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

TL;DR

This paper analyzes a complex fluid model in a perforated domain, proving existence of solutions and deriving effective macroscopic models through homogenization, revealing different behaviors depending on the capillarity strength.

Contribution

It establishes the existence of weak solutions for the Navier-Stokes-Cahn-Hilliard system in perforated domains and derives two distinct homogenized models based on capillarity regimes.

Findings

Existence of weak solutions for fixed microscopic scale.

Homogenized models differ for vanishing and balanced capillarity.

Convergence of microscopic free energy to a homogenized energy.

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 $\Omega_p^\varepsilon\subset\mathbb{R}^3$. 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 $\lambda^\varepsilon>0$ depends on the microscopic scale $\varepsilon>0$. The analysis consists of two main parts. First, for each fixed $\varepsilon>0$, we prove the 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 $\lambda^\varepsilon$). Second, we perform the periodic homogenization for the perforated setting, a limit $\varepsilon\to0$. Depending on the limit value $\lambda$ of the capillarity strength $\lambda^\varepsilon$, we obtain two distinct effective models: (i) in the vanishing capillarity regime $\lambda=0$, the limit system is of Stokes-Cahn-Hilliard type, with no macroscopic convection or advection; (ii) in the balanced regime $\lambda\in(0,+\infty)$, we derive a Navier-Stokes-Cahn-Hilliard system with nonlinear convection and advective transport of the phase field at the macroscopic scale. Finally, we establish the convergence of the microscopic free energy to a homogenized energy functional satisfying an analogous dissipation law.