Consider the motion of particles of S different components
on a regular lattice in D-dimensional space. The particles of the
th
component have the molecular mass of
.
The distribution function
is the amount of
th
particles at node
,
time t, and velocity
where a=0, ..., b. The evolution of the particle distribution
function
satisfies the Lattice-Boltzmann equation
where
is the relaxation time that controls the rate of approach to equilibrium.
The equilibrium distributions can be represented as:
where
,
,
)
for model D3Q19 in three-dimension;(
,
,
) for model D2Q9 in two-dimension, and
defined as
The macroscopic density
,
mass density
and fluid velocity
for each fluid component
are
In the presence of the interaction between fluid component (e.g.multi-phase fluid) and external force (e.g. the gravity), the new momentum at each site for each component then becomes
where
is the new velocity will be used in Eqs.(2-4).
is the external body force, and
is
Here it is assumed that the nearest-neighbor interaction in four-dimension face-center hypercubic(FCHC), that corresponds to a potential that couples nearest and next nearest neighbors interactions in the D3Q19 and D2Q9 lattice model. For D3Q19 model in three-dimension (Nicos S. Martys and Hudong Chen, Physical Rev E, vol 53, p. 743, 1996), we have,
For D2Q9 model in two-dimension, we have,
where G is a constant. The fluid-fluid interaction can be modeled by
using
with
and
for
so that
The macroscopic total mass density
and the kinematic viscosity
is
where
is the concentration of each fluid component.