Unconditionally Stable Numerical Methods for Cahn-Hilliard-Navier-Stokes-Darcy System with Different Densities and Viscosities


In this article we consider the numerical modeling and simulation via the phase field approach for coupled two-phase free flow and two-phase porous media flow of different densities and viscosities. 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 several domain interface conditions. It is showed that the coupled model satisfies an energy law. Then we first propose a coupled unconditionally stable finite element method for solving this model and analyze the energy stability for this method. Furthermore, 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 this decoupled scheme with the finite element spatial discretization is rigorously established. We verify numerically that our schemes are convergent and energy-law preserving. Numerical experiments are also performed to illustrate the features of two-phase flows in the coupled free flow and porous media setting.


Mathematics and Statistics


National Science Foundation, Grant DMS-1912715

Keywords and Phrases

Cahn-Hilliard-Navier-Stokes-Darcy Model; Different Densities; Energy Stability; Finite Element Method; Two-Phase Flow

International Standard Serial Number (ISSN)

1090-2716; 0021-9991

Document Type

Article - Journal

Document Version


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© 2022 Elsevier, All rights reserved.

Publication Date

01 Apr 2022