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

Cohen, D. S. and Fokas, A. and Lagerstrom, P. A. (1978) Proof of some asymptotic results for a model equation for low Reynolds number flow. SIAM Journal on Applied Mathematics, 35 (1). pp. 187-207. ISSN 0036-1399. doi:10.1137/0135015.

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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.

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Additional Information:© 1978 Society for Industrial and Applied Mathematics. Received by the editors August 29, 1977. Original research in 1966-67 was supported in part by the U.S. Air Force under Contract AF-AFOSR-338-65. Recent research supported in part by the National Science Foundation under Grant MCS 75-05414.
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Air Force Office of Scientific ResearchAF-AFOSR-338-65
National Science FoundationMCS 75-05414
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ID Code:12661
Deposited On:18 Dec 2008 02:46
Last Modified:08 Nov 2021 22:31

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