Proof of Some Asymptotic Results for a Model Equation for Low Reynolds Number Flow

Abstract

A two-point boundary value problem in the interval $[ {\varepsilon ,\infty } ]$, $\varepsilon > 0$ is studied. The problem contains additional parameters $\alpha \geqq 0$, $\beta \geqq 0$, $0 \leqq U < \infty $, $k = {\text{real}}$. 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 $\alpha > 0$; for $\alpha = 0$ an explicit construction shows that no solution exists unless $k > 1$. A special method, is used to show uniqueness. For $\varepsilon \downarrow 0$, $k \geqq 1$, 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 $ \geqq 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.