Property-preserving numerical approximations of a Cahn-Hilliard-Navier-Stokes model with variable densities and degenerate mobility

In this paper, we present a new computational framework using coupled and decoupled approximations for a Cahn-Hilliard-Navier-Stokes model with variable densities and degenerate mobility. In this sense, the coupled approximation is shown to conserve the mass of the fluid, preserve the point-wise bounds of the density and decrease an energy functional. In contrast, the decoupled scheme is presented as a more computationally efficient alternative but the discrete energy-decreasing property can not be assured. Both schemes are 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 contrasting both approaches are conducted. In particular, results for a convergence test, a simple qualitative comparison and some well-known benchmark problems are shown.