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

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

“…In this article, we aim to perform numerical experiments on the authors' previous work, cf. [1,15]. We observe that the upscaled model has more advantages than the micro one for numerical simulations, as it takes less time than the micro-model, and it will reduce the computational cost while monitoring real-world problems.…”

“…However, the existence of such model and the homogenization from the micro to macro-scale in authors' previous work, cf. [1,15], shows that there does exist such solutions analytically. We now state the assumptions made for the analysis, weak formulation, existence and homogenization results in a nutshell for the reader's convenience.…”

“…Assume that c0 ∈ C1(Ω ε p ), u0 ∈ U(Ω ε p ) with c0 L ∞ (Ω) ≤ 1 and |m(c0)| < 1. Also, F satisfies the assumptions A1-A4 stated in section 2.1 of [15], then for any T > 0, there exists a global weak solution (u ε , c ε , w ε ) to the problem (P ε ) in the sense of definition 3.1 of [15], which satisfies…”

“…We associate a pressure p ε := ∂tP ε with each weak solution (c ε , w ε , u ε ), which satisfies (1.6a) in the distributional sense, cf. [1].…”

“…We start with a microscopic model and then the homogenized or upscaled version to the same from author's previous work, cf. [1], 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.…”