A study of the barotropic quasi-hydrodynamic model for the two-phase mixture involving surface effects

A study of the barotropic quasi-hydrodynamic system of equations for the two-phase two-component mixture involving the surface tension and stationary potential force is accomplished. Under fairly general assumptions on the Helmholtz free energy of the mixture, the energy balance equation with non-positive energy production together with its corollary, the law of nonincreasing total energy, are derived; in particular, both the isothermal and isentropic cases are covered. Under additional assumptions, the necessary and sufficient conditions for the linearized stability of constant solutions are derived (it does not take place always).
A finite-difference approximation of the problem is constructed in the 2D periodic case for a non-uniform rectangular mesh. Approximation of the capillary stress tensor is improved in the momentum balance equation thus increasing the quality of numerical solutions.
The results of numerical experiments are presented. They demonstrate the ability of the model to ensure the qualitatively correct description for the dynamics of surface effects as well as applicability of the linearized stability criterion in the original nonlinear statement.