Phase separation in lava flow

In this paper we propose a model to study the phenomenon of phase separation during lava flow. Lava is considered as a mixture of two incompressible fluids with different density, in that the mass density of the mixture is determined by the concentrations of the two constituents. We consider as state variables the order parameter, describing the difference in concentration of the fluids, the velocity of the mixture and the absolute temperature. We assume that the order parameter satisfies a Cahn-Hilliard equation, where the chemical potential depends on the velocity and we model lava as a Bingham fluid whose apparent viscosity and yield stress increase exponentially as temperature decreases, according to experimental data. The heat equation provides the evolution equation for temperature. We prove that this model is consistent with the principles of thermodynamics.