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On transition to chaos in two-dimensional channel flow symmetrically driven by accelerating walls

Published online by Cambridge University Press:  26 April 2006

E. B. B. Watson ,
W. H. H. Banks ,
M. B. Zaturska  and
P. G. Drazin
E. B. B. Watson
Affiliation:
School of Mathematics, University Walk, Bristol BS8 1TW, UK
W. H. H. Banks
Affiliation:
School of Mathematics, University Walk, Bristol BS8 1TW, UK
M. B. Zaturska
Affiliation:
School of Mathematics, University Walk, Bristol BS8 1TW, UK
P. G. Drazin
Affiliation:
School of Mathematics, University Walk, Bristol BS8 1TW, UK
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Abstract

We study theoretically the flow of a viscous incompressible fluid in a parallel-walled channel: the flow is driven by uniform steady suction through the porous and accelerating walls of the channel. Previous authors have discussed special cases of such flows, confining attention to flows which are symmetric, steady and two-dimensional; a similarity form of solution is assumed, as used by Berman and originally due to Hiemenz, to reduce the Navier–Stokes equations to a nonlinear ordinary differential equation. We generalize their work by considering asymmetric flows, unsteady flows and three-dimensional perturbations. By use of numerical calculations, matched asymptotic expansions for large values of the Reynolds number, both positive and negative, and the theory of dynamical systems we find many more exact solutions of the Navier–Stokes equations, examine their stability and interpret them; although much of the theory is for the general case, most of the numerical calculations are for the case of zero suction. In particular we show that most of the previously found steady solutions are unstable to antisymmetric two-dimensional disturbances. This leads to a pitchfork bifurcation, stable asymmetric steady solutions, a Hopf bifurcation, stable time-periodic solutions, stable quasi-periodic solutions and chaos in succession as the Reynolds number increases from zero, and a pitchfork bifurcation, stable asymmetric steady solutions, a Hopf bifurcation, periodic solutions, chaos via period doubling, other periodic solutions and chaos in succession as the Reynolds number decreases from zero.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 212 , March 1990 , pp. 451 - 485
Copyright
© 1990 Cambridge University Press

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