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


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.

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Bertram, C. & Tscherry, J. 2006 The onset of flow-rate limitation and flow-induced oscillations in collapsible tubes. J. Fluids Struct. 22, 10291045.
Bertram, C. D. & Castles, R. J. 1999 Flow limitation in uniform thick-walled collapsible tubes. J. Fluids Struct. 13, 399418.
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.
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.
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.
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
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.
Heil, M. & Waters, S. 2006 Transverse flows in rapidly oscillating, elastic cylindrical shells. J. Fluid Mech. 547, 185214.
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.
Jensen, O. E. & Heil, M. 2003 High-frequency self-excited oscillations in a collapsible-channel flow. J. Fluid Mech. 481, 235268.
Luo, X. Y. & Pedley, T. J. 2000 Multiple solutions and flow limitation in collapsible channel flows. J. Fluid Mech. 420, 301324.
Soedel, W. 1993 Vibrations of Shells and Plates. Marcel Dekker.
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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