Phase separation in quasi-incompressible Cahn–Hilliard fluids

The paper provides a scheme for phase separation and transition by accounting for diffusion, dynamic equations and consistency with thermodynamics. The constituents are incompressible and hence the mass density of the mixture is determined by the concentration of a constituent. Such an approximation is realistic in many circumstances such as in the equilibrium between ice and water or in many mixtures of fluids. The mass densities of the constituents are independent of temperature. The evolution of concentration is described by the standard equation for mixtures but the balance of energy and entropy of the mixture are stated as for a single constituent. However, due to the non-simple character of the mixture, an extra-energy flux is allowed, in addition to the heat flux. Also motion and diffusion effects are considered by letting the stress in the mixture have additive viscous terms and, remarkably, the chemical potential contains a quadratic term in the (shear) stretching tensor. As a result a whole set of evolution equations is set up for the concentration, the velocity, and the temperature. A maximum theorem is proved which implies that the concentration of the mixture has values from 0 to 1 as is required from the physical standpoint.