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On the non-existence of periodic neutral-wave solutions to a complex-valued periodic differential equation

Part of: Incompressible viscous fluids

Published online by Cambridge University Press:  26 February 2010

A. G. Walton
A. G. Walton
Affiliation:
Department of Mathematics, Imperial College of Science, Technology & Medicine, 180 Queen's Gate, London SW7 2BZ.
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Abstract

The equation

where ℱ is a certain complex-valued function of the given real periodic function λ, is studied analytically and numerically. The equation is motivated physically by a boundary-layer stability problem in which λ represents the skin-friction of the undisturbed basic flow profile. It is proved that no periodic neutral solutions exist for any attached basic flow and the implications of this result for certain vortex-wave interactions are discussed.

MSC classification

Secondary: 76D10: Boundary-layer theory, separation and reattachment, higher-order effects
Type
Research Article
Information
Mathematika , Volume 43 , Issue 2 , December 1996 , pp. 371 - 381
Copyright
Copyright © University College London 1996

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References

1.Bennett, J. and Hall, P.. On the secondary instability of Taylor Görtler vortices to Tollmien Sehlichting waves in fully-developed flows. J. Fluid Mech., 186 (1988), 445–469.CrossRefGoogle Scholar
2.Eastham, M. S. P.. The spectral theory of periodic differential equations (Scottish Academic Press, 1973).Google Scholar
3.Hall, P. and Smith, F. T.. On strongly nonlinear vortex wave interactions in boundary-layer transition. J. Fluid Mech. 227 (1991). 641–666.CrossRefGoogle Scholar
4.Magnus, W. and Winkler, S.. Hill's equation (Interscience, 1966).Google Scholar
5.Smith, F. T.. Instability of flow through pipes of general cross-section, part 1. Malhematika, 26 (1979), 187–210.CrossRefGoogle Scholar
6.Walton, A. G.. Theory and computation of three-dimensional nonlinear effects in pipe flow transition. Ph.D. thesis (University of London, 1991).Google Scholar
7.Walton, A. G.Bowles, R. I. and Smith, F. T.. Vortex/wave interaction in separating flows. Eur. J. Mech., B/Fluids, 5 (1994), 629–655.Google Scholar
8.Walton, A. G. and Smith, F. T.. Properties of strongly nonlinear vortex/Tollmien-Schlichting wave interactions. J. Fluid Mech., 244 (1992), 649–676.CrossRefGoogle Scholar