On a New Spatial Discretization for a Regularized 3D Compressible Isothermal Navier–Stokes–Cahn–Hilliard System of Equations with Boundary Conditions

We construct a new spatial finite-difference discretization for a regularized 3D Navier–Stokes–Cahn–Hilliard system of equations. The system can be attributed to phase field type models and describes flows of a viscous compressible isothermal two-component two-phase fluid with surface effects; the potential body force is also taken into account. In the discretization, the main sought functions are defined on one and the same mesh, and an original approximation of the solid wall boundary conditions (homogeneous with the discretization of equations) is suggested. The discretization has an important property of the total energy dissipativity allowing one to eliminate completely the so-called spurious currents. The discrete total mass and component mass conservation laws hold as well, and the discretization is also well-balanced for the equilibrium solutions. To ensure that the concentration C remains within the physically meaningful interval (0, 1), the non-convex part of the Helmholtz free energy is taken in a special logarithmic form (the Flory–Huggins potential). The speed of sound can depend on C that leads to different equilibrium mass densities of the “pure” phases. The results of numerical 3D simulations are also presented including those with a gravitational-type force. The positive role of the relaxation parameter is discussed too.