Another model was proposed by Flekkøy [59]. This model simulates
two miscible fluids moving on the same grid. There is no interaction between
the two fluids and so there is no state in which the fluids are immiscible. A
brief description of the model is given for completeness.
In this model the
total density 
and the difference in densities,
are used. Two lattice Boltzmann equations
are defined
where 
and 
are relaxation parameters which determine
the viscosity 
and the diffusion coefficient D. For a hexagonal
lattice the equilibrium distributions are defined
and
where the density 
, the momentum 
and the density difference 
are defined
The tensor 
is defined
where 
is the FHP speed of sound and G = 4.5 is a
non-Galilean
invariant factor. With these definitions the model can be shown
[59] to satisfy the convection-diffusion equation
and the Navier-Stokes equation
where the diffusion coefficient, the viscosity and the pressure are given by
and
