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Applications of Leray–Schauder degree theory to problems of hydrodynamic stability

Published online by Cambridge University Press:  24 October 2008

T. Brooke Benjamin
T. Brooke Benjamin
Affiliation:
Fluid Mechanics Research Institute, University of Essex
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Abstract

A previous application of the theory, to the nonlinear boundary-value problem for steady flows of a viscous fluid in a bounded domain, is first retraced in order to verify a general theorem concerning the indices of multiple solutions. Then, in §5, the bearing of the theorem on questions of the bifurcation of steady flows is discussed, and the conclusion is drawn that a transcritical form of bifurcation is virtually universal in practice. In §6 it is proved that a flow represented by a solution with index i = – 1 is necessarily unstable, and hence it appears that lack of uniqueness generally implies the existence of an unstable steady flow.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 79 , Issue 2 , March 1976 , pp. 373 - 392
Copyright
Copyright © Cambridge Philosophical Society 1976

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