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The energetics of flow through a rapidly oscillating tube. Part 2. Application to an elliptical tube

Published online by Cambridge University Press:  07 April 2010

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

In Part 1 of this work, we derived general asymptotic results for the three-dimensional flow field and energy fluxes for flow within a tube whose walls perform prescribed small-amplitude periodic oscillations of high frequency and large axial wavelength. In the current paper, we illustrate how these results can be applied to the case of flow through a finite-length axially non-uniform tube of elliptical cross-section – a model of flow in a Starling resistor. The results of numerical simulations for three model problems (an axially uniform tube under pressure–flux and pressure–pressure boundary conditions and an axially non-uniform tube with prescribed flux) with prescribed wall motion are compared with the theoretical predictions made in Part 1, each showing excellent agreement. When upstream and downstream pressures are prescribed, we show how the mean flux adjusts slowly under the action of Reynolds stresses using a multiple-scale analysis. We test the asymptotic expressions obtained for the mean energy transfer E from the flow to the wall over a period of the motion. In particular, the critical point at which E = 0 is predicted accurately: this point corresponds to energetically neutral oscillations, the condition which is relevant to the onset of global instability in the Starling resistor.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 648 , 10 April 2010 , pp. 123 - 153
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Bertram, C. D. 2003 Experimental studies of collapsible tubes. In Flow Past Highly Compliant Boundaries and in Collapsible Tubes (ed. Carpenter, P. W. & Pedley, T. J.), chap. 3, pp. 51–65. Kluwer Academic.Google Scholar
Bertram, C. D. & Tscherry, J. 2006 The onset of flow-rate limitation and flow-induced oscillations in collapsible tubes. J. Fluids Struct. 22, 1029–1045.CrossRefGoogle Scholar
Davies, C. & Carpenter, P. W. 1997 Instabilities in a plane channel flow between compliant walls. J. Fluid Mech. 352, 205–243.CrossRefGoogle Scholar
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121–147.CrossRefGoogle Scholar
Heil, M. & Hazel, A. L. 2006 Oomph-lib – an object-oriented multi-physics finite-element library. In Fluid–Structure Interaction (ed. Bungartz, H.-J. & Schäfer, M.), pp. 19–49. Springer. (Oomph-lib is available as open-source software at http://www.oomph-lib.org/.)CrossRefGoogle Scholar
Heil, M. & Waters, S. L. 2006 Transverse flows in rapidly oscillating elastic cylindrical shells. J. Fluid Mech. 547, 185–214.CrossRefGoogle Scholar
Heil, M. & Waters, S. L. 2008 How rapidly oscillating collapsible tubes extract energy from a viscous mean flow. J. Fluid Mech. 601, 199–227.CrossRefGoogle Scholar
Jensen, O. E. 1990 Instabilities of flow in a collapsed tube. J. Fluid Mech. 220, 623–659.CrossRefGoogle Scholar
Jensen, O. E. & Heil, M. 2003 High-frequency self-excited oscillations on a collapsible-channel flow. J. Fluid Mech. 481, 235–268.CrossRefGoogle Scholar
Luo, X. Y., Cai, Z., Li, W. G. & Pedley, T. J. 2008 The cascade structure of linear instability in collapsible channel flows. J. Fluid Mech. 600, 45–76.Google Scholar
Mortensen, N. A. & Bruus, H. 2006 Universal dynamics in the onset of a Hagen–Poiseuille flow. Phys. Rev. E 74, 017301.CrossRefGoogle ScholarPubMed
Stewart, P. S., Waters, S. L. & Jensen, O. E. 2009 Local and global instabilities of flow in a flexible-walled channel. Eur. J. Mech. B 28 (4), 541–557.CrossRefGoogle Scholar
Troesch, B. A. & Troesch, H. R. 1973 Eigenfrequencies of an elliptic membrane. Math. Comp. 27 (124), 755–765.Google Scholar
Whittaker, R. J., Waters, S. L., Jensen, O. E., Boyle, J. & Heil, M. 2010 The energetics of flow through a rapidly oscillating tube. Part 1. General theory. J. Fluid Mech. 648, 83–121.CrossRefGoogle Scholar