Electronic Journal of Differential Equations,
Conference 03 (1999), pp. 108-118.

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

Author: Chunqing Lu (Southern Illinois Univ.,  Edwardsville, IL, USA) 
 
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.