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

Published online by Cambridge University Press:  25 April 2008

MATTHIAS HEIL
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
SARAH L. WATERS
Affiliation:
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, Recrit, at which the wall begins to extract energy from the flow. We conjecture that Recrit 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 Recrit 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.

Type
Papers
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Bertram, C. & Tscherry, J. 2006 The onset of flow-rate limitation and flow-induced oscillations in collapsible tubes. J. Fluids Struct. 22, 10291045.CrossRefGoogle Scholar
Bertram, C. D. & Castles, R. J. 1999 Flow limitation in uniform thick-walled collapsible tubes. J. Fluids Struct. 13, 399418.CrossRefGoogle Scholar
Bertram, C. D. & Raymond, C. J. 1991 Measurements of wave speed and compliance in a collapsible tube during self-excited oscillations: a test of the choking hypothesis. Med. Biol. Engng Comput. 29, 493500.CrossRefGoogle Scholar
Bertram, C. D., Raymond, C. J. & Pedley, T. J. 1990 Mapping of instabilities for flow through collapsible tubes of differering length. J. Fluids Struct. 4, 125153.CrossRefGoogle Scholar
Elman, H. C., Silvester, D. J. & Wathen, A. J. 2005 Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics. Oxford University Press.CrossRefGoogle Scholar
Heil, M. & Hazel, A. L. 2006 oomph-lib – an object-oriented multi-physics finite-element library. In Fluid–Structure Interaction (ed. Schäfer, M. & Bungartz, H.-J.), pp. 19–49. Springer. oomph-lib is available as open-source software at http://www.oomph-lib.org.CrossRefGoogle Scholar
Heil, M. & Jensen, O. E. 2003 Flows in deformable tubes and channels – theoretical models and biological applications. In Flow in Collapsible Tubes and Past Other Highly Compliant Boundaries (ed. Pedley, T. J. & Carpenter, P. W.), pp. 1550. Kluwer.CrossRefGoogle Scholar
Heil, M. & Waters, S. 2006 Transverse flows in rapidly oscillating, elastic cylindrical shells. J. Fluid Mech. 547, 185214.CrossRefGoogle Scholar
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.CrossRefGoogle Scholar
Jensen, O. E. & Heil, M. 2003 High-frequency self-excited oscillations in a collapsible-channel flow. J. Fluid Mech. 481, 235268.CrossRefGoogle Scholar
Luo, X. Y. & Pedley, T. J. 2000 Multiple solutions and flow limitation in collapsible channel flows. J. Fluid Mech. 420, 301324.CrossRefGoogle Scholar
Soedel, W. 1993 Vibrations of Shells and Plates. Marcel Dekker.Google Scholar
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic.Google Scholar