A regularized system of equations describing a flow of isothermal two-component mixture with diffuse interface is studied. The equation of energy balance and its corollary, i.e., the law of non-increasing of the total energy are derived under general assumptions on the Helmholtz free energy of the mixture. Necessary and sufficient conditions for linearized stability of constant solutions are obtained (in particular case). A difference approximation of the problem is constructed in the two-dimensional periodic case on a nonuniform rectangular grid. The results of numerical experiments demonstrate a qualitative well-posedness of the problem and the applicability of the criterion of linearized stabilization in the original nonlinear formulation.

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We consider the regularized 3D Navier-Stokes-Cahn-Hilliard equations describing isothermal flows of viscous compressible two-component fluids with interphase effects. We construct for them a new energy dissipative finite-difference discretization in space, i.e., with the non-increasing total energy in time. This property is preserved in the absence of a regularization. In addition, the discretization is well-balanced for equilibrium flows and the potential body force. The sought total density, mixture velocity and concentration of one of the components are defined at nodes of one and the same grid. The results of computer simulation of several 2D test problems are presented. They demonstrate advantages of the constructed discretization including the absence of the so-called parasitic currents.

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