Skip to main content

Reopening modes of a collapsed elasto-rigid channel

  • Lucie Ducloué (a1), Andrew L. Hazel (a2), Alice B. Thompson (a2) and Anne Juel (a1)

Motivated by the reopening mechanics of strongly collapsed airways, we study the steady propagation of an air finger through a collapsed oil-filled channel with a single compliant wall. In a previous study using fully compliant elastic tubes, a ‘pointed’ air finger was found to propagate at high speed and low pressure, which, if clinically accessible, offers the potential for rapid reopening of highly collapsed airways with minimal tissue damage (Heap & Juel Phys. Fluids, vol. 20 (8), 2008, 081702). The mechanism underlying the selection of that pointed finger, however, remained unexplained. In this paper, we identify the required selection mechanism by conducting an experimental study in a simpler geometry: a rigid rectangular Hele-Shaw channel with an elastic top boundary. The constitutive behaviour of this elasto-rigid channel is nonlinear and broadly similar to that of an elastic tube, but unlike the tube, the channel’s cross-section adopts self-similar shapes from the undeformed state to the point of first near wall contact. The ensuing simplification of the vessel geometry enables the systematic investigation of the reopening dynamics in terms of increasing initial collapse. We find that for low levels of initial collapse, a single centred symmetric finger propagates in the channel and its shape is set by the tip curvature. As the level of collapse increases, the channel cross-section develops a central region of near opposite wall contact, and the finger shape evolves smoothly towards a ‘flat-tipped’ finger whose geometry is set by the strong depth gradient near the channel walls. We show that the flat-tipped mode of reopening is analogous to the pointed finger observed in tubes. Its propagation is sustained by the vessel’s extreme cross-sectional profile at high collapse, while vessel compliance is necessary to stabilise it. A simple scaling argument based on the dissipated power reveals that this reopening mode is preferred at higher propagation speeds when it becomes favourable to displace the elastic channel wall rather than the viscous fluid.

Corresponding author
Present address: Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), UMR CNRS 7636, PSL – ESPCI Paris, Sorbonne Université – UPMC – Univ. Paris 06, Sorbonne Paris Cité – UPD – Univ. Paris 07, 10 rue Vauquelin, 75005 Paris, France. Email address for correspondence:
Hide All
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 (3), 034502.
Al-Housseiny, T. T. & Stone, H. A. 2013 Controlling viscous fingering in tapered Hele-Shaw cells. Phys. Fluids 25 (9), 092102.
Al-Housseiny, T. T., Tsai, P. A. & Stone, H. A. 2012 Control of interfacial instabilities using flow geometry. Nat. Phys. 8 (10), 747750.
Ben Amar, M., Combescot, R. & Couder, Y. 1993 Viscous fingering with adverse anisotropy: a new Saffman–Taylor finger. Phys. Rev. Lett. 70 (20), 30473050.
Bensimon, D. 1986 Stability of viscous fingering. Phys. Rev. A 33 (2), 13021308.
Bland, R. D. 2001 Loss of liquid from the lung lumen in labor: more than a simple squeeze. Am. J. Physiol.-Lung C 280 (4), L602L605.
Borno, R. T., Steinmeyer, J. D. & Maharbiz, M. M. 2006 Transpiration actuation: the design, fabrication and characterization of biomimetic microactuators driven by the surface tension of water. J. Micromech. Microengng 16 (11), 2375.
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10 (2), 166188.
Combescot, R. 1994 Saffman–Taylor fingers with adverse anisotropic surface tension. Phys. Rev. E 49 (5), 41724178.
Couder, Y., Gerard, N. & Rabaud, M. 1986 Narrow fingers in the Saffman–Taylor instability. Phys. Rev. A 34 (6), 51755178.
Flaherty, J. E., Keller, J. B. & Rubinow, S. I. 1972 Post buckling behavior of elastic tubes and rings with opposite sides in contact. SIAM J. Appl. Maths 23 (4), 446455.
Franco-Gómez, A., Thompson, A. B., Hazel, A. L. & Juel, A. 2016 Sensitivity of Saffman–Taylor fingers to channel depth perturbations. J. Fluid Mech. 794, 343368.
Gaver, D. P. III, Halpern, D., Jensen, O. E. & Grotberg, J. B. 1996 The steady motion of a semi-infinite bubble through a flexible-walled channel. J. Fluid Mech. 319, 2565.
Gaver, D. P. III, Samsel, R. W. & Solway, J. 1990 Effects of surface tension and viscosity on airway reopening. J. Appl. Phys. 69 (1), 7485.
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Ann. Rev. Fluid Mech. 36 (1), 121147.
Halpern, D. & Grotberg, J. B. 1992 Fluid-elastic instabilities of liquid-lined flexible tubes. J. Fluid Mech. 244, 615632.
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, 4770.
Heap, A.2008 The reopening of a collapsed, fluid-filled elastic tube. PhD thesis, University of Manchester.
Heap, A. & Juel, A. 2008 Anomalous bubble propagation in elastic tubes. Phys. Fluids 20 (8), 081702.
Heap, A. & Juel, A. 2009 Bubble transitions in strongly collapsed elastic tubes. J. Fluid Mech. 633, 485507.
Heil, M. 1999 Minimal liquid bridges in non-axisymmetrically buckled elastic tubes. J. Fluid Mech. 380, 309337.
Hoberg, T. B., Verneuil, E. & Hosoi, A. E. 2014 Elastocapillary flows in flexible tubes. Phys. Fluids 26 (12), 122103.
Hodson, W. A. 1991 The First Breath, pp. 16651675. Raven.
Jensen, M. H., Libchaber, A., Pelcé, P. & Zocchi, G. 1987 Effect of gravity on the Saffman–Taylor meniscus: theory and experiment. Phys. Rev. A 35 (5), 22212227.
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 (2), 443457.
Juel, A. & Heap, A. 2007 The reopening of a collapsed fluid-filled elastic tube. J. Fluid Mech. 572, 287310.
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 (15), 154501.
de Lózar, A., Heap, A., Box, F., Hazel, A. L. & Juel, A. 2009 Tube geometry can force switchlike transitions in the behavior of propagating bubbles. Phys. Fluids 21 (10), 101702.
Macklem, P. T., Proctor, D. F. & Hogg, J. C. 1970 The stability of peripheral airways. Respir. Physiol. 8 (2), 191203.
Ozsun, O., Yakhot, V. & Ekinci, K. L. 2013 Non-invasive measurement of the pressure distribution in a deformable micro-channel. J. Fluid Mech. 734, R1.
Park, C.-W. & Homsy, G. M. 1984 Two-phase displacement in Hele Shaw cells: theory. J. Fluid Mech. 139, 291308.
Perun, M. L. & Gaver, D. P. III 1995 Interaction between airway lining fluid forces and parenchymal tethering during pulmonary airway reopening. J. Appl. Phys. 79 (5), 17171728.
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 (7), 074502.
Pihler-Puzović, D., Juel, A., Peng, G. G., Lister, J. R. & Heil, M. 2015 Displacement flows under elastic membranes. Part 1. Experiments and direct numerical simulations. J. Fluid Mech. 784, 487511.
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, 162183.
Pokroy, B., Kang, S. H., Mahadevan, L. & Aizenberg, J. 2009 Self-organization of a mesoscale bristle into ordered, hierarchical helical assemblies. Science 323 (5911), 237240.
Roman, B. & Bico, J. 2010 Elasto-capillarity: deforming an elastic structure with a liquid droplet. J. Phys.: Condens. Matter 22 (49), 493101.
Saffman, P. G. & Taylor, G. 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 (1242), 312329.
Shapiro, A. H. 1977 Steady flow in collapsible tubes. Trans. ASME J. Biomech. Engng 99 (3), 126147.
Tabeling, P., Zocchi, G. & Libchaber, A. 1987 An experimental study of the Saffman–Taylor instability. J. Fluid Mech. 177, 6782.
Thompson, A. B., Juel, A. & Hazel, A. L. 2014 Multiple finger propagation modes in Hele-Shaw channels of variable depth. J. Fluid Mech. 746, 123164.
Van Honschoten, J. W., Escalante, M., Tas, N. R., Jansen, H. V. & Elwenspoek, M. 2007 Elastocapillary filling of deformable nanochannels. J. Appl. Phys. 101 (9), 094310.
Zhao, H., Casademunt, J., Yeung, C. & Maher, J. V. 1992 Perturbing Hele-Shaw flow with a small gap gradient. Phys. Rev. A 45, 24552460.
Zheng, Y., Fujioka, H., Bian, S., Torisawa, Y., Huh, D., Takayama, S. & Grotberg, J. B. 2009 Liquid plug propagation in flexible microchannels: a small airway model. Phys. Fluids 21 (7), 071903.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


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