The miscible binary fluid proposed by Flekkøy [59] is not suited to our aim of applying the lattice Boltzmann model to interfacial waves since we require two immiscible fluids. It also has a non-Galilean invariant factor associated with it. The colour model [56] and the local interaction model [60] both simulate immiscible fluids. In the colour model the fluid separation is driven by the local `colour' differences between the fluids. The method used involves maximising the scalar product of the colour gradient and the colour flux. This does indeed produce an interface between the two fluids and Laplace's law is seen to be obeyed. This colour based scheme does not directly mimic the physics of phase separation since there is no consideration of the thermodynamics of the process. This means that although the fluids separate there is no guarantee that they separate an a manner akin to a real fluid. The approach of Shan et al. [60] is based on the assumption that phase separation is produced by microscopic interactions on the scale of the lattice sites. This interaction introduces a momentum change into the Boltzmann equation. This model has also been shown to facilitate phase separation, however, like the colour model there is no certainty that the microscopic interactions are in fact mimicking real phase separation. The free energy model [9, 63, 38] does consider the thermodynamics of the problem. The approach to equilibrium is governed by the free energy which enters the model through the equilibrium distribution function. The model therefore simulates the phase separation of a real fluid mixture. Unfortunately the liquid-gas model is not Galilean invariant so its application is limited. The binary fluid model, on the other hand, does not suffer from a lack of Galilean invariance and is isotropic [38]. This makes it a suitable choice for the interfacial wave simulations considered in chapters 7 and 8.