Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-24T02:18:01.463Z Has data issue: false hasContentIssue false
  • English
  • Français

Divergence-driven oscillations in a flexible-channel flow with fixed upstream flux

Published online by Cambridge University Press:  16 April 2013

Feng Xu ,
John Billingham  and
Oliver E. Jensen
Feng Xu
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
John Billingham
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
Oliver E. Jensen*
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
*
†Email address for correspondence: oliver.jensen@manchester.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider flow in a finite-length channel, one wall of which contains a membrane under longitudinal tension. The upstream flux and downstream pressure are prescribed and an external linear pressure distribution is applied to the membrane such that the system admits uniform Poiseuille flow as a steady solution. The system is described using a one-dimensional model that accounts for viscous and inertial effects. A linear stability analysis reveals that the uniform state is unstable to static (or divergent) and oscillatory instabilities. Asymptotic analysis in the neighbourhood of a Takens–Bogdanov bifurcation point shows how, when the downstream rigid section of the channel is not substantially longer than the membrane, an oscillatory mode arises through an interaction between two static eigenmodes. Perturbations to the uniform state exhibit the dynamics of a weakly dissipative Hamiltonian system for which low-frequency self-excited oscillations are forced by the divergent instability of two nearby steady solutions, before ultimately growing to large amplitudes. Simulations show that the subsequent dynamics can involve slamming motion in which the membrane briefly comes into near-contact with the opposite rigid wall over short length scales.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Instability: Instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 723 , 25 May 2013 , pp. 706 - 733
Copyright
©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bertram, C. D. 2008 Flow-induced oscillation of collapsed tubes and airway structures. Respir. Physiol. Neurobiol. 163 (1–3), 256–265.CrossRefGoogle ScholarPubMed
Bertram, C. D., Raymond, C. J. & Pedley, T. J. 1990 Mapping of instabilities for flow through collapsed tubes of differing length. J. Fluids Struct. 4 (2), 125–153.Google Scholar
Bessems, D., Rutten, M. & van de Vosse, F. 2007 A wave propagation model of blood flow in large vessels using an approximate velocity profile function. J. Fluid Mech. 580, 145–168.CrossRefGoogle Scholar
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121–147.Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, vol. 42. Springer.CrossRefGoogle Scholar
Guneratne, J. C. & Pedley, T. J. 2006 High-Reynolds-number steady flow in a collapsible channel. J. Fluid Mech. 569, 151–184.CrossRefGoogle Scholar
Heil, M. & Hazel, A. L. 2011 Fluid-structure interaction in internal physiological flows. Annu. Rev. Fluid Mech. 43, 141–162.Google Scholar
Heil, M. & Jensen, O. E. 2003 Flows in deformable tubes and channels - theoretical models and biological applications. In Flow Past Highly Compliant Boundaries and in Collapsible Tubes (ed. Carpenter, P. W. & Pedley, T. J.), pp. 15–49. Kluwer.Google Scholar
Hughes, T. J. R. & Lubliner, J. 1973 On the one-dimensional theory of blood flow in the larger vessels. Math. Biosci. 18, 161–170.Google 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 in a collapsible-channel flow. J. Fluid Mech. 481, 235–268.Google Scholar
Kerswell, R. R. 2005 Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18, R17–R44.Google Scholar
Knowlton, F. P. & Starling, E. H. 1912 The influence of variations in temperature and blood-pressure on the performance of the isolated mammalian heart. J. Physiol. 44 (3), 206–219.Google Scholar
Kudenatti, R. B., Bujurke, N. M. & Pedley, T. J. 2012 Stability of two-dimensional collapsible-channel flow at high Reynolds number. J. Fluid Mech. 705, 371–386.CrossRefGoogle Scholar
Liu, H. F., Luo, X. Y. & Cai, Z. X. 2012 Stability and energy budget of pressure-driven collapsible channel flows. J. Fluid Mech. 705, 348–370.Google Scholar
Luo, X. Y., Cai, Z. X., Li, W. G. & Pedley, T. J. 2008 The cascade structure of linear instability in collapsible channel flows. J. Fluid Mech. 600, 45–76.CrossRefGoogle Scholar
Luo, X. Y. & Pedley, T. J. 1996 A numerical simulation of unsteady flow in a two-dimensional collapsible channel. J. Fluid Mech. 314, 191–225.Google Scholar
Pedley, T. J. 1992 Longitudinal tension variation in collapsible channels: a new mechanism for the breakdown of steady flow. J. Biomech. Engng 114, 60–67.Google Scholar
Pihler-Puzović, D. & Pedley, T. J. 2013 Stability of high-Reynolds-number flow in a collapsible channel. J. Fluid Mech. 714, 536–561.CrossRefGoogle Scholar
Stewart, P. S., Heil, M., Waters, S. L. & Jensen, O. E. 2010a Sloshing and slamming oscillations in a collapsible channel flow. J. Fluid Mech. 662, 288–319.Google Scholar
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 (Fluids) 28 (4), 541–557.CrossRefGoogle Scholar
Stewart, P. S., Waters, S. L. & Jensen, O. E. 2010b Local instabilities of flow in a flexible channel: asymmetric flutter driven by a weak critical layer. Phys. Fluids 22, 031902.CrossRefGoogle Scholar
Thomson, S. L., Mongeau, L. & Frankel, S. H. 2007 Flow over a membrane-covered, fluid-filled cavity. Comput. & Struct. 85, 1012–1019.Google Scholar
Whittaker, R. J., Heil, M., Boyle, J., Jensen, O. E. & Waters, S. L. 2010a The energetics of flow through a rapidly oscillating tube. Part 2. Application to an elliptical tube. J. Fluid Mech. 648, 123–153.Google Scholar
Whittaker, R. J., Heil, M., Jensen, O. E. & Waters, S. L. 2010b Predicting the onset of high-frequency self-excited oscillations in elastic-walled tubes. Proc. R. Soc. Lond. A 466, 3635–3657.Google Scholar
Whittaker, R. J., Heil, M., Jensen, O. E. & Waters, S. L. 2010c A rational derivation of a tube law from shell theory. Q. J. Mech. Appl. Maths. 63 (4), 465–496.Google Scholar
Whittaker, R. J., Waters, S. L., Jensen, O. E., Boyle, J. & Heil, M. 2010d The energetics of flow through a rapidly oscillating tube. Part 1. General theory. J. Fluid Mech. 648, 83–121.CrossRefGoogle Scholar

Xu et al. supplementary movie

This movie shows (in blue) the evolution of the channel width h(x,t) along the length of the channel, corresponding to the solution shown in Figure 8 of the paper. The red curve shows the steady solution that exists for the same parameter values.

Download Xu et al. supplementary movie(Video)
Video 3.5 MB