Thermodynamically consistent spatial discretization of the one-dimensional regularized system of the Navier–Stokes–Cahn–Hilliard equations

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This paper is devoted to the construction of a thermodynamically consistent (with non-increasing total energy on the discrete level) semi-discrete approximation (continuous in time and discrete in space) for the one-dimensional regularized system of the Navier–Stokes–Cahn–Hilliard equations that describes a compressible isothermal flow of two-phase binary fluid with surface effects. Thermodynamical consistency is achieved using special form of momentum equation with surface force term written in so called potential form. Results of numerical study are presented.