Homogenization of the Navier-Stokes-Cahn-Hilliard system in the small-hole regime

This paper investigates the homogenization of the 3D Navier--Stokes--Cahn--Hilliard (NSCH) system in domains containing a large number of solid obstacles (named holes).

Each hole has diameter of order $\varepsilon^{\alpha}(\alpha>3)$, where $\varepsilon > 0$ denotes the small length scale for inter-hole separation.

Both viscosity and mobility depend on the phase-field variable.

We establish two distinct asymptotic regimes: if the capillary strength $\lambda_\varepsilon\to \lambda>0$ as $\varepsilon\to 0$, the limit system coincides with the original NSCH system; if $\lambda_\varepsilon\to 0$ as $\varepsilon\to 0$, the scaled velocity, phase field and chemical potential converge to a weak solution to a Stokes--Cahn--Hilliard (SCH) system.

To the best of our knowledge, this work constitutes the first rigorous homogenization analysis for evolutionary NSCH flows with phase-dependent viscosity and mobility under the subcritical hole scaling.