Rigorous estimates for the 2D Oseen–Brinkman transmission problem in terms of the Stokes–Brinkman expansion

Abstract

AbstractIn this paper we consider the problem of two‐dimensional (2D) steady flow of a viscous incompressible fluid at low Reynolds number past a porous body of arbitrary shape whose boundary is a closed Lipschitz curve. Assuming that the flow inside the porous body is governed by the Brinkman equation, we consider indirect layer potential representations corresponding to Brinkman, Stokes and Oseen systems. We show that the boundary integral equations of the Oseen–Brinkman coupling turn out to be a regular perturbation of those of the Stokes–Brinkman coupling. This allows us to prove that the difference between the first terms of the matched asymptotic expansions of the Stokes system and of the Oseen system is of the order |ln Re|−3 uniformly in any compact region of ℝ2. Since it is not known whether the coupled Brinkman–Navier–Stokes 2D interface problem has a solution and whether it can be approximated (if exists) by the Brinkman–Oseen transmission problem, these estimates can be considered as a first step in the complete analysis of 2D low Reynolds number flow past a porous body. Copyright © 2010 John Wiley & Sons, Ltd.