Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-23T07:54:54.778Z Has data issue: false hasContentIssue false
  • English
  • Français

Modelling the suppression of viscous fingering in elastic-walled Hele-Shaw cells

Published online by Cambridge University Press:  14 August 2013

Draga Pihler-Puzović ,
Raphaël Périllat ,
Matthew Russell ,
Anne Juel  and
Matthias Heil
Draga Pihler-Puzović*
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Raphaël Périllat
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Matthew Russell
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Anne Juel
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Matthias Heil
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Mathematics, 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

Recent experiments by Pihler-Puzovic et al. (Phys. Rev. Lett., vol. 108, 2012, article 074502) have shown that the onset of viscous fingering in circular Hele-Shaw cells in which an air bubble displaces a viscous fluid is delayed considerably when the top boundary of the cell is replaced by an elastic membrane. Non-axisymmetric instabilities are only observed at much larger flow rates, and the large-amplitude fingers that develop are fundamentally different from the highly branched fingers in rigid-walled cells. We explain the mechanism for the suppression of the instability using a combination of linear stability analysis and direct numerical simulations, based on a theoretical model that couples a depth-averaged lubrication equation for the fluid flow to the Föppl–von Kármán equations, which describe the deformation of the elastic membrane. We show that fluid–structure interaction affects the instability primarily via two changes to the axisymmetric base flow: the axisymmetric inflation of the membrane prior to the onset of any instabilities slows down the expansion of the air bubble and forces the air–liquid interface to propagate into a converging fluid-filled gap. Both of these changes reduce the destabilizing viscous effects that drive the fingering instability in a rigid-walled cell. In contrast, capillary effects only play a very minor role in the suppression of the instability.

JFM classification

Interfacial Flows (free surface): Fingering instability Low-Reynolds-number flows: Hele-Shaw flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 731 , 25 September 2013 , pp. 162 - 183
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

Al Housseiny, T. T., Tsai, P. A. & Stone, H. A. 2012 Control of interfacial instabilities using flow geometry. Nat. Phys. 8, 747–750.Google Scholar
Ben Jacob, E., Schmueli, H., Shochet, O. & Tenenbaum, A. 1992 Adaptive self-organisation during growth of bacterial colonies. Physica A 187, 378–424.Google Scholar
Bunger, A. P. & Cruden, A. R. 2011 Modelling the growth of laccoliths and large mafic sills: roll of magma body forces. J. Geophys. Res. 116, B02203.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
Chong, Y. H., Gaskell, P. H. & Kapur, N. 2007 Coating with deformable rolls: an experimental investigation of ribbing instability. Chem. Engng Sci. 62, 4138–4145.Google Scholar
Cinar, Y., Riaz, A. & Tchelepi, H. A. 2009 Experimental study of ${\mathrm{CO} }_{2} $ injection into saline formations. Soc. Petrol. Engrs J. 14, 589–594.Google Scholar
Clanet, C. & Searby, G. 1998 First experimental study of the Darrieus-Landau instability. Phys. Rev. Lett. 80 (17), 3867–3870.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
Dias, E. O., Alvarez-Lacalle, E., Carvalho, M. S. & Miranda, J. A. 2012 Minimization of viscous fluid fingering: a variational scheme for optimal flow rates. Phys. Rev. Lett. 109, 144502.Google Scholar
Dias, E. O. & Miranda, J. A. 2010 Control of radial fingering patterns: a weakly nonlinear approach. Phys. Rev. E 81, 016312.Google Scholar
Dias, E. O. & Miranda, J. A. 2013 Taper-induced control of viscous fingering in variable-gap Hele-Shaw flows. Phys. Rev. E 87, 053015.Google Scholar
Fast, P., Kondic, L., Shelley, M. J. & Palffy-Muhoray, P. 2001 Pattern formation in non-Newtonian Hele-Shaw flow. Phys. Fluids 13, 1191–1212.Google Scholar
Gadêlha, H. & Miranda, J. A. 2009 Effects of normal viscous stresses on radial viscous fingering. Phys. Rev. E 79, 066312.CrossRefGoogle ScholarPubMed
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121–147.CrossRefGoogle Scholar
Heap, A. & Juel, A. 2008 Anomalous bubble propagation in elastic tubes. Phys. Fluids 20, 081702.CrossRefGoogle 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
Heil, M. & Hazel, A. 2011 Fluid-structure interaction in internal physiological flows. Annu. Rev. Fluid Mech. 43, 141–162.Google Scholar
Homsy, G. M. 1987 Viscous fingering in porous medium. Annu. Rev. Fluid Mech. 19, 271–311.CrossRefGoogle Scholar
Hosoi, A. E. & Mahadevan, L. 2004 Peeling, healing, and bursting in a lubricated elastic sheet. Phys. Rev. Lett. 93, 137802.Google Scholar
Hull, D. 1999 Fractology. Cambridge University Press.Google Scholar
Jensen, O. E., Horsburgh, M. K., Halpern, D. & Gaver III, D. P. 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. 2012 Flattened fingers. Nat. Phys. 8, 706–707.Google Scholar
Kagei, N., Kanie, D. & Kawaguchi, M. 2005 Viscous fingering in shear thickening silica suspensions. Phys. Fluids 17, 054103.Google Scholar
Kim, H., Funada, T., Joseph, D. D. & Homsy, G. M. 2009 Viscous potential flow analysis of radial fingering in a Hele-Shaw cell. Phys. Fluids 21, 074106.Google Scholar
Kondic, L., Shelley, M. J. & Palffy-Muhoray, P. 1998 Non-Newtonian Hele-Shaw flow and the Saffman-Taylor instability. Phys. Rev. Lett. 80, 1433–1436.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1970 Theory of Elasticity, 2nd edn. Pergamon.Google Scholar
Li, S., Lowengrub, J. S., Fontana, J. & Palffy-Muhoray, P. 2009 Control of viscous fingering patterns in a radial Hele-Shaw cell. Phys. Rev. Lett. 102, 174501.CrossRefGoogle Scholar
Lister, J. R., Peng, G. G. & Neufeld, J. A. 2013 Viscous peeling of an elastic sheet by bending and pulling. Phys. Rev. Lett. (submitted).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.CrossRefGoogle Scholar
Mullins, W. W. & Sekerka, R. F. 1964 Stability of a planar interface during solidification of a dilute binary alloy. J. Appl. Phys. 35, 444–451.Google Scholar
Nase, J., Derks, D. & Lindner, A. 2011 Dynamic evolution of fingering patterns in a lifted Hele-Shaw cell. Phys. Fluids 23, 123101.Google Scholar
Orr, F. M. Jr & Taber, J. J. 1984 Use of carbon dioxide in enhanced oil recovery. Science 224, 563–569.Google Scholar
Paterson, L. 1981 Radial fingering in a Hele-Shaw cell. J. Fluid Mech. 113, 513–529.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.Google Scholar
Reinelt, D. A. 1987 Interface conditions for two-phase displacement in Hele-Shaw cells. J. Fluid Mech. 183, 219–244.Google Scholar
Reinelt, D. A. 1995 The primary and inverse instabilities of directional viscous fingering. J. Fluid Mech. 285, 303–327.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
Taylor, G. I. 1963 Cavitation of a viscous fluid in narrow passages. J. Fluid Mech. 16, 595–619.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