Fluid flow - Binary flow
We use a Rothman-Keller type interface. Even though not the most used model to handle binary fluid, it has been found to be very handy for implementing multicomponents. Its main advantage over other interfaces models resides in that it does not need any embedded physics such as interface tracking and that it remains a purely local model. This crucial property really eased the implementation.
The Gunstensen interface
Recalling the standard densities of probability f's defined above in (I), the Gunstensen model uses R's and B's corresponding now to a density of red (R) and blue (B) fluid. On a pure node (red or blue), the fluid undergoes a normal LBGK collision. On an interfacial node (where blue and red fluid coexist), the new R's and B's collide similarly but through their sum - colourless f's (f=R+B) are collided. This says the LBGK collision model is well established, 'the collision of the red and blue fluid at the interface will follow the same correct hydrodynamics'.
After having collided the f's (sum of red and blue), a segregation process is applied to separate the two species. A re-colouring process maximises the colour flux across the interface, minimising the diffusion of the red fluid into the blue and vice versa (practically, it achieves it with a great efficiency). This step is very straight forward and the core is contained in a look up table allocating the order of re-colouring. This interfacial model taken alone does not have any surface tension embedded and would be of very low interrest and utility.
Surface tension is imposed by an additional term after the collision step, while treating with colour-blind densities. Two strategies exist.
The Guntensen surface tension method
Guntensen designed a way to impress surface tension to his model. As mentioned previously, it has to be forced in the algorithm as an additional step. It is added to the collided colourless densities before the re-colouring process (see previous section). The Guntensen approach is a very pragmatic one: the surface tension of a drop induces a diminution in the droplet's interfacial area which induces a pressure step across the interface (c.f. the Laplace law). From this statement, it is reasonable to assume that contracting artificially the interfacial area of a droplet not subjected to any surface tension would indeed produce the required interfacial pressure step and therefore surface tension.
Guntensen's approach consist therefore in depleting the links parallel to the interface and feeding to the links perpendicular to it through the following perturbation step:
where A is the surface tension parameter,
is the angle with the ith link, 
is the angle of the colourfield (assumed to be perpendicular to the interface), and cc is a modulator ensuring constant forcing throughout the interfacial domain.
The Lishchuk surface tension method
Lishchuk approach to impose surface tension to the Guntensen's interface is the reverse engineer from Guntensen's original method. It consists in forcing the inner pressure of the droplet to build up with respect to its surrounding though a net bodyforce acting on the interface, from the surrounding into the drop.
This pressure step causes the interfacial length to decrease slightly and produces the wanted surface tension as well. The body force imposed on the interface is as followed:
where F is the forcing term, s is the surface tension parameter, div(n) is the local curvature, rN is the interfacial function defined as follow: rN=(rB-rR)/(rB+rR).
It is important to note that the Lishchuck method provides much better drop's isotropy and reduced micro-current activity. This enhanced behaviour is probably due to its second order gradients quantities, that makes it less local than the Gunstensen method.