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On the stability of Poiseuille-Couette flow: a bifurcation from infinity

Published online by Cambridge University Press:  20 April 2006

S. J. Cowley  and
F. T. Smith
S. J. Cowley
Affiliation:
Mathematics Department, University College, Gower Street, London WC1E 6BT
F. T. Smith
Affiliation:
Mathematics Department, University College, Gower Street, London WC1E 6BT
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Abstract

The linear and weakly nonlinear stability of Poiseuille–Couette flow is considered for various values of the relative wall velocity 2uw, An account is given first of the asymptotic upper and lower branches of the linear neutral curve (s), followed by their disappearance, as uw is increased. Two main (and one minor) neutral curves are found to exist for smaller O(1) (or lesser) values of uw, then one for moderate O(1) values of uw, and none for larger O(1) values of uw. The cut-off velocity at which each main neutral curve disappears is determined, and in each case the whole neutral curve for uw just below the cut-off value is determined in closed form. Secondly, weakly nonlinear solutions are found to bifurcate subcritically from the neutral curve for uw just below cut-off, but to ‘bifurcate from infinity’ just above cut-off. This identifies a minimum threshold amplitude at the entry to the regime where no linear neutral curve exists.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 156 , July 1985 , pp. 83 - 100
Copyright
© 1985 Cambridge University Press

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