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Mathematics and Computers in Simulation 72 (2006) 141–146

Lattice Boltzmann simulation of the dispersionof aggregated particles under shear flows

T. Inamuro∗, T. Ii

 Department of Aeronautics and Astronautics, and Advanced Research Institute of Fluid Science and Engineering,

Graduate School of Engineering, Kyoto University, Kyoto 606-8501, Japan

Available online 3 July 2006

Abstract

The lattice Boltzmann method (LBM) for multicomponent immiscible fluids is applied to simulations of the deformation and

breakup of a particle-cluster aggregate in shear flows. In the simulations, the solid particle is modeled by a droplet with strong

interfacial tension and large viscosity. The van der Waals attraction force is taken into account for the interaction between the

particles. The ratio of the hydrodynamic drag force to cohesive force,  I , is introduced, and the effect of  I  on the aggregate defor-

mation and breakup in shear flows is investigated. It is found that the aggregate is easier to deform and to be dispersed when  I  is

over 100.

© 2006 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords:   Lattice Boltzmann method; Particle simulation; Dispersion

1. Introduction

The dispersion of small particles in a liquid is important for making new functional materials such as ceramics,

polymers, and electronic products. Usually, small particles are easy to be aggregated by an attraction force between the

particles. Thus, it is difficult to disperse a large number of particles uniformly in a liquid. However, the characteristics of 

the aggregated particles in fluid flows have been unclear. In the present paper, we numerically investigate the dispersion

of aggregated particles under shear flows. From a numerical point of view, this subject is a moving boundary problem

and so there are some difficulties in dealing with many moving particles in a liquid, though a few numerical methods

have been proposed [7,2,1].

In order to overcome the difficulties, we use the lattice Boltzmann method (LBM) for multicomponent fluidswith the same density [4,6]. In the LBM, it does not track interfaces, but can maintain sharp interfaces without any

artificial treatments. Also, the LBM is accurate for the mass conservation of each component fluid. Making use of 

these advantages, we apply the LBM to the simulation of aggregated particles under shear flows. In the simulation the

particle is represented by a hard droplet with large viscosity and strong surface tension. In addition, colored droplets

are introduced to avoid merging of droplets.

∗ Corresponding author. Tel.: +81 75 753 5791; fax: +81 75 753 4947.

 E-mail address: inamuro@kuaero.kyoto-u.ac.jp (T. Inamuro).

0378-4754/$32.00 © 2006 IMACS. Published by Elsevier B.V. All rights reserved.

doi:10.1016/j.matcom.2006.05.022

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142   T. Inamuro, T. Ii / Mathematics and Computers in Simulation 72 (2006) 141–146 

In the following sections, we present a numerical method for simulating particles in a liquid and then apply the

method to the simulation of the behaviors of aggregated particles under shear flows. The calculated results are arranged

with a parameter which is the ratio between shear rate and inter-particle force.

2. Numerical method

Non-dimensional variables are used as in   [5]. The lattice kinetic scheme (LKS)   [3], which is an extension

method of LBMs, is used for the formulation of the method. In the LKS, macroscopic variables are calcu-

lated without particle velocity distribution functions, and thus the scheme can save computer memory, since

there is no need to store the particle velocity distribution functions. In addition, in order to represent many hard

droplets which cannot merge into bigger droplets, we introduce colored order parameters to make different col-

ored droplets. Note that the color is physically meaningless and is used only for distinguishing each droplet

from the others. In the present paper, the Stokes flow is assumed, since the diameter of particle is very small

(e.g., 1m).

The fifteen-velocity model with particle velocities ci  (i = 1, 2, . . . , 15) is used in the present paper. The physical

space is divided into a cubic lattice, and the colored order parameter φl(x, t ) (l = 1, 2, . . . , N  ) where N  is the number

of colors, the pressure  p(x, t ) and the velocity  u(x, t ) of whole fluid at the lattice point  x  and at time  t  are computed

as follows:

φl(x, t + t ) =

15i=1

f eq

li   (x− cix,t ),   (1)

p(x, t + t ) =1

3

15i=1

geqi   (x− cix, t ),   (2)

u(x, t + t ) =

15i=1

cigeqi   (x− cix,t ),   (3)

where f eqli   and geq

i   are the equilibrium distribution functions, x  a spacing of the cubic lattice, and  t  is a time stepduring which the particles travel the lattice spacing.

The equilibrium distribution functions in Eqs. (1)–(3) are given by

f eqli   = H iφl + F i

p0(φl) − κf φl∇ 

2φl −κf 

6|∇ φl|

2+ 3Eiφlciαuα + Eiκf Gαβ(φl)ciαciβ,   (4)

geqi   = Ei

3p + 3ciαuα + Ax

∂uβ

∂xα

+∂uα

∂xβ

ciαciβ

+ EiκgGαβ(φl)ciαciβ + 3Eiciαx

N m=1

N l=1

f vlmαl,

(5)

where

E1  =  29

, E2  = E3  = E4  = · · · = E7  =  19

,

E8  = E9  = E10  = · · · = E15  =  172 ,

H 1  = 1, H 2  = H 3  = H 4  = · · · = H 15  = 0,

F 1  =  −7

3   , F i  = 3Ei(i = 2, 3, 4, . . . , 15),

(6)

and

Gαβ(φ) =9

2

∂φ

∂xα

∂φ

∂xβ

−3

2

∂φ

∂xγ 

∂φ

∂xγ 

δαβ,   (7)

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T. Inamuro, T. Ii / Mathematics and Computers in Simulation 72 (2006) 141–146    143

with α, β, γ  = x ,y ,z (subscripts α, β, and γ  represent Cartesian coordinates and the summation convention is used).

In the above equations, δαβ is the Kronecker delta, κf  is a constant parameter determining the width of the interface, κg

is a constant parameter determining the strength of the surface tension, and  A  is a constant parameter related to fluid

viscosity. In Eq. (4), p0(φ) is given by

p0

(φ) = φT φ

1

1 − bφ− aφ2,   (8)

where a, b, and T φ  are free parameters determining the maximum and minimum values of the order parameter  φ. The

van der Waals attraction force between particles is represented by the last term of Eq. (5) in which the attraction force

per unit mass from the mth droplet to the lth droplet f vlm  is given by [8]

f vlm  =

Ah

πD3

  D2rlm

(r2lm − D2)2

 +D2

r3lm

−2rlm

r2lm − D2

 +2

rlm

 xGm − xGl

rlm

,   for   rlm  ≥ 1.005D

0,   otherwise

(9)

and l  is unity if  φl  is inside a droplet and zero if  φl  is outside a droplet. In Eq.  (9)  D  is the diameter of the particle,

xGl and xGm  are the positions of the centers of the  lth and mth droplets, rlm  = |xGm − xGl|, and Ah  is the Hamarkerconstant.

The viscosity µ  and the surface tension σ s are given by

µ =

1

6 −

2

9A

x   (10)

and

σ s  = κg

   ∞−∞

∂φ

∂ξ 

2

dξ,   (11)

with ξ  being the coordinate normal to the interface.

3. Results and discussion

Aggregated particles with the diameter D are in a liquid inside a rectangular domain, and at t  = 0 the top and bottom

walls begin to move in the  x -direction with velocities  uw  and  −uw, respectively (see Fig. 1). Both the particles and

the liquid have the same density. The periodic boundary condition is used on the other sides of the domain. Four cases

with 2, 6, 18, and 36 particles are calculated. For the cases with 2, 6, and 18 particles, the same number of colored

order parameters are used, but for the case with 36 particles, 18 colored order parameters are used in order to save

computation time. The domain is divided into a 240 × 100 × 100 cubic lattice. The viscosity ratio of the droplet to the

surrounding liquid is η = µd/µc  = 10. We chose a = 9/49, b = 2/21, and T φ  = 0.55 in Eq. (8). The other parameters

are fixed at κf  = 0.01(x)2 and κg  = 0.01(x)2.

Fig. 1. Computational domain.

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144   T. Inamuro, T. Ii / Mathematics and Computers in Simulation 72 (2006) 141–146 

Fig. 2. Deformation of aggregate with 18 particles for I  = 2. (a) t ∗ = 8.3; (b) t ∗ = 16.5; (c) t ∗ = 24.8; (d) t ∗ = 33.0 where t ∗ = 2uwt/Lz.

In order to classify calculated results, we introduce a parameter  I  which is the ratio of hydrodynamic drag force to

cohesive force defined by

I  =µc(uwD/Lz)D

Ah/D

=µcuwD3

AhLz

.   (12)

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T. Inamuro, T. Ii / Mathematics and Computers in Simulation 72 (2006) 141–146    145

Fig. 3. Deformation of aggregate with 36 particles for I  = 1000. (a) t ∗ = 8.4; (b) t ∗ = 26.0; (c) t ∗ = 42.0; (d) t ∗ = 50.0 where t ∗ = 2uwt/Lz.

The calculated results with 18 particles for I  = 2 are shown in Fig. 2. In this case, the arrangement of the particles is

a little changed, but the aggregated particles rotates together and are not separated.  Fig. 3 shows the calculated results

with 36 particles for I  = 1000. It is seen that as the time goes on, the aggregated particles deform into an ellipsoidal

shape and then are separated into two aggregates. The state of the dispersion of the aggregate is measured by the

standard deviation of the lengths between the center of the aggregation and those of the particles.  Fig. 4 shows the

standard deviation σ  against the parameter I  at t ∗ = 50 for the four cases with 2, 6, 18, and 36 particles. It is found that

in spite of the number of particles the aggregates are dispersed when  I  is over 100. Note that the deviation  σ  changes

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146   T. Inamuro, T. Ii / Mathematics and Computers in Simulation 72 (2006) 141–146 

Fig. 4. Standard deviation  σ  vs.  I  at t ∗ = 50 for the cases with 2, 6, 18, and 36 particles.  σ 0  is the standard deviation at  t ∗ = 0.

little where  I > 1000, since the length of  Lx  is limited and thus particles going beyond the left side of the domain

come into the domain through the right side of the domain.

4. Concluding remarks

The simulations of the deformation and breakup of particle-cluster aggregates have been performed by using the

lattice Boltzmann method for multicomponent immiscible fluids. We have found that the parameter  I  is dominant for

the behavior of the aggregate and that the aggregate is deformed and dispersed when I  is over 100. The present method

has two advantages: one is that the solid boundary condition is not required, and the other is that the computation

time is proportional not to the number of the particles but to that of the colored order parameters. On the other hand,

a disadvantage of the present method is that the interface between the particle and the liquid has a non-zero thickness

(about three times x), and the thick interface has some unexpected effects on calculated results. In addition, we have

to compare the calculated results with experimental data, but very few reliable data are available at present. Furtherwork remains in order to verify the results.

We are now computing the behavior of the aggregate with 100 particles with 20 colored order parameters. In future

work, we will compute 1000 particles by using parallel computers.

Acknowledgements

This work is partly supported by the Grant-in-Aid (No. 16560145) for Scientific Research from the Ministry of 

Education, Culture, Sports, Science, and Technology in Japan and by a Research Program from NEDO (New Energy

and Industrial Technology Development Organization) in Japan.

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