A generalization of the Navier-Stokes equations to two-phase flows

A modified Allen-Cahn equation is combined with the compressible Navier-Stokes system. We show that, after a modification of the stress tensor, the second law of thermodynamics is valid for the resulting equations. We give a physical motivation of this alteration of the stress tensor and compare the new equations with the well known phase-field approach. The model can be used to describe cavitation in a flowing liquid.