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

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 domain \(\epsilon \Omega ^s\). 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 \(\epsilon ^{-1}\), \(\log \epsilon\)