Colour Model

The first model proposed for simulating immiscible, binary fluids [56] is based on the colour lattice gas model of Rothman and Keller [28]. As in the lattice gas model a colour gradient is defined

where and are the distribution functions of the red and blue fluids respectively on link i. The distribution function of the whole fluid on link i, , is the sum of the two colour distribution functions

An angle is also defined

The fluid is then evolved in the following manner.

1. The fluid distribution function is found for each link at each site: .
   
2. The total fluid distribution function is collided, but not propagated, according to the lattice Boltzmann equation

    

   to give the new fluid distribution function f'.

3. At all sites were , where is a small number, the distribution function is perturbed to an alternative value such that

   

   where is the angle makes with the x-axis and A is a constant which sets the strength of the surface tension. If then f'' = f'.

4. The new distribution functions for the red and blue fluid, and are then found by solving the maximisation problem

   

   where

   

   and and are all possible functions which satisfy the conservation of mass

   

   and the conservation of colour

   

5. The red and blue distribution functions are then propagated along the lattice

   

   and

   

The collision term in equation (4.68) will depend on the lattice Boltzmann model being used. In reference [56] a collision term similar to the linearised collision operator [40, 41] is employed. A single relaxation time lattice Boltzmann model has also be used [57, 58]. The surface tension produced in this manner can be shown [56] to satisfy Laplace's equation.