VOLUME 79, NUMBER 14 PHYSICAL REVIEW LETTERS 6OCTOBER 1997
A Lattice BGK Model for Viscoelastic Media
Yue-Hong Qian
1
and Yue-Fan Deng
2
1
Department of Applied Physics, Columbia University, New York, New York 10027
2
Department of Applied Mathematics and Statistics, SUNY-Stony Brook, Stony Brook, New York 11794
(Received 11 December 1996)
A two- and three-dimensional lattice Bhatnager-Gross-Krook model is proposed to simulate
viscoelastic media. Large-scale equations are derived using the Chapman-Enskog expansion. A simple
linear relationship between the parameter E which is introduced to characterize the elastic behavior and
the transverse velocity is obtained. Numerical simulations further confirm the analytical predictions.
[S0031-9007(97)04196-3]
PACS numbers: 83.50.Fc, 05.50. + q
We investigate a new lattice-based model for simulat-
ing viscoelastic media with the Bhatnager-Gross-Krook
(BGK) approximation [1]. A decade ago, lattice gas
models for hydrodynamics were introduced by Frisch
et al. [2–5] and much research effort has led to encourag-
ing progress [6–9]. These models have several appealing
advantages over conventional methods for complex flows
such as multiphase flows [10,11], flows in porous media
[12], and reaction-diffusion systems [13]. The inconve-
niences such as statistical noise [14,15] and non-Galilean
invariance [4,5,16] in the original lattice gas models have
been overcome by the use of lattice Boltzmann equations
[17–20]. Another difficulty in lattice gas models, the
existence of extra conserved quantities [21–24] which are
not physically meaningful, was recently solved by using
a fractional propagation procedure [25]. The introduc-
tion of the BGK approximation to lattice-based models
[20,26–28] simplified the complicated collision pro-
cesses, increased computing efficiency, and offered new
flexibility. These models have been quite effective for
solving fluid problems. However, they have not been paid
enough attention to in tackling solid or fluid-solid prob-
lems [29–31]. The study of viscoelastic behavior of ma-
terials done by d’Humières and Lallemand [32] based on a
lattice gas model successfully produced transverse waves
and the propagation speed. Their prediction was also con-
firmed nicely by numerical simulations. However, their
model requires an internal variable to describe “legal”
collisions responsible for propagation of transverse
waves and is clumsy in extension of their model to three
dimensions.
The equation used in lattice BGK models has the
following form [20,26–28]:
f
i
x 1 c
i
, t 1 1 f
i
x, t 1v f
e
i
x, t 2 f
i
x, t ,
(1)
where f
i
is the particle distribution density with pre-
defined discrete velocity c
i
and v the relaxation parameter
0 #v# 2 and i runs over the discrete velocity set.
A suitable equilibrium f
e
i
leading to the Navier-Stokes
equations is [20,26]
f
e
i
t
p
r
∑
1 1
c
ia
u
a
c
2
s
1
c
i a
c
i b
2 c
2
s
d
ab
2c
4
s
u
a
u
b
∏
,
(2)
where t
p
is a lattice weight factor (the index p is equal
to c
2
i
) and c
s
a constant. Greek subscripts a and b
denote the space directions in Cartesian coordinates. The
hydrodynamic quantities p and u are defined as
r
X
i
f
i
X
i
f
e
i
, r u
X
i
f
i
c
i
X
i
f
e
i
c
i
.
(3)
We propose a simple lattice BGK model for viscoelas-
tic materials. Unlike the internal variable needed in [32],
a parameter E is introduced to characterize the elastic-
ity. Choosing an equilibrium distribution is much more
flexible in lattice BGK models than in the lattice gas
models. In fact, adding a term to the equilibrium (2) to
model the transverse waves is convenient,
f
e
i
t
p
r
∑
1 1
c
i a
u
a
c
2
s
1
c
i a
c
i b
2 c
2
s
d
ab
2c
4
s
u
a
u
b
1 E
c
i a
c
i b
2 c
2
s
d
ab
2c
4
s
u
a
I
b
1 u
b
I
a
2
∏
.
(4)
The last term involving E stems from the fact that one
simple elastic behavior is wavelike propagation, like a
vibrating string. The vector symbol I
a
appearing in the
above equilibrium is defined as
I
a
1, ;a x, y, z .
The Chapman-Enskog expansion [16,20,33] is used to
derive the large-scale dynamical equations up to second
order of the Knudsen number,
≠
t
r1≠
a
ru
a
0, (5)
2742 0031-9007 97 79(14) 2742(4)$10.00 © 1997 The American Physical Society