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Linear instability of annular Poiseuille flow

Published online by Cambridge University Press:  08 August 2008

C. J. HEATON
C. J. HEATON*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
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Abstract

The linear stability of flow along an annular pipe formed by two coaxial circular cylinders is considered. We find that the flow is unstable above a critical Reynolds number for all 0 < η ≤ 1, where η is the ratio between the radii of the inner and outer cylinders. This contradicts a recent claim that the flow is stable at all Reynolds numbers for radius ratio η less than a finite critical value. We find that non-axisymmetric disturbances become stable at all Reynolds numbers for η < 0.11686215, and we are able to study this ‘bifurcation from infinity’ asymptotically. However, axisymmetric disturbances remain unstable, with critical Reynolds number tending to infinity as η → 0. A second asymptotic analysis is performed to show that the critical Reynolds number Rec ∝ η−1 log(η−1) as η → 0, with the form of the mean flow profile causing the appearance of the logarithm. The stability of Hagen–Poiseuille flow (η = 0) at all Reynolds numbers is therefore interpreted as a limit result, and there are no annular pipe flows which share this stability.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 610 , 10 September 2008 , pp. 391 - 406
Copyright
Copyright © Cambridge University Press 2008

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