Numerical Validation for a Stokes-Cahn-Hilliard System in a Porous Medium

Having a finite interfacial thickness, the phase-field models supply a way to model the fluid interfaces, which allows the calculations of the interface movements and deformations on the fixed grids. Such modeling is applied to the computation of two-phase incompressible Stokes flows in this paper, leading to a system of Stokes-Cahn-Hilliard equations. The Stokes equation is modified by adding the continuum force $ - c \nabla w $, where $ c $ is the order parameter and $ w $ is the chemical potential of $ c $. Similarly, the advection effects are modeled by addition of the term $ \vec{u} \cdot \nabla c $ in the Cahn-Hilliard equation. We hereby discuss how the solutions to the above equations approach the original sharp interface Stokes equation as the interfacial thickness $ \varepsilon$ tends to zero. We start with a microscopic model and then the homogenized or upscaled version to the same from author's previous work, cf. \cite{lakhmara2022}, where the analysis and homogenization of the system have been performed in detail. Further, we perform the numerical computations to compare the outcome of the effective model with the original heterogeneous microscale model.