Singular perturbations for the two-dimensional viscous flow problem

Abstract

This paper is concerned with the validity of the method of matched inner and outer expansions for treating the two-dimensional steady flow of a viscous, incompressible fluid past an arbitrary obstacle. In particular, it is shown that the force exerted on the obstacle by the fluid admits the asymptotic representation:

$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{F} = 4\pi \{ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A} _1 (logR)^{ - 1} + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A} _2 (logR)^{ - 2} \} + {\rm O}((logR)^{ - 3} )$$

as the Reynolds number R→0⁺, where A_i's are constant vectors which are the same as those obtained by the matching procedure formulated previously by the authors. This asymptotic representation formula agrees also, up to terms of O((log R)⁻²), with the expression from the solution of the complete Oseen boundary-value problem; in fact, it is seen that these calculations are as accurate as those from the Oseen solution, since the Oseen solution is no longer a valid approximation to the solution of the viscous flow problem for terms of order higher than (log R)⁻². Proofs involve simple layer potentials and asymptotic estimates for solutions of various linearized Navier-Stokes equations.

The work of this author was supported in part by the Alexander von Humboldt Foundation, Germany.

The work of this author was supported in part by the National Science Foundation under Grant MCS-800-1944.