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Subcritical bifurcation of plane Poiseuille flow

Published online by Cambridge University Press:  29 March 2006

T. S. Chen  and
D. D. Joseph
T. S. Chen
Affiliation:
Department of Mechanical and Aerospace Engineering, University of Missouri-Rolla
D. D. Joseph
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota
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Abstract

We apply the perturbation theory which was recently developed and justified by Joseph & Sattinger (1972) to determine the form of the time-periodic solutions which bifurcate from plane Poiseuille flow. The results a t lowest significant order seem to be in good agreement with those following from the formal perturbation method of Stuart (1960) as extended by Reynolds & Potter (1967). Given the numerical results of the present calculation, the rigorous theory guarantees that the only time-periodic solution which bifurcates from laminar Poiseuille flow is a two-dimensional wave. The wave which bifurcates at the lowest Reynolds number exists, but it is unstable when its amplitude is small. Solutions which escape the small domain of attraction of laminar Poiseuille flow snap through this unstable time-periodic solution with a small amplitude to solutions of larger amplitudes.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 58 , Issue 2 , 17 April 1973 , pp. 337 - 351
Copyright
© 1973 Cambridge University Press

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References

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