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Stability of two-dimensional collapsible-channel flow at high Reynolds number

Published online by Cambridge University Press:  16 February 2012

Ramesh B. Kudenatti ,
N. M. Bujurke  and
T. J. Pedley
Ramesh B. Kudenatti
Affiliation:
Department of Mathematics, Bangalore University, Bangalore-560 001, India Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
N. M. Bujurke
Affiliation:
Department of Mathematics, Karnatak University, Dharwad-580 003, India
T. J. Pedley*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: t.j.pedley@damtp.cam.ac.uk
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Abstract

We study the linear stability of two-dimensional high-Reynolds-number flow in a rigid parallel-sided channel, of which part of one wall has been replaced by a flexible membrane under longitudinal tension . Far upstream the flow is parallel Poiseuille flow at Reynolds number ; the width of the channel is and the length of the membrane is , where . Steady flow was studied using interactive boundary-layer theory by Guneratne & Pedley (J. Fluid Mech., vol. 569, 2006, pp. 151–184) for various values of the pressure difference across the membrane at its upstream end. Here unsteady interactive boundary-layer theory is used to investigate the stability of the trivial steady solution for . An unexpected finding is that the flow is always unstable, with a growth rate that increases with . In other words, the stability problem is ill-posed. However, when the pressure difference is held fixed () at the downstream end of the membrane, or a little further downstream, the problem is well-posed and all solutions are stable. The physical mechanisms underlying these findings are explored using a simple inviscid model; the crucial factor in the fluid dynamics is the vorticity gradient across the incoming Poiseuille flow.

JFM classification

Instability: Instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 705: Special issue dedicated to Professor T. J. Pedley on his 70th birthday , 25 August 2012 , pp. 371 - 386
Copyright
Copyright © Cambridge University Press 2012

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