Fourth Mississippi State Conference on Differential Equations and Computational Simulations,
Electron. J. Diff. Eqns., Conf. 03, 1999, pp. 108-118.

On matched asymptotic analysis for laminar channel flow with a turning point

Chunqing Lu

Abstract:
This paper presents a formal analysis of the asimptotic behaviour of solutions of type III for the Berman equation
$$ \epsilon f^{iv}=ff'''-f'f'' ,\quad f(0)=f''(0)=f'(1)=f(1)-1=0\,, $$
where $f$ describes a laminar flow in a channel with porous walls. A solution has a nonlinear turning point $(1-\Delta )$, i.e.\ $f(1-\Delta) = 0$ for some $\Delta(\epsilon)$. It is shown that
$$ f(\eta )\sim -\frac{1-\Delta }{\pi \Delta }\sin \frac{\pi \eta }{1-\Delta }, $$
as
$\epsilon \to 0^{+}$, for $\eta \in [0,1-\Delta )$ where $\Delta $ satisfies
$$ \frac{\Delta }{\epsilon } e^{\Delta/\epsilon }\sim \frac{1}{2e\pi^{9}\epsilon ^{8}}. $$

Published July 10, 2000.
Math Subject Classifications: 34B15, 34E05, 34E20.
Key Words: singular perturbations, turning point, laminar flow, transcendental terms.

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Department of Mathematics and Statistics
Southern Illinois University at Edwardsville
Edwardsville, IL 62026 USA
e-mail: clu@siue.edu
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