Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-17T08:29:24.167Z Has data issue: false hasContentIssue false

Resonance-driven oscillations in a flexible-channel flow with fixed upstream flux and a long downstream rigid segment

Published online by Cambridge University Press:  03 April 2014

Feng Xu
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
John Billingham
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
Oliver E. Jensen*
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Email address for correspondence:


Flow driven through a planar channel having a finite-length membrane inserted in one wall can be unstable to self-excited oscillations. In a recent study (Xu, Billingham & Jensen J. Fluid Mech., vol. 723, 2013, pp. 706–733), we identified a mechanism of instability arising when the inlet flux and outlet pressure are held constant, and the rigid segment of the channel downstream of the membrane is sufficiently short to have negligible influence on the resulting oscillations. Here we identify an independent mechanism of instability that is intrinsically coupled to flow in the downstream rigid segment, which becomes prominent when the downstream segment is much longer than the membrane. Using a spatially one-dimensional model of the system, we perform a three-parameter unfolding of a degenerate bifurcation point having four zero eigenvalues. Our analysis reveals how instability is promoted by a 1:1 resonant interaction between two modes, with the resulting oscillations described by a fourth-order amplitude equation. This predicts the existence of saturated sawtooth oscillations, which we reproduce in full Navier–Stokes simulations of the same system.

© 2014 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.)


Bertram, C. D. & Butcher, K. S. A. 1992 A collapsible-tube oscillator is not readily enslaved to an external resonator. J. Fluids Struct. 6 (2), 163180.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 (8), 10291045.Google Scholar
Cai, Z. X. & Luo, X. Y. 2003 A fluid–beam model for flow in a collapsible channel. J. Fluids Struct. 17, 125146.Google Scholar
Chakraborty, D., Prakash, J. R., Friend, J. & Yeo, L. 2012 Fluid-structure interaction in deformable microchannels. Phys. Fluids 24, 102002.Google Scholar
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121147.Google Scholar
Guneratne, J. C. & Pedley, T. J. 2006 High-Reynolds-number steady flow in a collapsible channel. J. Fluid Mech. 569, 151184.Google Scholar
Heil, M. & Hazel, A. L.2006 oomph-lib An object-oriented multi-physics finite-element library. In Fluid-Structure Interaction (ed. M. Schäfer & H.-J. Bungartz), pp. 19–49. Springer (oomph-lib is available as open-source software at Scholar
Heil, M. & Hazel, A. L. 2011 Fluid–structure interaction in internal physiological flows. Annu. Rev. Fluid Mech. 43, 141162.Google Scholar
Holmes, P. J. 1980 Averaging and chaotic motions in forced oscillations. SIAM J. Appl. Maths 38 (1), 6580.Google Scholar
Jensen, O. E. & Heil, M. 2003 High-frequency self-excited oscillations in a collapsible-channel flow. J. Fluid Mech. 481, 235268.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), 206219.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, 371386.Google 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, 348370.Google Scholar
Luo, X. Y. & Pedley, T. J. 1996 A numerical simulation of unsteady flow in a two-dimensional collapsible channel. J. Fluid Mech. 314, 191225.Google Scholar
Luo, X. Y. & Pedley, T. J. 1998 The effects of wall inertia on flow in a two-dimensional collapsible channel. J. Fluid Mech. 363, 253280.Google Scholar
Luo, X. Y. & Pedley, T. J. 2000 Multiple solutions and flow limitation in collapsible channel flows. J. Fluid Mech. 420, 301324.Google Scholar
Mandre, S. & Mahadevan, L. 2010 A generalised theory of viscous and inviscid flutter. Proc. R. Soc. Lond. A 466, 141156.Google Scholar
Pedley, T. J. 1992 Longitudinal tension variation in collapsible channels - a new mechanism for the breakdown of steady flow. Trans. ASME: J. Biomech. Engng 114 (1), 6067.Google Scholar
Pihler-Puzović, D. & Pedley, T. J. 2013 Stability of high-Reynolds-number flow in a collapsible channel. J. Fluid Mech. 714, 536561.Google Scholar
Stewart, P. S., Heil, M., Waters, S. L. & Jensen, O. E. 2010 Sloshing and slamming oscillations in a collapsible channel flow. J. Fluid Mech. 662, 288319.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), 541557.Google Scholar
Wang, J. W., Chew, Y. T. & Low, H. T. 2009 Effects of downstream system on self-excited oscillations in collapsible tubes. Commun. Numer. Meth. Engng 25 (5), 429445.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, 123153.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, 36353657.Google Scholar
Whittaker, R. J., Waters, S. L., Jensen, O. E., Boyle, J. & Heil, M. 2010c The energetics of flow through a rapidly oscillating tube. Part 1. General theory. J. Fluid Mech. 648, 83121.CrossRefGoogle Scholar
Xu, F., Billingham, J. & Jensen, O. E. 2013 Divergence-driven oscillations in a flexible-channel flow with fixed upstream flux. J. Fluid Mech. 723, 706733.CrossRefGoogle Scholar

Xu et al. supplementary movie

A saturated oscillation corresponding to figure 11(a), showing the axial flow field, computed using oomph-lib. The inlet profile is parabolic. The vertical axis is rescaled by a factor of 10. The constriction at the downstream end of the flexible membrane opens more quickly than it closes.

Download Xu et al. supplementary movie(Video)
Video 164.4 KB