Singular perturbations for the exterior three-dimensional slow viscous flow problem

Abstract

In this paper the rigorous justification of the formal asymptotic expansions constructed by the method of matched inner and outer expansions is established for the three-dimensional steady flow of a viscous, incompressible fluid past an arbitrary obstacle. The justification is based on the series representation of the solution to the Navier-Stokes equations due to Finn, and it involves the reductions of various exterior boundary value problems for the Stokes and Oseen equations to boundary integral equations of the first kind from which existence as well as asymptotic error estimates for the solutions are deduced. In particular, it is shown that the force exerted on the obstacle by the fluid admits the asymptotic representation F = A 0 + A 1 Re + O( Re 2 ln Re −1) as the Reynolds number Re → 0 +, where the vectors A 0 and A 1 can be obtained from the method of matched inner and outer expansions.