Admissible equations of state for immiscible and miscible mixtures

This paper addresses the construction of admissible Equations of State (EoS) for compressible two-phase ows. We investigate two approaches. In the first one, the mixture is treated as a single uid with a complex thermodynamic. Most of the time the available EoS are determined experimentally and are often incomplete EoS, i.e. we know only the pressure as a function of the volume and the temperature. We present here a general framework to compute a complete EoS based on such an incomplete EoS. In the second approach, each phase is depicted by its own EoS. Following the Gibbs formalism, the mixture entropy is the sum of the phasic entropies which achieves its maximum at equilibrium. Depending on the miscibility of the mixture, one gets different geometrical properties on the resulting mixture entropy. Eventually we address the coupling of mixture EoS with the dynamic of the uid. Homogeneous Equilibrium and Relaxation Models (HEM and HRM) are introduced for an immiscible and a miscible two-phase mixture. Hyperbolicity is ensured taking advantage of the concavity properties of the mixture entropies.