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

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 $epsilon o 0^{+}$, for $eta in [0,1-Delta )$ where $Delta $ satisfies $$ frac{Delta }{epsilon } e^{Delta/epsilon }sim frac{1}{2epi^{9} epsilon ^{8}}. $$

Keywords

• Singular perturbations
• turning point
• laminar flow
• transcendental terms.