Thermodynamical analysis and constitutive equations for a mixture of viscous Korteweg fluids

A complete thermodynamical analysis for a binary mixture of viscous Korteweg fluids with two velocities and two temperatures is developed. The constitutive functions are allowed to depend on the diffusion velocity and the specific internal energies of both constituents, together with their first gradients, on the symmetric part of the gradient of barycentric velocity as well as on the mass density of the mixture and the concentration of one of the constituents, together with their first and second gradients. Compatibility with the entropy principle is analyzed by applying the extended Liu procedure, and a complete solution of the set of thermodynamical restrictions is recovered in three space dimensions. Finally, the equilibrium configurations are investigated, and it is proved that no restrictions arise on the admissible phase boundaries. The theoretical results here provided may serve as a basis for experimental and/or numerical investigations, in particular for determining the surface levels of phase boundaries at equilibrium and making a comparison with the experimental profiles.