On the homogenization of the Stokes problem in a perforated domain

TL;DR

This paper studies the asymptotic behavior of the Stokes equations in a perforated domain with many small holes, showing they can be approximated by a Stokes-Brinkman model as the number of holes increases.

Contribution

It provides a rigorous homogenization result for the Stokes problem in perforated domains with a large number of small, identical spherical holes.

Findings

Solution approximates a Stokes-Brinkman problem as holes increase

No concentration in hole distribution is assumed

Asymptotic analysis confirms the validity of the homogenized model

Abstract

We consider the Stokes equations on a bounded perforated domaincompleted with non-zero constant boundary conditions on the holes. We investigate configurations forwhich the holes are identical spheres and their number N goes to infinity while their radius1/N tends to zero. We prove that, under the assumption that there is no concentrationin the distribution of holes, the solution is well approximated asymptotically by solving aStokes-Brinkman problem.