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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 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 in the 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 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 $\lambda\in(0,+\infty)$, we derive a Navier--Stokes--Cahn--Hilliard system with nonlinear convection and advective transport of the phase field at the macroscopic scale, 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.

Submission history

From: Amartya Chakrabortty Dr [view email]
[v1] Wed, 24 Dec 2025 13:39:34 UTC (52 KB)
[v2] Sat, 14 Mar 2026 11:39:36 UTC (51 KB)
[v3] Wed, 17 Jun 2026 16:16:14 UTC (48 KB)