Papers

How rapidly oscillating collapsible tubes extract energy from a viscous mean flow

MATTHIAS HEIL^a1 and SARAH L. WATERS^a2 p1

^a1 School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK

^a2 Division of Applied Mathematics, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

Abstract

We present a combined theoretical and computational analysis of three-dimensional unsteady finite-Reynolds-number flows in collapsible tubes whose walls perform prescribed high-frequency oscillations which resemble those typically observed in experiments with a Starling resistor. Following an analysis of the flow fields, we investigate the system's overall energy budget and establish the critical Reynolds number, Re_crit, at which the wall begins to extract energy from the flow. We conjecture that Re_crit corresponds to the Reynolds number beyond which collapsible tubes are capable of performing sustained self-excited oscillations. Our computations suggest a simple functional relationship between Re_crit and the system parameters, and we present a scaling argument to explain this observation. Finally, we demonstrate that, within the framework of the instability mechanism analysed here, self-excited oscillations of collapsible tubes are much more likely to develop from steady-state configurations in which the tube is buckled non-axisymmetrically, rather than from axisymmetric steady states, which is in agreement with experimental observations.

(Received August 21 2007)

(Revised January 08 2008)

Correspondence:

^p1 Present address: Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, 24–29 St Giles', Oxford, OX1 3LB, UK.