Journal of Computational Science and Technology

Journal of Computational Science and Technology

Vol.2, No.1, 2008

Numerical Simulations of Circular Particles in Parallel-plate Channel Flow Using Lattice Boltzmann Method∗ Toru HYAKUTAKE∗∗ , Takeshi MATSUMOTO∗∗∗ and Shinichiro YANASE∗∗ ∗∗

Graduate School of Natural Science and Technology, Okayama University, 3–1–1 Tsushima-naka, Okayama 700–8530, Japan E-mail: [email protected] ∗∗∗ Graduate School of Engineering Science, Osaka University, 1–3 Machikaneyama-cho, Toyonaka, Osaka 560–8531, Japan

Abstract Numerical simulations are performed using the lattice Boltzmann method for particulate suspension in a plane channel flow at low and moderate Reynolds numbers in order to investigate the blood cell behavior in microvascular flows. The simulation results of three types of particle volume fractions indicate the existence of an important relationship between the Reynolds number and the variance of the particles. When the particle volume fraction is small, it is found that the particles are concentrated between the centerline and the wall, that is, the Segr´e-Silberberg effect occurs. On the other hand, as the particle volume fraction becomes larger, this effect disappears and the variance of the particles increases. In the case of an inelastic collision, the number of the particles that flow near the wall increases and the variance of the particle distribution decreases in comparison with the case of the elastic collision. Key words : Lattice Boltzmann Method, Microvascular Flow, Blood Cells, Segr´eSilberberg Effect, Inelastic Collision

1. Introduction

∗ Received 21 Jan., 2008 (No. T1-05-0997) Japanese Original: Trans. Jpn. Soc. Mech. Eng., Vol.72, No.718, B (2006), pp.1434–1441 (Received 20 Sep., 2005) [DOI: 10.1299/jcst.2.56]

In the past, many studies have been carried out on the behavior of particulate suspension in a Poiseuille flow; this flow has attracted considerable attention because of its practical importance in mechanical, chemical, and biological engineering. Blood flow is given as a typical example. Blood must be considered as a particulate suspension of red and white blood cells and small platelets. In particular, in microcirculation, where the vessel diameter is less than several tens of micrometers, the rheological characteristic of blood is not negligible since the vessel diameter is on the same order as that of various particles(1) . Therefore, for such a microvascular flow, it is necessary to investigate the mechanical behavior of the particles in the flow and the effect of the interaction between particles in detail. Apart from these blood cells, the development of liposome-encapsulated hemoglobin (LEH) is being vigorously promoted and has been recognized as a useful alternative for red blood cells(2) ; therefore, the investigation of flow characteristics such as LEH and other cells in the microcirculation becomes more important(3), (4) . On the basis of this background, numerical simulations of circular particles in a parallel-plate channel flow at low and moderate Reynolds numbers are performed in this study in order to clarify the flow behavior of such suspended particles. There have been many studies on the flow behavior of particles suspended in a fluid. Segr´e et al.(5) have experimentally examined the movement of particles that are suspended in a tube and observed the Segr´e-Silberberg effect, in which the particles move to the equilibrium position existing between the tube axis and the tube wall. Ogino et al.(6), (7) have conducted an experiment on the spherical and disk particle flow in a circular tube and investigated the flow and heat transfer in a two-phase flow of neutrally buoyant particles and liquid in de56

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tail. On the other hand, Feng et al.(8), (9) have simulated the movement of a particle that is suspended in the plane Poiseuille flow using the finite element method, and they have reproduced the Segr´e-Sliberberg effect. For such a multiphase flow, the lattice Boltzmann method (LBM)(10), (11) is effective; recently, several studies(12) – (16) have been carried out using this method. For example, Ladd(12) has simulated the solid-fluid suspensions of particles using the LBM. Inamuro et al.(13) have numerically investigated the motions of a single line and two lines of neutrally buoyant circular particles in a fluid between flat parallel walls, and they have reproduced the Segr´e-Silberberg effect. The simulation of non-circular particles, that is, elliptic and rectangular particles, has also been performed(14), (15) . Moreover, the simulation of multi-particles in the Poiseuille flow has also been performed(16) . However, for the multi-particle simulation, the collision between particles is treated only as an elastic collision. Considering the real flow of the neutrally buoyant particles, it is expected that the particulate behaviors would change according to the coefficient of restitution and coefficient of kinetic friction between particles. Furthermore, very few studies on the particulate flow with a low Reynolds number have addressed microcirculation. Therefore, in the present study, simulations of the flow of the multi-particles that undergo inelastic collisions are performed in the wide range of 0.5 ≤ Re ≤ 500.

Nomenclature ci D Dp e f fi Fm I L M np P p Re Rp Tm u Vm ym Δt Δx ρ ρp σ τ φ Ωm

: : : : : : : : : : : : : : : : : : : : : : : : : : :

velocity vector of nine-velocity model distance between parallel plates diameter of circular particle coefficient of restitution coefficient of kinetic friction particle distribution function force acting on circular particle m inertial moment of circular particle length of plate in the computational region mass of circular particle number of particles in the computational region impulsive force between particles pressure Reynolds number radius of circular particle torque acting on circular particle m velocity of fluid velocity vector of circular particle m distance from centerline to center of circular particle m time step lattice spacing density of fluid density of circular particle stress tensor single relaxation time particle volume fraction angular velocity of circular particle m

2. Numerical method 2.1. Lattice Boltzmann method In the present study, we adopt the LBM(10), (11) for the computation of flows. The LBM is a relatively new and promising numerical scheme for simulating complex flows and has

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attracted a great deal of attention as an alternative approach to conventional Navier-Stokes equations for computational fluid dynamics. One advantage of the LBM is the simplicity of handling complex moving geometries. Therefore, its use is very effective for the analysis of multiphase fluid flows, which are addressed in the present study. For simplicity, a two-dimensional simulation is performed in the present study. In the LBM, a modeled fluid is composed of identical particles, and their velocities are restricted to a finite set of vectors. Since the present study considers a two-dimensional problem, the ninevelocity model(17) is used in the following calculations. The velocity vectors of the model are defined as follows: c0 = 0,

(1)

ci = [cos(π(i − 1)/2), sin(π(i − 1)/2)] for i = 1, 2, 3, 4, √ 11 11 ci = 2[cos(π(i − )/2), sin(π(i − )/2)] for i = 5, 6, 7, 8. 2 2

(2) (3)

The evolution of the particle velocity distribution function fi (x, t) with the velocity ci at point x and time t is computed by the following equations: fi (x + ci Δt, t + Δt) − fi (x, t) = −

 1 fi (x, t) − fieq (x, t) for i = 0, 1, . . . , 8, (4) τ

where fieq (x, t) is the equilibrium distribution function, τ is a single relaxation time, Δx is the lattice spacing, and Δt is the time step. The Bhatnagar-Gross-Krook model(18) is used for the collision terms on the right-hand side of Eq. (4). A suitable equilibrium distribution function of the model is given by Qian et al.(17) , as follows:   9 3 eq 2 (5) fi (x, t) = Ei ρ 1 + 3ci · u + (ci · u) − ui · u , 2 2 1 where E0 = 92 , Ei = 19 (i = 1, 2, 3, 4), and Ei = 36 (i = 5, 6, 7, 8). The fluid density ρ and fluid velocity u are calculated in terms of the particle velocity distribution functions by

ρ=

8 

fi ,

(6)

8 1 fi ci , ρ i=0

(7)

i=0

u=

and the pressure p is related to the density ρ as p = 13 ρ. The kinematic viscosity ν is given by   1 1 ν= τ− Δx. (8) 3 2 2.2. Movement of circular particles In the LBM, the physical space is divided into square lattices. Thus, the surface of a circular particle m is represented by square lattice grids, as shown in Fig. 1. On the boundary nodes, the unknown particle velocity distribution functions are determined by the no-slip boundary condition. The velocity Vm and angular velocity Ωm of a particle m can be obtained by solving the following equations of motion: dVm = Fm , (9) dt dΩm = Tm, I (10) dt



where M = ρp πD2p /4 and I = M D2p /8 are the mass and moment of inertia, respectively, of a particle m. Dp and ρp are the diameter and density, respectively, of a particle m. Fm and T m are the force and torque, respectively, acting on a particle m. We consider a closed surface M

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Fig. 1

Lattice grids surrounding a circular particle m and closed surface S

Fig. 2 Collision between particles m and n

S separated from the particle m and calculate the force and torque by integrating the stress tensor and momentum flux over S . 

Fm = (11) σ · n − ρu{(u − Vm ) · n} dS , S 

T m = r × σ · n − ρu{(u − Vm ) · n} dS , (12) S

where n is the unit outward normal vector on S and r is a vector from the center of the particle to the point on S . To determine the diameter of S , we calculate the force and torque by changing the value of the diameter DS and find that by using values between DS = 1.16 and 1.32Dp , we obtain almost the same force and torque. In fact, at DS < 1.28Dp , some lattice grids representing the wall is located outside S ; therefore, in the following calculation, we select DS = 1.32Dp . This circle S is approximated by the quadrature of 360 points. ρ, u, σ on S are calculated from the linearly interpolated values of fi at the point of S . We can calculate the stress tensor σ = {σαβ } as follows: σαβ = −

τ− 1 p δαβ − 2τ τ

1 8 2 

fi (ciα − uα )(ciβ − uβ )

(13)

i=0

where δαβ is the Kronecker delta. At each time step, the force and torque acting on the circular particle are obtained from Eqs. (11) and (12), respectively. The motions of the particle are then explicitly updated. 2.3. Inter-particle collision With regard to collisions between particles, we assume a binary collision, as shown in Fig. 2. We adopt Tanaka’s method(19) , which considers the coefficient of restitution and the coefficient of kinetic friction between particles as described below. We assume that particles m and n collide when surface S m touches surface S n . At this instant, assuming the circular particles as a rigid cylinder, the impulsive equations between m

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Fig. 3

Boundary condition for parallel-plate channel flow

and n are   M V∗m − Vm = P, M(V∗n − Vn ) I(Ω∗m − Ωm ) I(Ω∗n − Ωn )

(14)

= −P,

(15)

= Rp nc × P,

(16)

= Rp nc × P,

(17)

where Rp denotes the particle radius, nc denotes the normal unit vector directed from m to n, and P denotes the impulsive force exerted on m. The superscript ∗ represents the quantities determined after the collision. The impulsive force P is obtained by the coefficient of restitution e and the kinetic friction coefficient f between two particles as follows: P = Pn nc + Pt tc , 1 Pn = (1 + e)MG · nc , 2   1 Pt = min − f Pn , M | Gct | , 7

(18) (19) (20)

where G is the relative velocity of the centers of mass, Gct is the slip velocity between particle surfaces, and tc is the tangential unit vector of the slip velocity from particle n to particle m. As a consequence, we can obtain the velocity and angular velocity of particles m and n after collision from Eqs. (13)-(16). For the collisions of the particles with the wall, Tanaka’s method is also applied. 2.4. Boundary conditions Figure 3 shows the boundary conditions for the parallel-plate channel flow in the present simulation. A periodic boundary condition with a constant pressure difference Δp is used at the inlet and outlet. The Reynolds number (Re), a dimensionless parameter for characterizing the flow behavior, is defined as follows: Re =

um D ρD3 = Δp, μ 12ν2 L

(21)

where um is the mean flow velocity assuming a plane Poiseuille flow between the parallel plates, and μ is the kinematic viscosity of the fluid. In the present simulation, the pressure difference is fixed as Δp = 2.50 × 10−3 , and the simulations of various Reynolds numbers (0.5 ≤ Re ≤ 500) are performed by changing the kinematic viscosity and the number of lattice grids. The no-slip boundary condition(13), (20) is applied to the boundary nodes representing the upper and lower walls and the particle wall. The unknown distribution functions on the nodes are determined by this boundary condition. The number of particles in the computational 60

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Fig. 4

Distributions of circular particles for three different particle volume fractions

region, np , is 2 − 80. The present study employs three np values, that is, np = 8, 28, and 56, as the representative cases. Figure 4 shows the distribution of the particles for the three different np values. The particle volume fractions in the computational region, φ = np πD2S /(4LD), are 4.4, 15, and 31%, respectively. The circular particles are randomly arranged in the computational region at t = 0. After a sufficient number of time steps, the distance between the center of the particle m and the centerline, ym , is measured when the particles cross at x = L. The number of sampled particles is 2000 − 6000. As the function of the normalized value of ym , ηm (= ym /(D/2)), the average distance η¯ , the particle distribution p(η), and the variance S η2 are calculated as follows: η¯

=

p(η) = S η2

1 N

N 

ηm ,

(22)

m=1

Nη , − (η − W/2)2 }

N{(η + N 1  = (ηm − η¯ )2 , N − 1 m=1 W/2)2

(23) (24)

where N is the number of sampling particles, W is the interval width (= D/50), and Nη is the number of sampling particles in the range of η ± W/2. In the present simulation, since the results may change slightly depending upon the initial position of the particles, an ensemble average of 5 − 10 times is considered. We assume the coefficient of kinetic friction f to be 0.3. With regard to the coefficient of restitution e, two types of e, that is, e = 0 (perfectly inelastic collision) and e = 1 (perfectly elastic collision), are considered. In the case of the perfectly inelastic collision, it is possible that the particles move as a cluster after the interparticle collision. However, in the present simulation, each particle is considered separately. The normalized diameter of the circular particles, Dp /D, is set to 0.1, and the mass density of the particles is equal to that of the fluid.

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Fig. 5 Traces of particles (φ = 4.4%, e = 0, Re = 10)

Fig. 6

Traces of particles (φ = 15%, e = 0, Re = 10)

Fig. 7

Traces of particles (φ = 31%, e = 0, Re = 10)

3. Results and discussion 3.1. Comparison of particle volume fraction Figure 5 shows the traces of circular particles at φ = 4.4%, e = 0, and Re = 10. Here, the traces of only those particles whose y-coordinate of the center at t = 0 is negative are indicated. At y/D = −0.418, the particle touches the wall; therefore, there the center of

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Fig. 8

Fig. 9

Velocity distributions for three different particle volume fractions (e = 0, Re = 10)

Particle distribution from centerline for three different particle volume fractions φ (Re = 10)

the particles does not exist at y/D < −0.418. When the particle volume fraction is low, a collision between particles occurs rarely. Therefore, the force acting on the particle from the fluid is dominant; accordingly, the Segr´e-Silberberg effect is observed to a significant extent. From Fig. 5, it is found that no particles collide with the wall and they are concentrated at the equilibrium position existing between the wall and the centerline with the passage of time. On the other hand, for a high particle volume fraction, the change in the trace due to the inter-particle collision is larger than the change in the trace due to the force acting on the particles from the fluid. In the case of φ = 15% (Fig. 6), the number of inter-particle collisions increases, and some particles collide with the wall; therefore, the Segr´e-Silberberg effect disappears gradually. By a further increase in the particle volume fraction (Fig. 7),

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Fig. 10

Particle distribution from centerline at three different Reynolds numbers (φ = 15%)

some particles cross the centerline and move to y/D > 0. Figure 8 shows the velocity distributions in the x-direction for the three different particle volume fractions, where e = 0 and Re = 10. The transverse axis η indicates the normalized distance from the centerline, and the vertical axis indicates the normalized velocity. In the case of φ = 4.4%, the velocity becomes smaller than that in the other cases in the vicinity of η = 0.5, because the particles are concentrated at the equilibrium position existing between the wall and the centerline. On the other hand, at φ = 31%, the velocity becomes smaller than that in the other cases at η = 0 − 0.2, because the particles are concentrated near the centerline. The particle distribution from the centerline for three different particle volume fractions is illustrated in Fig. 9. The transverse axis η indicates the normalized distance from the centerline to the center of the particles, and the vertical axis p(η) indicates the particle distribution. The Reynolds number is 10 in all cases. The figures on the left-hand side are for e = 0 (perfectly inelastic collision), whereas those on the right-hand side are for e = 1 (perfectly elastic collision). At η = 0.836, the particles touch the wall; therefore, at η > 0.836, the center of the particles does not exist. For a low particle volume fraction (φ = 4.4%), there are few particle collisions, and the change in the particle trace due to the inter-particle collision is small. As a result, the particles easily move to the equilibrium position, which is decided by the force acting on the particles from the fluid. Therefore, the particles are concentrated at the equilibrium position existing between the wall and the centerline, and the Segr´e-Silberberg effect occurs to a significant extent. On the other hand, at φ = 15%, the Segr´e-Silberberg effect is smaller than that in the case of φ = 4.4%, and the number of particles that are distributed near the centerline (η = 0) increases. The number of particles that are suspended near the wall (η = 0.836) also increases. By a further increase in the particle volume fraction, these tendencies become more remarkable. Next, we consider the comparison between figures on the left- and right-hand sides, that is, the effect of the coefficient of restitution. In the case of

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the inelastic collision, the particles lose energy due to the collision with the wall; therefore, it is difficult to separate them from the wall. Consequently, for a high particle volume fraction, it is found that the number of particles suspended near the wall in the case of the inelastic collision increases in comparison with that in the case of the elastic collision. 3.2. Comparison of the Reynolds number We investigate the effect of the Reynolds number on the particle distribution. Figure 10 shows the particle distribution from the centerline at three different Reynolds numbers (Re = 5, 50, and 500). The particle volume fraction is 15% in all cases. The transverse axis η indicates the normalized distance from the centerline to the center of the particles, and the vertical axis p(η) indicates the particle distribution. The figures on the left-hand side are for e = 0 (perfectly inelastic collision), and those on the right-hand side are for e = 1 (perfectly elastic collision). At Re = 5, two features of the particle distribution are observed; one is the particles that flow near the wall and the other is the particles that are distributed almost uniformly at η > 0.6. Therefore, the Segr´e-Silberberg effect is not observed at all. As the Reynolds number increases, the number of particles that flow near the wall decreases and the particles are concentrated at the equilibrium position existing between the centerline and the wall (Segr´e-Silberberg effect). At Re = 500, the peak of the particle distribution moves slightly toward the centerline and the Segr´e-Silberberg effect starts to disappear again. For both the values of the coefficient of restitution, the separation of the particles from the wall is more difficult in the case of the inelastic collision than that in the case of the elastic collision; this is due to the decrease in the kinetic energy by the collision of the particles with the wall, as discussed in the previous subsection. Therefore, for all Reynolds numbers, the number of particles distributed near the wall at e = 0 increases in comparison with those at e = 1. Next, we consider the particle distribution near η = 0. At Re = 5 and 50, almost no difference is observed between the elastic and inelastic collisions, whereas, at Re = 500, the number of particles that flow near η = 0 increases for the elastic collision (e = 1). The increase in the Reynolds number causes an increase in the effect of the inertia force. As a result, the difference in the velocity after the collision greatly influences the particle distribution between the parallel plates. Accordingly, there is a significant difference in the particle distribution near the centerline. 3.3. Variance of particle distributions Finally, the relationship between the Reynolds number and the variance S η2 of the particles is shown in Fig. 11 for three different particle volume fractions. As the variance decreases, the particles are concentrated at the equilibrium position between the wall and the centerline, that is, the effect of the Segr´e-Silbereberg effect increases. At φ = 4.4%, as the Reynolds number increases, S η2 decreases, and then at Re > 20, S η2 becomes almost zero. Namely, all the particles move to the equilibrium position because the number of particles is less (np = 8). At φ = 15%, as the Reynolds number increases, the variance decreases at Re < 100; in contract, at Re > 100, the variance increases slightly as the Reynolds number increases. The reason for this behavior can be explained from Fig. 10. As shown in this figure, at Re = 50, the Segr´eSilberberg effect is observed to a significant extent. On the other hand, at Re = 500, the peak of the particle distribution approaches the centerline, and the particles tend to be dispersed. Therefore, at a high Reynolds number (Re > 100), the variance of the particle distribution increases slightly. At φ = 31%, the variance is minimum at Re = 100 − 200; however, the change in the variance is small. The comparison of the two collision types (e = 0 and 1) reveals that the variance of the inelastic collision is higher than that of the elastic one, except for the case of S η2 = 0 (φ = 4.4% and Re > 20).

4. Conclusions In the present study, the behavior of the circular particle flow in the parallel-plate channel

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Fig. 11 Relation between Reynolds number and variance of particles (S η2 )

has been simulated using the LBM. The Reynolds number is assumed in the wide range of 0.5 − 500, and the particle volume fraction is considered for three particle volume fractions, φ = 4.4, 15, 31%. Under each condition, the simulation of the perfectly inelastic collision and the perfectly elastic collision is performed. The following conclusions are obtained from simulation results. • As the particle volume fraction decreases, the particles are concentrated at the equilibrium position exiting between the wall and the centerline (Segr´e-Silberberg effect). On the other hand, as the particle volume fraction increases, the Segr´e-Silberberg effect disappears, and the variance of the particle distribution increases. Additionally, the number of the particles that flow near the wall increases. • A difference is observed in the particle distribution between elastic and inelastic collisions. In the case of the inelastic collision, the number of particles that flow near the wall increases and the variation of the particle distribution decreases in comparison with the case of the elastic collision. • The variance in the particle distribution changes according to the Reynolds number. At φ = 4.4%, the variance is almost zero at Re > 20, and the particles exist only at the equilibrium position. At φ = 15%, the variance becomes the smallest at Re = 100, and the Segr´e-Silberberg effect is observed to a significant extent. At higher particle volume fractions, the effect of variance on the Reynolds number reduces.

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( 7 ) Ogino, F., Inamuro, T., Suzuki, T. and Kagimoto, T., Flow and heat transfer of solidliquid two-phase flow including large disc particles as solid component, Kagaku Kogaku Ronbunshu Vol. 25 (1999), pp. 106-111. ( 8 ) Feng, J., Hu, H. H. and Joseph, D. D., Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 1: Sedimentation, J. Fluid Mech. Vol. 261 (1994), pp. 95-134. ( 9 ) Feng, J., Hu, H. H. and Joseph, D. D., Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 2: Couette and Poiseuille flows, J. Fluid Mech. Vol. 277 (1994), pp. 271-301. (10) Tsutahara, M., Takada, N. and Kataoka, T., Lattice Gas and Lattice Boltzmann Methods (1999), Corona. (11) Succi, S., The Lattice Boltzmann Equation (2001), Oxford. (12) Ladd, A. J. C., Numerical simulations of particulate suspensions via a discretized Boltzmann equation, J. Fluid Mech. Vol. 271 (1994), pp. 285-309. (13) Inamuro, T., Maeba, K. and Ogino, F., Flow between parallel walls containing the lines of neutrally buoyant circular cylinders, Int. J. Multiphase Fluid Vol. 26 (2000), pp. 19812004. (14) Qi, D. W., Lattice-Boltzmann simulations of particles in non-zero-Reynolds-number flows, J. Fluid Mech. Vol. 385 (1999), pp. 41-62. (15) Qi, D. W., Lattice-Boltzmann simulations fluidization of rectangular particles, Int. J. Multiphase Fluid Vol. 26 (2000), pp. 412-433. (16) Qi, D. W., Simulations of fluidization of cylindrical multi-particles in a threedimensional space, Int. J. Multiphase Fluid Vol. 27 (2001), pp. 107-118. (17) Qian, T. H., d’Humi´eres, D. and Lallemand, P., Lattice BGK models for Navier-Stokes equation, Europhys. Lett. Vol. 17 (1992), pp. 479-484. (18) Bhatnagar, P.L., Gross, E.P., and Krook, M., A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev. Vol. 94 (1954) pp. 511-525. (19) Tanaka, T., Kadono, K. and Tsuji, Y., Numerical Simulation of gas-solid two-phase flow in a vertical pipe: On the effect of inter-particle collision, T. Jpn Soc. Mech. Eng. Series B Vol. 56, No. 531 (1990), pp. 3210-3216. (20) Inamuro, T., Yoshino, M. and Ogino, F., A non-slip boundary condition for lattice Boltzmann simulations, (Erratum: 8, 1124), Phys. Fluid Vol. 7 (1995), pp. 2928-2930.

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