Navier-Stokes-Cahn-Hilliard system in a $3$D perforated domain with free slip and source term: Existence and homogenization

Abstract

We study a diffuse--interface model for a binary incompressible mixture in a periodically perforated porous medium, described by a time-dependent Navier--Stokes--Cahn--Hilliard (NSCH) system posed on the pore domain $Ω_p^\varepsilon\subset\mathbb{R}^3$. The microscopic model involves a variable viscosity tensor, a non-conservative source term in the Cahn--Hilliard equation, and mixed boundary conditions: no-slip on the outer boundary and Navier slip with zero tangential stress on the surfaces of the solid inclusions. The capillarity strength $λ^\varepsilon>0$ depends on the microscopic scale $\varepsilon>0$. The analysis consists of two main parts. First, for each fixed $\varepsilon>0$ we prove existence of a weak solution on a finite time interval $(0,T)$ and derive a priori estimates that are uniform with respect to $\varepsilon$ (and $λ^\varepsilon$). Second, we perform the periodic homogenization for the perforated setting in the limit $\varepsilon\to0$. Depending on the limit value $λ$ of the capillarity strength $λ^\varepsilon$, we obtain two distinct effective models: (i) in the vanishing capillarity regime $λ=0$, the limit system decouples completely into a standalone linear Stokes system for the velocity--pressure pair and a standalone Cahn--Hilliard system with source term $G$ for the phase field and chemical potential, with no macroscopic convection, advection, or capillary coupling between the two; (ii) in the balanced regime $λ\in(0,+\infty)$, we derive a Navier--Stokes--Cahn--Hilliard system with nonlinear convection and advective transport of the phase field at the macroscopic scale, coupled through a capillary forcing term. Finally, we establish the convergence of the microscopic free energy to a homogenized energy functional satisfying an analogous dissipation law.