Energy integral of the Stokes flow in a singularly perturbed exterior domain

Abstract

We consider a pair of domains $\Omega^{b}$ and $\Omega^{s}$ in $\mathbb{R}^n$ and we assume that the closure of $\Omega^{b}$ does not intersect the closure of $\epsilon \Omega ^s$ for $\epsilon \in (0,\epsilon _0)$. Then for a fixed $\epsilon \in (0,\epsilon_0)$ we consider a boundary value problem in $\mathbb{R}^n \setminus (\Omega ^b \cup \epsilon \Omega ^s)$ which describes the steady state Stokes flow of an incompressible viscous fluid past a body occupying the domain $\Omega^{b}$ and past a small impurity occupying the $\epsilon \Omega ^s$. The unknown of the problem are the velocity field u and the pressure field $p$, and we impose the value of the velocity field $u$ on the boundary both of the body and of the impurity. We assume that the boundary velocity on the impurity displays an arbitrarily strong singularity when $\epsilon$ tends to $0$. The goal is to understand the behaviour of the strain energy of $(u, p)$ for $\epsilon$ small and positive. The methods developed aim at representing the limiting behaviour in terms of analytic maps and possibly singular but completely known functions of $\epsilon$, such as $\epsilon ^{-1}$, log $\epsilon$.