  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.
