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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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