Proof of some asymptotic results for a model equation for low Reynolds number flow

Abstract

A two-point boundary value problem in the interval [ε, ∞], ε > 0 is studied. The problem contains additional parameters α ≥ 0, β ≥ 0, 0 ≤ U < ∞, k real. It was originally proposed by Lagerstrom as a model for viscous flow at low Reynolds numbers. A related initial value problem is transformed into an integral equation which is shown to have a unique solution by a pincer method. The integral representation is used for a simple proof of the existence of a solution of the boundary value problem for a α > 0; for α = 0 an explicit construction shows that no solution exists unless k > 1. A special method is used to show uniqueness. For ε ↓ 0, k ≥ 1, various results had previously been obtained by the method of matched asymptotic expansions. Examples of these results are verified rigorously using the integral representation. For k < 1, the problem is shown not to be a layer-type problem, a fact previously demonstrated explicitly for k = 0. If k is an integer ≥ 0 the intuitive understanding of the problem is aided by regarding it as spherically symmetric in k + 1 dimensions. In the present study, however, k may be any real number, even negative.