Relaxation Two-Phase Flow Models and the Subcharacteristic Condition

The subcharacteristic condition for hyperbolic relaxation systems states that wave velocities of an equilibrium system cannot exceed the corresponding wave velocities of its relaxation system. This condition is central to the stability of hyperbolic relaxation systems, and is expected to hold for most such models describing natural phenomena.

In this paper, we study a hierarchy of two-phase flow models. We consider relaxation with respect to volume transfer, heat transfer and mass transfer. We formally verify that our relaxation processes are consistent with the first and second laws of thermodynamics, and present analytical expressions for the wave velocities for each model in the hierarchy. Through an appropriate choice of variables, we prove directly by sums-of-squares that for all relaxation processes considered, the subcharacteristic condition holds for any thermodynamically stable equation of state.