Elaborating equations of state of a liquid and its vapor for two-phase flow models

Some two-phase flow models have shown an excellent ability for the resolution of a wide class of applications ranging from interface problems to mixtures with several velocities. These models account for waves propagation (acoustic and convective) and consist in hyperbolic systems of partial differential equations. In this context, each phase is compressible and necessitates the knowledge of an appropriate pure fluid equation of state. The literature abounds in equations of state (Van der Waals for example) that consider the phases as a mixture and not as a separated phases flow in thermodynamical non-equilibrium, which makes them unsuited to such models. Moreover, their formulation leads to ill-posed problems for thermodynamic states inside the saturation dome (speed of sound squared is negative). In the present approach, each fluid is governed by a `Stiffened Gas' EOS (3). Its particularly simple analytical form allows explicit mathematical calculations of important flow relations which are at the centre of theoretical analysis and building of modern numerical methods (acoustic properties, Riemann problems, reactive Riemann solvers) while retaining with a high accuracy the main physical properties of the matter (attractive and repulsive molecular effects). The determination of the corresponding parameters is complexified when the liquid is in presence of its vapor. In this case, the EOS parameters of each phase are strongly linked. The determination of the analytical forms of the EOS and their associated coefficients for miscible and non-miscible fluids is the subject of this article.