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On Benjamin's theory of conjugate vortex flows

Published online by Cambridge University Press:  28 March 2006

L. E. Fraenkel
L. E. Fraenkel
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Cambridge
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Abstract

Benjamin (1962) introduced the idea that a given ‘primary’ swirling flow, with cylindrical stream surfaces, may have associated with it ‘conjugate flows’, also swirling and cylindrical, which in a certain sense are equivalent to the primary one. He deduced that, in cases where such conjugate flows exist and where the primary flow cannot support standing waves of small amplitude, the conjugate flow nearest the primary one (a) can support such waves, and (b) has a ‘flow force’ greater than that of the primary flow. In the present paper these two results are proved rigorously by a method which differs from Benjamin's.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 28 , Issue 1 , 12 April 1967 , pp. 85 - 96
Copyright
© 1967 Cambridge University Press

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References

Benjamin, T. B. 1962 Theory of the vortex breakdown phenomenon J. Fluid Mech. 14, 593.Google Scholar
Bolza, O. 1961 Lectures on the Calculus of Variations. New York: Dover.
Burkill, J. C. 1956 The Theory of Ordinary Differential Equations. Edinburgh and London: Oliver and Boyd.
Coddington, E. A. & Levinson, N. 1955 Theory of Ordinary Differential Equations. New York: McGraw-Hill.
Squire, H. B. 1956 Rotating fluids. Art. in Surveys in Mechanics (ed. Batchelor and Davies). Cambridge University Press.