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

Abstract

We study a time-dependent Navier--Stokes--Cahn--Hilliard system for a binary incompressible mixture in a periodically perforated domain $Ω_p^\varepsilon\subset\mathbb{R}^3$. The obstacles have diameter of order $\varepsilon^α$, with $α>3$, and mutual distance of order $\varepsilon$. Hence $\varepsilon^α/\varepsilon^3\to0$, which corresponds to the subcritical dilute regime. The model includes a periodically oscillating viscosity tensor, a nonconservative source term in the Cahn--Hilliard equation, no-slip conditions on the outer boundary, and free-slip conditions on the obstacle surfaces. The capillary coefficient $λ^\varepsilon>0$ depends on $\varepsilon$. For every fixed $\varepsilon>0$, we prove existence of a weak solution and derive estimates uniform in $\varepsilon$ with explicit $λ^\varepsilon$-scaling. Assuming $λ^\varepsilon\toλ\in[0,\infty)$, we derive the homogenized system on the whole domain. The subcritical obstacles leave no additional resistance term, and the cell problems are posed on the full periodic cell. The scalar correctors vanish, so the scalar diffusion operators remain unchanged, while the oscillating viscosity gives a time-dependent effective viscosity tensor. If $λ=0$, the limit decouples into an effective unsteady Stokes system and a Cahn--Hilliard system with source. If $λ>0$, the limit retains the Navier--Stokes--Cahn--Hilliard coupling, with convection, phase transport, and capillary forcing weighted by $\sqrtλ$. We also prove convergence of the time-integrated normalized microscopic energy to the corresponding macroscopic energy.