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The energetics of flow through a rapidly oscillating tube. Part 1. General theory

Published online by Cambridge University Press:  07 April 2010

ROBERT J. WHITTAKER ,
SARAH L. WATERS ,
OLIVER E. JENSEN ,
JONATHAN BOYLE  and
MATTHIAS HEIL
ROBERT J. WHITTAKER
Affiliation:
Oxford Centre for Industrial and Applied Mathematics, University of Oxford, 24–29 St Giles, Oxford OX1 3LB, UK
SARAH L. WATERS
Affiliation:
Oxford Centre for Industrial and Applied Mathematics, University of Oxford, 24–29 St Giles, Oxford OX1 3LB, UK
OLIVER E. JENSEN
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
JONATHAN BOYLE
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
MATTHIAS HEIL
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
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Abstract

We examine the effect of prescribed wall-driven oscillations of a flexible tube of arbitrary cross-section, through which a flow is driven by prescribing either a steady flux at the downstream end or a steady pressure difference between the ends. A large-Womersley-number large-Strouhal-number regime is considered, in which the oscillations of the wall are small in amplitude, but sufficiently rapid to ensure viscous effects are confined to a thin boundary layer. We derive asymptotic expressions for the flow fields and evaluate the energy budget. A general result for the conditions under which there is zero net energy transfer from the flow to the wall is provided. This is presented as a critical inverse Strouhal number (a dimensionless measure of the background flow rate) which is expressed only in terms of the tube geometry, the fluid properties and the profile of the prescribed wall oscillations. Our results identify an essential component of a fundamental mechanism for self-excited oscillations in three-dimensional collapsible tube flows, and enable us to assess how geometric and flow properties affect the stability of the system.

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Papers
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Journal of Fluid Mechanics , Volume 648 , 10 April 2010 , pp. 83 - 121
Copyright
Copyright © Cambridge University Press 2010

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