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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

References

[1] Agarwal, R. P. and O’Regan, D., Singular integral equations arising in Homann flow. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 9(2002), no. 4, 481–488.Google Scholar
[2] Acheson, D. J., Elementary Fluid Dynamics. Oxford Univ. Press, New York, 1990.Google Scholar
[3] Brown, S. N. and Stewartson, K., On the reversed flow solutions of the Falkner-Skan equations. Mathematika 13(1966), 1–6.Google Scholar
[4] Coppel, W. A., On a differential equation of boundary-layer theory. Phil. Trans. Roy. Soc. London, Ser. A 253(1960), 101–136.Google Scholar
[5] Craven, A. H. and Peletier, L. A., On the uniqueness of solutions of the Falkner-Skan equation. Mathematika 19(1972), 129–133.Google Scholar
[6] Hartman, P., Ordinary Differential Equations. Classics in Applied Mathematics 38. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002.Google Scholar
[7] Hartman, P., On the existence of similar solutions of some boundary layer problems. SIAM J. Math. Anal. 3(1972), 120–147.Google Scholar
[8] Hastings, S. P., An existence theorem for a class of nonlinear boundary value problems including that of Falkner and Skan.. J. Differential Equations 9(1971), 580–590.Google Scholar
[9] Hastings, S. P., Reversed flow solutions of the Falkner-Skan equations. SIAM J. Appl. Math. 22(1972), 329–334.Google Scholar
[10] Hastings, S. P. and Troy, W. C., Osillating solutions of the Falkner-Skan equation for negative β. SIAM J. Math. Anal. 18(1987), no. 2, 422–429.Google Scholar
[11] Libby, P. A. and Liu, T. M., Further solutions of the Falkner-Skan equation. AIAA J. 5(1967), no. 5, 1040–1042.Google Scholar
[12] Na, T. Y., Computational Methods in Engineering Boundary Value Problems. Mathematics in Science and Engineering 145. Academic Press, New York, 1979.Google Scholar
[13] Riley, N. and Weidman, P. D., Multiple solutions of the Falkner-Skan equation for flow past a stretching boundary. SIAM J. Appl. Math. 49(1989), no. 5, 1350–1358.Google Scholar
[14] Schlichting, H. and Gersten, K., Boundary-Layer Theory. Eighth edition. Springer-Verlag, Berlin, 2000.Google Scholar
[15] Schrader, K., A generalization of the Helly selection theorem. Bull. Amer. Math. Soc. 78(1972), 415–419.Google Scholar
[16] Tam, K. K., A note on the existence of a solution of the Falkner-Skan equation. Canad. Math. Bull. 13(1970), 125–127.Google Scholar
[17] Wang, J., Gao, W. and Zhang, Z., Singular nonlinear boundary value problems arising in boundary layer theory. J. Math. Anal. Appl. 233(1999), no. 1, 246–256.Google Scholar
[18] Weyl, H., On the differential equations of the simplest boundary-layer problems. Ann. of Math. 43(1942), 381–407.Google Scholar
[19] Yang, G. C., Existence of solutions to the third-order nonlinear differential equations arising in boundary layer theory. Appl. Math. Lett. 16(2003), no. 6, 827–832.Google Scholar
[20] Yang, G. C., A note on f′′′ + f f′′ + λ(1 – f′2) = 0 with λ ∈ (–½, 0) arising in boundary layer theory. Appl. Math. Lett. 17(2004), no. 11, 1261–1265.Google Scholar
[21] Yang, G. C. and Li, J., A new result on f′′′ + f f′′ + λ(1 – f′2) = 0 arising in boundary layer theory. Nonlinear Func. Anal. Appl. 10(2005), no. 1, 117–122.Google Scholar