Homogenization of evolutionary Stokes–Cahn–Hilliard equations for two-phase porous media flow

Abstract

We consider homogenization of a phase-field model for two-phase immiscible, incompressible porous media flow with surface tension effects. The pore-scale model consists of a strongly coupled system of time-dependent Stokes-Cahn-Hilliard equations. In the considered model the fluids are separated by an evolving diffuse interface of a finite width, which is assumed to be independent of the scale parameter ε. We obtain upscaled equations for the considered model by a rigorous two-scale convergence approach.

Homogenization of a coupled incompressible Stokes-Cahn-Hilliard system modeling binary fluid mixture in a porous medium

“…The terms containing ε are bounded and the limits converge to 0. Hence, we get =⇒ µ We follow the existence of a pressure P 1 ∈ L ∞ (S; L 2 0 (Ω; L 2 # (Y p ))) and two-scale convergence results as in [7] for the last step of our proof. (5.23)…”

“…) n and ξ ∈ C ∞ 0 (S) and proceed as in [7]. Then, using lemma (5.1) and lemma (5.2) we obtain for ε → 0 Next, we pass ε → 0 in the two-scale sense.…”

“…Phase-field approach is a popular tool for the modeling and simulation of multiphase flow problems, cf. [7,25,21,8,5]. The diffused interface and Cahn-Hilliard formulation of a quasi-compressible binary fluid mixture allows for topological changes of the interface, cf.…”

“…Equations (1.3a), (1.3c)-(1.3d) represent the unsteady Stokes equations for incompressible fluid and Cahn-Hilliard equations, respectively, cf. [7].…”

The effect of root exudates on rhizosphere water dynamics

“…Equation (2.1c) ensures that the mass of both the water and air is conserved. The Cahn-Hilliard model has been used here as it has previously been homogenized allowing us to build on existing theory [8,[26][27][28]. Here we extend from the previous application to a fuel cell in [26,27] to consider fluid flow in soil.…”