Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-27T05:28:35.378Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiple finger propagation modes in Hele-Shaw channels of variable depth

Published online by Cambridge University Press:  28 March 2014

Alice B. Thompson ,
Anne Juel  and
Andrew L. Hazel
Alice B. Thompson*
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Anne Juel
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Andrew L. Hazel
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
*
†Present address: Department of Mathematics, Imperial College London, London SW7 2AZ, UK. Email address for correspondence: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider the propagation of an air finger into a wide fluid-filled channel with a spatially varying depth profile. Our aim is to understand the origin of the multiple coexisting families of both steady and oscillatory propagating fingers previously observed in experiments in axially uniform channels each containing a centred step-like occlusion. We find that a depth-averaged model can reproduce all the finger propagation modes observed experimentally. In addition, the model reveals new modes for symmetric finger propagation. The inclusion of a spatially variable channel depth in the depth-averaged equations leads to: (i) a variable mobility coefficient within the fluid domain due to variations in viscous resistance of the channel; and (ii) a variable transverse curvature term in the dynamic boundary condition that modifies the pressure jump over the air–liquid interface. We use our model to examine the roles of these two distinct effects and find that both contribute to the steady bifurcation structure, while the transverse curvature term is responsible for the distinctive oscillatory propagation modes.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Drops and Bubbles: Bubble dynamics Low-Reynolds-number flows: Hele-Shaw flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 746 , 10 May 2014 , pp. 123 - 164
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© 2014 Cambridge University Press

References

Abbyad, P., Dangla, R., Alexandrou, A. & Baroud, C. 2011 Rails and anchors: guiding and trapping droplet microreactors in two dimensions. Lab on a Chip 11, 813–821.CrossRefGoogle ScholarPubMed
Al-Housseiny, T. T., Tsai, P. A. & Stone, H. A. 2012 Control of interfacial instabilities using flow geometry. Nat. Phys. 8, 747–750.CrossRefGoogle Scholar
Casademunt, J. 2004 Viscous fingering as a paradigm of interfacial pattern formation: recent challenges and new results. Chaos 14, 809–824.CrossRefGoogle Scholar
Cliffe, K. A., Spence, A. & Tavener, S. J. 2000 The numerical analysis of bifurcation problems with application to fluid mechanics. Acta Numerica 9, 39–131.CrossRefGoogle Scholar
Couder, Y., Cardoso, O., Dupuy, D., Tavernier, P. & Thom, W. 1986a Dendritic growth in the Saffman–Taylor experiment. Europhys. Lett. 2, 437–443.CrossRefGoogle Scholar
Couder, Y., Gerard, N. & Rabaud, M. 1986b Narrow fingers in the Saffman–Taylor instability. Phys. Rev. A 34, 5175–5178.CrossRefGoogle ScholarPubMed
Dangla, R., Lee, S. & Baroud, C. N. 2011 Trapping microfluidic drops in wells of surface energy. Phys. Rev. Lett. 107, 124501.CrossRefGoogle ScholarPubMed
Dawson, G., Lee, S. & Juel, A. 2013 The trapping and release of bubbles from a linear pore. J. Fluid Mech. 722, 437–460.CrossRefGoogle Scholar
Hazel, A. L., Pailha, M., Cox, S. J. & Juel, A. 2013 Multiple states of finger propagation in partially occluded tubes. Phys. Fluids 25, 062106.CrossRefGoogle Scholar
Heap, A. & Juel, A. 2008 Anomalous bubble propagation in elastic tubes. Phys. Fluids 20, 081702.CrossRefGoogle Scholar
Heil, M. 1999 Minimal liquid bridges in non-axisymmetrically buckled elastic tubes. J. Fluid Mech. 380, 309–337.CrossRefGoogle Scholar
Heil, M. & Hazel, A. L. 2006 oomph-lib: An object-oriented multi-physics finite-element library. In Fluid–Structure Interaction, Lecture Notes on Computational Science and Engineering, vol. 53, pp. 19–49. Springer.CrossRefGoogle Scholar
Heroux, M., Bartlett, R., Howle, V., Hoekstra, R., Hu, J., Kolda, T., Lehoucq, R., Long, K., Pawlowski, R., Phipps, E., Salinger, A., Thornquist, H., Tuminaro, R., Willenbring, J. & Williams, A.2003 An overview of Trilinos. Tech. Rep. SAND2003-2927. Sandia National Laboratories.Google Scholar
Homsy, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271–311.CrossRefGoogle Scholar
Hong, D. C. 1989 Wire perturbations in the Saffman–Taylor problem: pattern selection of asymmetric fingers. Phys. Rev. A 39, 2042–2049.CrossRefGoogle ScholarPubMed
Jisiou, M., Dawson, G., Thompson, A. B., Mohr, S., Fielden, P. R., Hazel, A. L. & Juel, A. 2014 Geometry-induced oscillations of finite bubbles in microchannels. Procedia IUTAM 11, 81–88.CrossRefGoogle Scholar
Kessler, D. & Levine, H. 1986 Theory of the Saffman–Taylor ‘finger’ pattern. II. Phys. Rev. A 33, 2634–2639.CrossRefGoogle ScholarPubMed
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, 101702.CrossRefGoogle Scholar
de Lózar, A., Juel, A. & Hazel, A. L. 2008 The steady propagation of an air finger into a rectangular tube. J. Fluid Mech. 614, 173–195.CrossRefGoogle Scholar
Maxworthy, T. 1987 The nonlinear growth of a gravitationally unstable interface in a Hele-Shaw cell. J. Fluid Mech. 177, 207–232.CrossRefGoogle Scholar
McCloud, K. V. & Maher, J. V. 1995 Experimental perturbations to Saffman–Taylor flow. Phys. Rep. 260, 139–185.Google Scholar
McLean, J. W. & Saffman, P. G. 1981 The effect of surface tension on the shape of fingers in a Hele-Shaw cell. J. Fluid Mech. 102, 455–469.CrossRefGoogle Scholar
Nagel, M., Brun, P.-T. & Gallaire, F. 2014 A numerical study of droplet trapping in microfluidic devices. Phys. Fluids 26, 032002.CrossRefGoogle Scholar
Pailha, M., Hazel, A. L., Glendinning, P. A. & Juel, A. 2012 Oscillatory bubbles induced by geometrical constraint. Phys. Fluids 24, 021702.CrossRefGoogle Scholar
Park, C.-W. & Homsy, G. M. 1984 Two-phase displacement in Hele-Shaw cells: theory. J. Fluid Mech. 139, 291–308.CrossRefGoogle Scholar
Park, C. W. & Homsy, G. M. 1985 The instability of long fingers in Hele-Shaw flows. Phys. Fluids 28, 1583.CrossRefGoogle 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
Rabaud, M., Couder, Y. & Gerard, N. 1988 Dynamics and stability of anomalous Saffman–Taylor fingers. Phys. Rev. A 37, 935–947.CrossRefGoogle ScholarPubMed
Reinelt, D. A. 1987 Interface conditions for two-phase displacement in Hele-Shaw cells. J. Fluid Mech. 183, 219–234.CrossRefGoogle Scholar
Reynolds, O. 1886 On the theory of lubrication and its application to Mr. Beauchamp Tower’s experiments, including an experimental determination of the viscosity of olive oil. Phil. Trans. R. Soc. 177, 157–234.Google Scholar
Romero, L. A.1982 The fingering problem in a Hele-Shaw cell. PhD thesis, California Institute of Technology.Google Scholar
Sackinger, P. A., Schunk, P. R. & Rao, R. R. 1996 A Newton–Raphson pseudo-solid domain mapping technique for free and moving boundary problems: a finite element implementation. J. Comput. Phys. 125, 83–103.CrossRefGoogle Scholar
Saffman, P. G. 1986 Viscous fingering in Hele-Shaw cells. J. Fluid Mech. 173, 73–94.CrossRefGoogle 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
Tabeling, P., Zocchi, G. & Libchaber, A. 1987 An experimental study of the Saffman–Taylor instability. J. Fluid Mech. 177, 67–82.CrossRefGoogle Scholar
Tanveer, S. 1987 Analytic theory for the linear stability of the Saffman–Taylor finger. Phys. Fluids 30, 2318–2329.CrossRefGoogle Scholar
Tanveer, S. 2000 Surprises in viscous fingering. J. Fluid Mech. 409, 273–308.CrossRefGoogle Scholar
Taylor, G. & Saffman, P. G. 1959 A note on the motion of bubbles in a Hele-Shaw cell and porous medium. Q. J. Mech. Appl. Maths 12, 265–277.CrossRefGoogle Scholar
Vanden-Broeck, J. -M. 1983 Fingers in a Hele-Shaw cell with surface tension. Phys. Fluids 26, 2033–2034.CrossRefGoogle Scholar
Wong, H., Radke, C. J. & Morris, S. 1995 The motion of long bubbles in polygonal capillaries. Part 1. Thin films. J. Fluid Mech. 292, 71–94.CrossRefGoogle 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, 1331–1364.CrossRefGoogle Scholar
Zocchi, G., Shaw, B. E., Libchaber, A. & Kadanoff, L. P. 1987 Finger narrowing under local perturbations in the Saffman–Taylor problem. Phys. Rev. A 36 (4), 1894–1900.CrossRefGoogle ScholarPubMed