A Multiscale Model of Stokes–Cahn–Hilliard Equations in a Porous Medium: Modeling, Analysis and Homogenization

Abstract

A diffused interface model describing the evolution of two conterminous incompressible fluids in a porous medium is discussed. The system consists of the Cahn-Hilliard equation with Flory-Huggins logarithmic potential, coupled via surface tension term with the evolutionary Stokes equation at the pore scale. An evolving diffused interface of finite thickness, depending on the scale parameter ε separates the fluids. The model is studied in a bounded domain Ω with a sufficiently smooth boundary ∂Ω in R d for d = 2, 3. At first, we investigate the existence of the system at the micro-scale and derive the essential a-priori estimates. Then, using the two-scale convergence approach and unfolding operator technique, we obtain the homogenized model for the microscopic one.