A decoupled numerical method for two-phase flows of different densities and viscosities in superposed fluid and porous layers

In this article we consider the numerical modeling and simulation via the phase field approach of two-phase flows of different densities and viscosities in superposed fluid and porous layers. The model consists of the Cahn-Hilliard-Navier-Stokes equations in the free flow region and the Cahn-Hilliard-Darcy equations in porous media that are coupled by seven domain interface boundary conditions. We show that the coupled model satisfies an energy law. Based on the ideas of pressure stabilization and artificial compressibility, we propose an unconditionally stable time stepping method that decouples the computation of the phase field variable, the velocity and pressure of free flow, the velocity and pressure of porous media, hence significantly reduces the computational cost. The energy stability of the scheme effected with the finite element spatial discretization is rigorously established. We verify numerically that our schemes are convergent and energy-law preserving. Ample numerical experiments are performed to illustrate the features of two-phase flows in the coupled free flow and porous media setting.