Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-23T13:04:12.035Z Has data issue: false hasContentIssue false
  • English
  • Français

Displacement flows under elastic membranes. Part 1. Experiments and direct numerical simulations

Published online by Cambridge University Press:  06 November 2015

Draga Pihler-Puzović ,
Anne Juel ,
Gunnar G. Peng ,
John R. Lister  and
Matthias Heil
Draga Pihler-Puzović*
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Anne Juel
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Gunnar G. Peng
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
John R. Lister
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
Matthias Heil
Affiliation:
School of Mathematics and Manchester Centre for Nonlinear Dynamics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
*
†Email address for correspondence: draga.pihler-puzovic@manchester.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

The injection of fluid into the narrow liquid-filled gap between a rigid plate and an elastic membrane drives a displacement flow that is controlled by the competition between elastic and viscous forces. We study such flows using the canonical set-up of an elastic-walled Hele-Shaw cell whose upper boundary is formed by an elastic sheet. We investigate both single- and two-phase displacement flows in which the localised injection of fluid at a constant flow rate is accommodated by the inflation of the sheet and the outward propagation of an axisymmetric front beyond which the cell remains approximately undeformed. We perform a direct comparison between quantitative experiments and numerical simulations of two theoretical models. The models couple the Föppl–von Kármán equations, which describe the deformation of the thin elastic membrane, to the equations describing the flow, which we model by (i) the Navier–Stokes equations or (ii) lubrication theory. We identify the dominant physical effects that control the behaviour of the system and critically assess modelling assumptions that were made in previous studies. The insight gained from these studies is then used in Part 2 of this work, where we formulate an improved lubrication model and develop an asymptotic description of the key phenomena.

JFM classification

Low-Reynolds-number flows: Hele-Shaw flows Low-Reynolds-number flows: Low-Reynolds-number flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 784 , 10 December 2015 , pp. 487 - 511
Check for updates
Copyright
© 2015 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

Al-Housseiny, T. T., Christov, I. C. & Stone, H. A. 2013 Two-phase fluid displacement and interfacial instabilities under elastic membranes. Phys. Rev. Lett. 111, 034502.Google Scholar
Carvalho, M. S. & Scriven, L. E. 1997 Deformable roll coating flows: steady state and linear perturbation analysis. J. Fluid Mech. 339, 143–172.Google Scholar
Carvalho, M. S. & Scriven, L. E. 1999 Three-dimensional stability analysis of free surface flows: application to forward deformable roll coating. J. Comput. Phys. 151, 534–562.Google Scholar
Chen, J. D. 1989 Growth of radial viscous fingers in a Hele-Shaw cell. J. Fluid Mech. 201, 223–242.Google Scholar
Chopin, J., Vella, D. & Boudaoud, A. 2008 The liquid blister test. Proc. R. Soc. Lond. A 464, 2887–2906.Google Scholar
Demmel, J. W., Eisenstat, S. C., Gilbert, J. R., Li, X. S. & Liu, J. W. H. 1999 A supernodal approach to sparse partial pivoting. SIAM J. Matrix Anal. Applics. 20 (3), 720–755.CrossRefGoogle Scholar
Gaver III, D. P., Halpern, D., Jensen, O. E. & Grotberg, J. B. 1996 The steady propagation of a semi-infinite bubble through a flexible-walled channel. J. Fluid Mech. 319, 25–65.Google Scholar
Gaver III, D. P., Samsel, R. W. & Solway, J. 1990 Effects of surface tension and viscosity on airway reopening. J. Appl. Physiol. 69, 74–85.Google Scholar
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121–147.Google Scholar
Halpern, D., Naire, S., Jensen, O. & Gaver, D. 2005 Unsteady bubble propagation in a flexible channel: predictions of a viscous stick–slip instability. J. Fluid Mech. 528, 53–86.Google Scholar
Hazel, A., Heil, M., Waters, S. L. & Oliver, J. M. 2012 On the liquid lining in fluid-conveying curved tubes. J. Fluid Mech. 705, 213–233.Google Scholar
Hazel, A.L. & Heil, M. 2005 Finite-Reynolds-number effects in steady, three-dimensional airway reopening. Trans. ASME J. Biomech. Engng 128, 573–578.Google Scholar
Hazel, A. L. & Heil, M. 2003 Three-dimensional airway reopening: the steady propagation of a semi-infinite bubble into a buckled elastic tube. J. Fluid Mech. 478, 47–70.CrossRefGoogle Scholar
Hazel, A. L. & Heil, M. 2008 The influence of gravity on the steady propagation of a semi-infinite bubble into a flexible channel. Phys. Fluids 20, 092109.CrossRefGoogle Scholar
Heap, A. & Juel, A. 2008 Anomalous bubble propagation in elastic tubes. Phys. Fluids 20, 081702.Google Scholar
Heil, M. 2000 Finite Reynolds number effects in the propagation of an air finger into a liquid-filled flexible-walled channel. J. Fluid Mech. 424, 21–44.Google Scholar
Heil, M. 2001 The Bretherton problem in elastic-walled channels: finite Reynolds number effects. In IUTAM Symposium on Free Surface Flows (ed. King, A. C. & Shikmurzaev, Y. D.), pp. 113–120. Kluwer.Google Scholar
Heil, M. & Hazel, A. 2011 Fluid–structure interaction in internal physiological flows. Annu. Rev. Fluid Mech. 43, 141–162.Google Scholar
Heil, M. & Hazel, A. L. 2006 oomph-lib – an object-oriented multi-physics finite-element library. In Fluid–Structure Interaction (ed. Schäfer, M. & Bungartz, H.-J.), pp. 19–49. Springer; oomph-lib is available as open-source software at http://www.oomph-lib.org.Google Scholar
Hewitt, I. J., Balmforth, N. J. & de Bruyn, J. R. 2015 Elastic-plated gravity currents. Eur. J. Appl. Maths 26, 1–31.Google Scholar
Homsy, G. M. 1987 Viscous fingering in porous medium. Annu. Rev. Fluid Mech. 19, 271–311.Google Scholar
Hosoi, A. E. & Mahadevan, L. 2004 Peeling, healing, and bursting in a lubricated elastic sheet. Phys. Rev. Lett. 93, 137802.Google Scholar
Jensen, O. E., Horsburgh, M. K., Halpern, D. & Gaver, D. P. III 2002 The steady propagation of a bubble in a flexible-walled channel: asymptotic and computational models. Phys. Fluids 14, 443–457.Google Scholar
Juel, A. & Heap, A. 2007 The reopening of a collapsed fluid-filled elastic tube. J. Fluid Mech. 572, 287–310.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1970 Theory of Elasticity, 2nd edn. Pergamon.Google Scholar
Lister, J. R., Peng, G. G. & Neufeld, J. A. 2013 Viscous control of peeling an elastic sheet by bending and pulling. Phys. Rev. Lett. 111, 154501.Google Scholar
McEwan, A. D. & Taylor, G. I. 1966 The peeling of a flexible strip attached by a viscous adhesive. J. Fluid Mech. 26, 1–15.Google Scholar
Michaut, C. 2011 Dynamics of magmatic intrusions in the upper crust: theory and applications to laccoliths on Earth and the Moon. J. Geophys. Res. 116, B05205.Google Scholar
Miranda, J. A. & Widom, M. 1998 Radial fingering in a Hele-Shaw cell: a weakly nonlinear analysis. Physica D 120, 315–328.Google Scholar
Paterson, L. 1981 Radial fingering in a Hele-Shaw cell. J. Fluid Mech. 113, 513–529.Google Scholar
Peng, G. G., Pihler-Puzović, D., Juel, A., Heil, M. & Lister, J. R. 2015 Displacement flows under elastic membranes. Part 2. Lubrication theory for two-phase flow. J. Fluid Mech. 784, 512–547.Google Scholar
Perun, M. L. & Gaver III, D. P. 1985 Interaction between airway lining fluid forces and parenchymal tethering during pulmonary airway reopening. J. Appl. Physiol. 79, 1717–1728.Google Scholar
Perun, M. L. & Gaver III, D. P. 1995 An experimental model investigation of the opening of a collapsed untethered pulmonary airway. Trans. ASME J. Biomech. Engng 117, 245–253.Google Scholar
Pihler-Puzović, D., Illien, P., Heil, M. & Juel, A. 2012 Suppression of complex fingerlike patterns at the interface between air and a viscous fluid by elastic membranes. Phys. Rev. Lett. 108, 074502.CrossRefGoogle Scholar
Pihler-Puzović, D., Juel, A. & Heil, M. 2014 The interaction between viscous fingering and wrinkling in elastic-walled Hele-Shaw cells. Phys. Fluids 26, 022102.Google Scholar
Pihler-Puzović, D., Périllat, R., Russell, M., Juel, A. & Heil, M. 2013 Modelling the suppression of viscous fingering in elastic-walled Hele-Shaw cells. J. Fluid Mech. 731, 162–183.Google Scholar
Reinelt, D. A. & Saffman, P. G. 1985 The penetration of a finger into a viscous fluid in a channel and tube. SIAM J. Sci. Stat. Comput. 6, 542–561.Google Scholar
Reynolds, O. 1886 On the theory of lubrication and its application to Beauchamp Tower’s experiment. Phil. Trans. R. Soc. Lond. A 117, 157–234.Google Scholar
Saffman, P. G. & Taylor, G. I. 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245, 312–329.Google Scholar
Shewchuk, J. R. 1996 Engineering a 2D quality mesh generator and Delaunay triangulator. In Applied Computational Geometry: Towards Geometric Engineering (ed. Lin, M. C. & Manocha, D.), Lecture Notes in Computer Science, vol. 1148, pp. 203–222. Springer. (From the First ACM Workshop on Applied Computational Geometry).Google Scholar
Slim, A. C., Balmforth, N. J., Craster, R. V. & Miller, J. C. 2009 Surface wrinkling of a channelized flow. Proc. R. Soc. Lond. A 465, 123–142.Google Scholar
Thomé, T., Rabaud, M., Hakim, V. & Couder, Y. 1989 The Saffman–Taylor instability: from the linear to the circular geometry. Phys. Fluids A 1, 224–240.Google Scholar
Zienkiewicz, O. C. & Zhu, J. Z. 1992 The superconvergent patch recovery and a posteriori error estimates. Part 1: the recovery technique. Intl J. Numer. Meth. Engng 33 (7), 1331–1364.Google Scholar
Supplementary material: File

Pihler-Puzović supplementary material

Figures

Download Pihler-Puzović supplementary material(File)
File 2 MB