Shan et al. [60, 61, 62] consider a fluid with S different components on a regular lattice with b links in D-dimensional space. The direction of the b links are given by the vectors , i = 1, ... b and each link has length unity. The distribution functions of the S components on link at site at time t is given by , i = 0, ..., b, a = 1, ..., S and is described by the Boltzmann equations
The collision term has the form
where each component has a single relaxation time . Here is the equilibrium distribution of the ath component on link at site at time t and is given by where and is the `equilibrium' velocity of the ath component which has still to be defined,
and is a positive constant. If we set to a common value for each component,
then the equilibrium distribution in equation (4.88) is simply that of S ideal gases, see equation (4.45). To incorporate inter-component interactions Shan et al. [60] introduce a nonlocal interaction with potential
where is a Green's function and , the `effective mass', is a function of the density of the ath component, . G is restricted to act only on nearest neighbours and so can be simplified to
The force acting on the ath component can be found, by summing over all the components and all the neighbouring sites, and is
The rate of change of momentum of the ath component induced at each site by the additional interactions is given by
This momentum change can be combined with the momentum change of each component due to the collisions at each site. This gives the equilibrium velocity
Combining equations (4.86) and (4.87) and summing over all directions gives
Substituting the expression for given in equation (4.88) we see that
The change in momentum at each site is found by combining equations (4.86) and (4.87), multiplying by and summing over all components and all directions,
where is given by
The conservation of mass, equation (4.96), and the momentum equation,
can be Taylor expanded and a Chapman-Enskog expansion performed. This gives the `diffusion' equation [62]
and the Navier-Stokes equation [62, 60, 44]
where the kinematic shear viscosity is , the bulk viscosity is , the pressure is and is defined by
and is the average of the fluid velocity before and after the inter-particle force acts. Summing equation (4.100) over a gives the usual continuity equation for the whole fluid
as expected. It can also be shown [62] that for a two component system with the following diffusion equation is satisfied,
where is a function of the deviation of the density from its equilibrium value. A binary fluid can be simulated using this model when S = 2. If the effective mass and the strength of the inter-component interaction are suitably chosen the two components are observed to separate [60]. One component forms circular drops inside the other component and the density difference across the boundary of the drops is seen to obey Laplace's Law. When S = 1 a single component fluid is modelled. This fluid is seen to phase separate into a liquid and gaseous phase with surface tension between the two phases, again for a suitable choice of and [61]. Full details of the properties of the binary fluid model and the liquid-gas model are given in references [60, 62] and [60, 61] respectively.