fluids streamlines and the Newtonian fluids streamlines in the stenotic artery for Re = 30, Re = 200, Re = 1000. The parameters are: latticesize 120 � 40, dx = 1/120, dt = dx/c, t = 2000dt, c = 2, e = dt. The streamlines number is 40.

Y. Liu / Applied Mathematical Modelling 36 (2012) 2890–2899 2897

difference of two kinds of fluids is similar with that given in the Ref. [39], and in Ref. [39], the streamlines of two kinds offluids are computed by the finite difference method (FDM).

In Fig. 7(a), the pressure comparison of the blood flows at Re = 30 by the Marker and Cell method (MAC) of FDM and theresults of the LBM is given. In this figure, the computation region is x = 80dx, 0 < r < 40dr. We find that the result of theLBM agreed with that of the FDM. By further comparison, we give the absolute error in Fig. 7(b). The absolute error is de-fined as ~e ¼ jPLð80; rÞ � PFð80; rÞj, here PL is the pressure of the LBM result and PF is the numerical result of the FDM. Thescope of absolute error is (�8.6 � 10�5, 8.6 � 10�5). Thus, from Fig. 7(b), we can see that the LBM results agree well withthe FDM.

Fig. 7. (a) Is the comparison between pressure of the Marker and Cell method (MAC) of the finite difference method (FDM) and the results of LBM versusvertical the position r at x = 80dx. (b) is the error curve of the finite difference method and the LBM.

2898 Y. Liu / Applied Mathematical Modelling 36 (2012) 2890–2899

5. Conclusions

In this paper, a lattice Boltzmann method for blood flow inside the stenotic artery, which is of fundamental interest andpractical importance in science as well as in medical care, is proposed. As the same time, we get the results as follows:

(1) A realistic rheology model and the control dynamics equations of blood flows have been developed and validated inthe simple cases.

(2) Qualitative agreements are reached when we compare the velocity profiles of the lattice Boltzmann Bi-viscosity modelwith that of the lattice Boltzmann Casson’s model, however, the lattice Boltzmann Bi-viscosity model can still workwell whether the fluid is yield or not, so this model is more suitable to study blood flow problems.

(3) To demonstrate the potential of this approach and its suitability for the application, based on the validate model, asexample, the blood flow inside the stenotic artery is investigated, and the numerical results are similar to the otherinvestigator’s finding, thus showing the potential of this numerical algorithm for future studies of blood flow.

In our future work, the effect of RBC needs to be analyzed, and the method should be efficiently extended to threedimensions.

Acknowledgements

This work is supported by the 985 Project of Jilin University, and the ChuangXin Foundation of Jilin University (No.2004CX041). We would like to thank Prof. Guangwu Yan, Dr. Jianying Zhang, Dr. Yinfeng Dong, Dr. Huimin Wang, and Dr.Bo Yan for their many helpful suggestions.

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