Property-preserving numerical approximation of a Cahn–Hilliard–Navier–Stokes model with variable density and degenerate mobility

In this paper, we present a new computational framework to approximate a Cahn–Hilliard–Navier–Stokes model with variable density and degenerate mobility that preserves the mass of the mixture, the pointwise bounds of the density and the decreasing energy. This numerical scheme is based on a finite element approximation for the Navier–Stokes fluid flow with discontinuous pressure and an upwind discontinuous Galerkin scheme for the Cahn–Hilliard part. Finally, several numerical experiments such as a convergence test and some well-known benchmark problems are conducted.