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 ${\epsilon }^{\alpha }(\alpha >3)$, where $\epsilon >0$ 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 ${\lambda }_{\epsilon }\to \lambda >0$ as $\epsilon \to 0$, the limit system coincides with the original NSCH system; if ${\lambda }_{\epsilon }\to 0$ as $\epsilon \to 0$, 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.