Reactive Flow around Spherically Evolving Particles in Critical and Supercritical Dilute Regimes

Abstract

We study a coupled Stokes-reaction-diffusion-advection system in a three dimensional domain perforated by a large number of evolving spherical inclusions with radii of order ${\epsilon }^{\alpha }$ for $\alpha \in (1,3]$. Their centers are separated on the order of $\epsilon$ but are not assumed to form a periodic lattice. The inclusions grow or shrink through an interfacial adsorption-desorption mechanism, so that the evolution of the geometry is coupled to the Stokes-reaction-diffusion-advection system. We first establish well-posedness on arbitrary large time intervals for sufficiently small $\epsilon$. We then derive quantitative homogenization limits in terms of the limiting empirical measure describing the spatial distribution and size of the inclusions. Its evolution is governed by a continuity equation in the radius variable, and the convergence of the empirical measures is controlled in the $2$-Wasserstein distance. In the critical case, $\alpha =3$, the Stokes system converges to a Brinkman system, while non-vanishing concentration boundary layers modify the microscopic exchange law for the reaction-diffusion-advection equation. For $\alpha \in (1,3)$, the limit is of Darcy type and the original exchange law is retained. The homogenization results are quantitative and provide explicit error estimates.

64 pages, Keywords: Homogenization, evolving microstructures, dilute perforated domains, reactive transport, Stokes-Brinkman-Darcy limits, interfacial exchange