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A non-linear instability theory for a wave system in plane Poiseuille flow

Published online by Cambridge University Press:  29 March 2006

K. Stewartson  and
J. T. Stuart
K. Stewartson
Affiliation:
Department of Mathematics, University College, London
J. T. Stuart
Affiliation:
Department of Mathematics, University College, London Department of Mathematics, Imperial College, London
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Abstract

The initial-value problem for linearized perturbations is discussed, and the asymptotic solution for large time is given. For values of the Reynolds number slightly greater than the critical value, above which perturbations may grow, the asymptotic solution is used as a guide in the choice of appropriate length and time scales for slow variations in the amplitude A of a non-linear two-dimensional perturbation wave. It is found that suitable time and space variables are εt and ε½(x+a1rt), where t is the time, x the distance in the direction of flow, ε the growth rate of linearized theory and (−a1r) the group velocity. By the method of multiple scales, A is found to satisfy a non-linear parabolic differential equation, a generalization of the time-dependent equation of earlier work. Initial conditions are given by the asymptotic solution of linearized theory.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 48 , Issue 3 , 16 August 1971 , pp. 529 - 545
Copyright
© 1971 Cambridge University Press

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