BGK source terms for out-of-equilibrium two-phase flow models

A modified form of BGK source terms is proposed for modeling two-phase flows with thermodynamical disequilibrium. The novelty is that three independent timescales allow to manage the return to the thermodynamical equilibrium while remaining in agreement with the second law of thermodynamics. This is achieved thanks to the definition of a "local-in-time" equilibrium state which tends towards the asymptotic equilibrium state when time increases. The thermodynamical paths of the system are then modified with respect to the classical BGK source terms used for two-phase flow modeling, and the relaxation process of the system towards the asymptotic equilibrium state can be defined with additional degrees of freedom. In a numerical point of view, both the classical and the modified BGK source terms have advantages. The choice between these two forms strongly depends on the numerical strategy used to perform simulations.