Skip to main content Accessibility help
×
Home

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

  • ROBERT J. WHITTAKER (a1), MATTHIAS HEIL (a2), JONATHAN BOYLE (a2), OLIVER E. JENSEN (a3) and SARAH L. WATERS (a1)...

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.

Copyright

References

Hide All
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. 5165. Kluwer Academic.
Bertram, C. D. & Tscherry, J. 2006 The onset of flow-rate limitation and flow-induced oscillations in collapsible tubes. J. Fluids Struct. 22, 10291045.
Davies, C. & Carpenter, P. W. 1997 Instabilities in a plane channel flow between compliant walls. J. Fluid Mech. 352, 205243.
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121147.
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. 1949. Springer. (Oomph-lib is available as open-source software at http://www.oomph-lib.org/.)
Heil, M. & Waters, S. L. 2006 Transverse flows in rapidly oscillating elastic cylindrical shells. J. Fluid Mech. 547, 185214.
Heil, M. & Waters, S. L. 2008 How rapidly oscillating collapsible tubes extract energy from a viscous mean flow. J. Fluid Mech. 601, 199227.
Jensen, O. E. 1990 Instabilities of flow in a collapsed tube. J. Fluid Mech. 220, 623659.
Jensen, O. E. & Heil, M. 2003 High-frequency self-excited oscillations on a collapsible-channel flow. J. Fluid Mech. 481, 235268.
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, 4576.
Mortensen, N. A. & Bruus, H. 2006 Universal dynamics in the onset of a Hagen–Poiseuille flow. Phys. Rev. E 74, 017301.
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), 541557.
Troesch, B. A. & Troesch, H. R. 1973 Eigenfrequencies of an elliptic membrane. Math. Comp. 27 (124), 755765.
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, 83121.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Related content

Powered by UNSILO

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

  • ROBERT J. WHITTAKER (a1), MATTHIAS HEIL (a2), JONATHAN BOYLE (a2), OLIVER E. JENSEN (a3) and SARAH L. WATERS (a1)...

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.