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Positive Solutions of the Falkner–Skan Equation Arising in the Boundary Layer Theory

Published online by Cambridge University Press:  20 November 2018

K. Q. Lan  and
G. C. Yang
K. Q. Lan
Affiliation:
Department of Mathematics, Ryerson University, Toronto, ON, M5B 2K3. e-mail: [email protected]
G. C. Yang
Affiliation:
Department of Computation Science, Chengdu University of Information Technology, Chengdu, Sichuan 610225, P. R. China. e-mail: [email protected]
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Abstract

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The well-known Falkner–Skan equation is one of the most important equations in laminar boundary layer theory and is used to describe the steady two-dimensional flow of a slightly viscous incompressible fluid past wedge shaped bodies of angles related to $\text{ }\!\!\lambda\!\!\text{ }\!\!\pi\!\!\text{ /2}$, where $\text{ }\!\!\lambda\!\!\text{ }\,\in \,\mathbb{R}$ is a parameter involved in the equation. It is known that there exists ${{\text{ }\!\!\lambda\!\!\text{ }}^{*}}\,<\,0$ such that the equation with suitable boundary conditions has at least one positive solution for each $\text{ }\!\!\lambda\!\!\text{ }\,\ge \,{{\text{ }\!\!\lambda\!\!\text{ }}^{*}}$ and has no positive solutions for $\text{ }\!\!\lambda\!\!\text{ }\,<\,{{\text{ }\!\!\lambda\!\!\text{ }}^{*}}$ . The known numerical result shows ${{\text{ }\!\!\lambda\!\!\text{ }}^{*}}\,=\,-0.1988$ . In this paper, ${{\text{ }\!\!\lambda\!\!\text{ }}^{*}}\,\in \,[-0.4,\,-0.12]$ is proved analytically by establishing a singular integral equation which is equivalent to the Falkner–Skan equation. The equivalence result provides new techniques to study properties and existence of solutions of the Falkner–Skan equation.

Keywords

34B1634B1834B4076D10Falkner–Skan equationboundary layer problemssingular integral equationpositive solutions
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 3 , 01 September 2008 , pp. 386 - 398
Copyright
Copyright © Canadian Mathematical Society 2008

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