Homogenization for Rigid Suspensions with Random Velocity-Dependent Interfacial Forces

TL;DR

This paper develops a homogenization framework for suspensions of rigid particles in viscous fluids with velocity-dependent interfacial forces, deriving effective macroscopic equations and viscosity in the limit of microstructure size.

Contribution

It introduces a homogenization method for suspensions with random velocity-dependent surface forces, using $ ext{Gamma}$-convergence to derive effective medium properties.

Findings

Weak convergence of solutions to a homogenized problem with constant coefficients

Existence of a strong $H^1$-convergence via a corrector

Effective viscosity and interfacial force effects characterized

Abstract

We study suspensions of solid particles in a viscous incompressible fluid in the presence of highly oscillatory velocity-dependent surface forces. The flow at a small Reynolds number is modeled by the Stokes equations coupled with the motion of rigid particles arranged in a periodic array. The objective is to perform homogenization for the given suspension and obtain an equivalent description of a homogeneous (effective) medium, the macroscopic effect of the interfacial forces and the effective viscosity are determined using the analysis on a periodicity cell. In particular, the solutions $\bm{u}^\e_\omega$ to a family of problems corresponding to the size of microstructure $\e$ and describing suspensions of rigid particles with random surface forces imposed on the interface, converge $H^1$-- weakly as $\e \to 0$ a.s. to a solution of the so-called homogenized problem with constant coefficients. It is also shown that there is a corrector to a homogenized solution that yields a strong $H^1$-- convergence. The main technical construct is built upon the $\Gamma$-- convergence theory.