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Instabilities in flexible channel flow with large external pressure

Published online by Cambridge University Press:  24 July 2017

Peter S. Stewart Open the ORCID record for Peter S. Stewart [Opens in a new window]
Peter S. Stewart*
Affiliation:
School of Mathematics and Statistics, The Mathematics and Statistics Building, University of Glasgow, University Place, GlasgowG12 8SQ, UK
*
†Email address for correspondence: [email protected]
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Abstract

We examine the stability of laminar high-Reynolds-number flow through an asymmetric flexible-walled channel driven by a fixed upstream flux and subject to a (large) uniform external pressure. We construct a long-wavelength, spatially one-dimensional model using a flow profile assumption, modelling the flexible wall as a thin tensioned membrane subject to a large axial pre-stress. We numerically construct the non-uniform static shape of the flexible wall and consider its stability using both a global eigensolver and numerical simulation of the nonlinear governing equations. The system admits multiple static solutions, including a highly collapsed steady state where the membrane has a single constriction which increases with increasing external pressure. We demonstrate that the non-uniform static state is unstable to two distinct (infinite) families of normal modes which we characterise in the limit of large external pressure. In particular, there is a family of low-frequency oscillatory modes which each persist to low membrane tensions, where the most unstable mode has an oscillating membrane profile which is outwardly bulged at the centre of the domain with a narrow constriction at the downstream end. In addition, there is a family of high-frequency oscillatory modes which are each unstable beyond a critical value of the tension within a two-branch neutral curve. Unstable modes along the lower branch of the neutral curve are sustained by a leading-order balance between unsteady inertia and the restoring force of membrane tension along the channel. In addition, we elucidate the mechanism of energy transfer to sustain the self-excited oscillations: oscillations decrease the mean maximal constriction of the channel over a period, which reduces the overall dissipation of the mean flow and releases energy to sustain the instability. Fully nonlinear simulations indicate that as the Reynolds number increases these unstable normal modes can grow supercritically into sustained large-amplitude ‘slamming’ oscillations, where the membrane is periodically drawn very close to the opposite rigid wall before recovering.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 825 , 25 August 2017 , pp. 922 - 960
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Copyright
© 2017 Cambridge University Press 

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References

Armitstead, J. P., Bertram, C. D. & Jensen, O. E. 1996 A study of the bifurcation behaviour of a model of flow through a collapsible tube. Bull. Math. Biol. 58 (4), 611–641.CrossRefGoogle Scholar
Bertram, C. D. 2003 Experimental studies of collapsible tubes. In Flow in Collapsible Tubes and Past Other Highly Compliant Boundaries (ed. Carpenter, P. W. & Pedley, T. J.), pp. 51–65. Kluwer.CrossRefGoogle 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.Google Scholar
Cancelli, C. & Pedley, T. J. 1985 A separated-flow model for collapsible-tube oscillations. J. Fluid Mech. 157, 375–404.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Elad, D., Kamm, R. D. & Shapiro, A. H. 1987 Choking phenomena in a lung-like model. Trans. ASME J. Biomech. Engng 109, 1–9.CrossRefGoogle Scholar
Gavriely, N. & Grotberg, J. B. 1988 Flow limitation and wheezes in a constant flow and volume lung preparation. J. Appl. Phys. 64 (1), 17–20.Google Scholar
Gavriely, N. & Jensen, O. E. 1993 Theory and measurements of snores. J. Appl. Phys. 74 (6), 2828–2837.Google ScholarPubMed
Glendinning, P. 1994 Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations. Cambridge University Press.CrossRefGoogle Scholar
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121–147.Google Scholar
Heil, M. & Boyle, J. 2010 Self-excited oscillations in three-dimensional collapsible tubes: simulating their onset and large-amplitude oscillations. J. Fluid Mech. 652, 405–426.Google Scholar
Heil, M. & Hazel, A. L. 2011 Fluid-structure interaction in internal physiological flows. Annu. Rev. Fluid Mech. 43, 141–162.Google Scholar
Jensen, O. E. 1990 Instabilities of flow in a collapsed tube. J. Fluid Mech. 220, 623–659.Google Scholar
Jensen, O. E. 1998 An asymptotic model of viscous flow limitation in a highly collapsed channel. Trans. ASME 120, 544–546.Google Scholar
Jensen, O. E. & Heil, M. 2003 High-frequency self-excited oscillations in a collapsible-channel flow. J. Fluid Mech. 481, 235–268.CrossRefGoogle Scholar
Kececioglu, I., Mcclurken, M. E., Kamm, R. D. & Shapiro, A. H. 1981 Steady, supercritical flow in collapsible tubes. Part 1. Experimental observations. J. Fluid Mech. 109, 367–389.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, 348–370.CrossRefGoogle Scholar
Liu, H. F., Luo, X. Y., Cai, Z. X. & Pedley, T. J. 2009 Sensitivity of unsteady collapsible channel flows to modelling assumptions. Commun. Numer. Meth. Engng 25 (5), 483–504.CrossRefGoogle 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
Luo, X. Y. & Pedley, T. J. 2000 Multiple solutions and flow limitation in collapsible channel flows. J. Fluid Mech. 420, 301–324.CrossRefGoogle Scholar
Mittal, R., Erath, B. D. & Plesniak, M. W. 2013 Fluid dynamics of human phonation and speech. Annu. Rev. Fluid Mech. 45, 437–467.Google Scholar
Pedley, T. J. 1992 Longitudinal tension variation in a collapsible channel: a new mechanism for the breakdown of steady flow. Trans. ASME J. Biomech. Engng 114, 60–67.CrossRefGoogle Scholar
Pihler-Puzović, D. & Pedley, T. J. 2013 Stability of high-Reynolds-number flow in a collapsible channel. J. Fluid Mech. 714, 536–561.Google Scholar
Reyn, J. W. 1987 Multiple solutions and flow limitation for steady flow through a collapsible tube held open at the ends. J. Fluid Mech. 174, 467–493.Google Scholar
Siviglia, A. & Toffolon, M. 2013 Steady analysis of transcritical flows in collapsible tubes with discontinuous mechanical properties: implications for arteries and veins. J. Fluid Mech. 736, 195–215.Google Scholar
Siviglia, A. & Toffolon, M. 2014 Multiple states for flow through a collapsible tube with discontinuities. J. Fluid Mech. 761, 105–122.Google Scholar
Stewart, P. S.2010 Flow in flexible-walled channels and airways. PhD thesis, University of Nottingham.Google Scholar
Stewart, P. S., Heil, M., Waters, S. L. & Jensen, O. E. 2010 Sloshing and slamming oscillations in collapsible channel flow. J. Fluid Mech. 662, 288–319.CrossRefGoogle Scholar
Stewart, P. S., Jensen, O. E. & Foss, A. J. E. 2014 A theoretical model to allow prediction of the CSF pressure from observations of the retinal venous pulse. Invest. Ophthalmol. Vis. Sci. 55 (10), 6319–6323.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.Google Scholar
Whittaker, R. J. 2015 A shear-relaxation boundary layer near the pinned ends of a buckled elastic-walled tube. IMA J. Appl. Maths 80 (6), 1932–1967.Google Scholar
Whittaker, R. J., Heil, M., Jensen, O. E. & Waters, S. L. 2010 Predicting the onset of high-frequency self-excited oscillations in elastic-walled tubes. Proc. R. Soc. Lond. A 466 (2124), 3635–3657.Google 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, 706–733.CrossRefGoogle Scholar
Xu, F., Billingham, J. & Jensen, O. E. 2014 Resonance-driven oscillations in a flexible-channel flow with fixed upstream flux and a long downstream rigid segment. J. Fluid Mech. 746, 368–404.CrossRefGoogle Scholar
Xu, F. & Jensen, O. E. 2015 A low-order model for slamming in a flexible-channel flow. Q. J. Mech. Appl. Maths 68 (3), 299–319.Google Scholar