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A phase-field model of two-phase Hele-Shaw flow

Published online by Cambridge University Press:  09 October 2014

Luis Cueto-Felgueroso  and
Ruben Juanes
Luis Cueto-Felgueroso*
Affiliation:
Department of Civil Engineering: Hydraulics, Energy and the Environment, Universidad Politécnica de Madrid, 28014 Madrid, Spain Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Ruben Juanes
Affiliation:
Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: luis.cueto@upm.es
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Abstract

We propose a continuum model of two-phase flow in a Hele-Shaw cell. The model describes the multiphase three-dimensional flow in the cell gap using gap-averaged quantities such as fluid saturation and Darcy flux. Viscous and capillary coupling between the fluids in the gap leads to a nonlinear fractional flow function. Capillarity and wetting phenomena are modelled within a phase-field framework, designing a heuristic free energy functional that induces phase segregation at equilibrium. We test the model through the simulation of bubbles and viscously unstable displacements (viscous fingering). We analyse the model’s rich behaviour as a function of capillary number, viscosity contrast and cell geometry. Including the effect of wetting films on the two-phase flow dynamics opens the door to exploring, with a simple two-dimensional model, the impact of wetting and flow rate on the performance of microfluidic devices and geological flows through fractures.

JFM classification

Interfacial Flows (free surface): Fingering instability Low-Reynolds-number flows: Hele-Shaw flows Multiphase and Particle-laden Flows: Multiphase flow

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Type
Papers
Information
Journal of Fluid Mechanics , Volume 758 , 10 November 2014 , pp. 522 - 552
Copyright
© 2014 Cambridge University Press 

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References

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
Almgren, R., Dai, W.-S. & Hakim, V. 1993 Scaling behavior in anisotropic Hele-Shaw flow. Phys. Rev. Lett. 71, 3461–3464.CrossRefGoogle ScholarPubMed
Alvarez-Lacalle, E., Ortín, J. & Casademunt, J. 2004a Low viscosity contrast fingering in a rotating Hele-Shaw cell. Phys. Fluids 16, 908–924.CrossRefGoogle Scholar
Alvarez-Lacalle, E., Ortín, J. & Casademunt, J. 2004b Nonlinear Saffman–Taylor instability. Phys. Rev. Lett. 92, 054501.CrossRefGoogle ScholarPubMed
Alvarez-Lacalle, E., Ortín, J. & Casademunt, J. 2006 Relevance of dynamic wetting in viscous fingering patterns. Phys. Rev. E 74, 025302.CrossRefGoogle ScholarPubMed
Anderson, D. M., McFadden, G. B. & Wheeler, A. A. 1998 Diffuse-interface methods in fluid mechanics. Annu. Rev. Mater. Res. 30, 139–165.Google Scholar
Anjos, P. H. A. & Miranda, J. A. 2013 Radial viscous fingering: wetting film effects on pattern-forming mechanisms. Phys. Rev. E 88, 053003.CrossRefGoogle ScholarPubMed
Antanovskii, L. K. 1995 A phase-field model of capillarity. Phys. Fluids 7, 747–753.CrossRefGoogle Scholar
Arnéodo, A., Couder, Y., Grasseau, G., Hakim, V. & Rabaud, M. 1989 Uncovering the analytical Saffman–Taylor finger in unstable viscous fingering and diffusion-limited aggregation. Phys. Rev. Lett. 63, 984–987.CrossRefGoogle Scholar
Badalassi, V. E., Ceniceros, H. D. & Banerjee, S. 2003 Computation of multiphase systems with phase field models. J. Comput. Phys. 190, 371–397.CrossRefGoogle Scholar
Banpurkar, A. G., Limaye, A. V. & Ogale, S. B. 2000 Occurrence of coexisting dendrite morphologies: immiscible fluid displacement in an anisotropic radial Hele-Shaw cell under a high flow rate regime. Phys. Rev. E 61, 5507–5511.CrossRefGoogle Scholar
Baroud, C. N., Gallaire, F. & Dangla, R. 2010 Dynamics of microfluidic droplets. Lab on a Chip 10, 2032–2045.CrossRefGoogle ScholarPubMed
Bear, J. 1972 Dynamics of Fluids in Porous Media. Wiley.Google Scholar
Ben-Jacob, E., Deitscher, G., Garik, P., Goldenfeld, N. G. & Lareah, Y. 1986 Formation of a dense branching morphology in interfacial growth. Phys. Rev. Lett. 57, 1903–1906.CrossRefGoogle ScholarPubMed
Ben-Jacob, E. & Garik, P. 1990 The formation of patterns in non-equilibrium growth. Nature 343, 923–930.CrossRefGoogle Scholar
Bensimon, D., Kadanoff, L. P., Liang, S., Shraiman, B. I. & Tang, C. 1986 Viscous flow in two dimensions. Rev. Mod. Phys. 58, 977–999.CrossRefGoogle Scholar
Benzi, R., Sbragaglia, M., Bernaschi, M. & Succi, S. 2011 Phase-field model of long-time glasslike relaxation in binary fluid mixtures. Phys. Rev. Lett. 106, 164501.CrossRefGoogle ScholarPubMed
Bertozzi, A. L., Ju, N. & Lu, H.-W. 2011 A biharmonic-modified forward time stepping method for fourth order nonlinear diffusion equations. J. Discrete Continuous Dyn. Syst. 29, 1367–1391.CrossRefGoogle Scholar
Bertozzi, A. L., Münch, A., Fanton, X. & Cazabat, A. M. 1998 Contact line stability and ‘undercompressive shocks’ in driven thin film flow. Phys. Rev. Lett. 81, 5169–5172.CrossRefGoogle Scholar
Bertozzi, A. L., Münch, A. & Shearer, M. 1999 Undercompressive shocks in thin film flows. Physica D 134, 431–464.CrossRefGoogle Scholar
Bertozzi, A. L. & Shearer, M. 2000 Existence of undercompressive traveling waves in thin film equations. SIAM J. Math. Anal. 32, 194–213.CrossRefGoogle Scholar
Boyer, F. 2002 A theoretical and numerical model for the study of incompressible mixture flows. Comput. Fluids 31, 41–68.CrossRefGoogle Scholar
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166–188.CrossRefGoogle Scholar
Buka, A., Kertész, J. & Vicsek, T. 1986 Transitions of viscous fingering patterns in nematic liquid crystals. Nature 323, 424–425.CrossRefGoogle Scholar
Buka, A., Palffy-Muhoray, P. & Rácz, Z. 1987 Viscous fingering in liquid crystals. Phys. Rev. A 36, 3984–3989.CrossRefGoogle ScholarPubMed
Cahn, J. W. 1961 On spinodal decomposition. Acta Metall. 9, 795–801.CrossRefGoogle Scholar
Cahn, J. W. & Hilliard, J. E. 1958 Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267.CrossRefGoogle Scholar
Carrillo, L., Magdaleno, F. X., Casademunt, J. & Ortin, J. 1996 Experiments in a rotating Hele-Shaw cell. Phys. Rev. E 54, 6260–6267.CrossRefGoogle Scholar
Carrillo, L., Soriano, J. & Ortín, J. 1999 Radial displacement of a fluid annulus in a rotating Hele-Shaw cell. Phys. Fluids 11, 778–785.CrossRefGoogle Scholar
Carrillo, L., Soriano, J. & Ortín, J. 2000 Interfacial instabilities of a fluid annulus in a rotating Hele-Shaw cell. Phys. Fluids 12, 1685–1698.CrossRefGoogle Scholar
Casademunt, J. 2004 Viscous fingering as a paradigm of interfacial pattern formation: recent results and new challenges. Chaos 14, 809–824.CrossRefGoogle ScholarPubMed
Casademunt, J. & Jasnow, D. 1991 Defect dynamics in viscous fingering. Phys. Rev. Lett. 67, 3677–3680.CrossRefGoogle ScholarPubMed
Casademunt, J. & Magdaleno, F. X. 2000 Dynamics and selection of fingering patterns. Recent developments in the Saffman–Taylor problem. Phys. Rep. 337, 1–35.CrossRefGoogle Scholar
Ceniceros, H. D. & Villalobos, J. M. 2002 Topological reconfiguration in expanding Hele-Shaw flow. J. Turbul. 3, N37.CrossRefGoogle Scholar
Chen, C.-Y., Huang, Y.-S. & Miranda, J. A. 2011 Diffuse-interface approach to Hele-Shaw flows. Phys. Rev. E 84, 046302.CrossRefGoogle ScholarPubMed
Combescot, R., Dombre, T., Hakim, V., Pomeau, Y. & Pumir, A. 1986 Shape selection of Saffman–Taylor fingers. Phys. Rev. Lett. 56, 2036–2039.CrossRefGoogle ScholarPubMed
Corvera-Poiré, E. & Amar, M. B. 1998 Finger behavior of a shear thinning fluid in a Hele-Shaw cell. Phys. Rev. Lett. 81, 2048–2051.CrossRefGoogle Scholar
Cueto-Felgueroso, L. & Juanes, R. 2008 Nonlocal interface dynamics and pattern formation in gravity-driven unsaturated flow through porous media. Phys. Rev. Lett. 101, 244504.CrossRefGoogle ScholarPubMed
Cueto-Felgueroso, L. & Juanes, R. 2012 Macroscopic phase-field model of partial wetting: bubbles in a capillary tube. Phys. Rev. Lett. 108, 144502.CrossRefGoogle Scholar
Dangla, R., Lee, S. & Baroud, C. N. 2011 Trapping microfluidic drops in wells of surface energy. Phys. Rev. Lett. 107, 124501.CrossRefGoogle ScholarPubMed
Decker, E. L., Ignés-Mullol, J. & Maher, J. V. 1999 Effect of lattice defects on Hele-Shaw flow over an etched lattice. Phys. Rev. E 60, 1767–1774.CrossRefGoogle ScholarPubMed
Degregoria, A. J. & Schwartz, L. W. 1986 A boundary-integral method for two-phase displacement in Hele-Shaw cells. J. Fluid Mech. 164, 383–400.CrossRefGoogle Scholar
Dias, E. O. & Miranda, J. A. 2013 Wavelength selection in Hele-Shaw flows: a maximum-amplitude criterion. Phys. Rev. E 88, 013016.Google ScholarPubMed
Ding, H., Spelt, P. D. M. & Shu, C. 2007 Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 226, 2078–2095.CrossRefGoogle Scholar
van Duijn, C. J., Peletier, L. A. & Pop, I. S. 2007 A new class of entropy solutions of the Buckley–Leverett equation. SIAM J. Math. Anal. 39, 507–536.CrossRefGoogle Scholar
E, W. & Palffy-Muhoray, P. 1997 Phase separation in incompressible systems. Phys. Rev. E 55, R3844.CrossRefGoogle Scholar
Eck, W. & Siekmann, J. 1978 On bubble motion in Hele-Shaw cell, a possibility to study two-phase flows under reduced gravity. Ing.-Arch. 47, 153–168.CrossRefGoogle Scholar
Emmerich, H. 2008 Advances of and by phase-field modelling in condensed-matter physics. Adv. Phys. 57, 1–87.CrossRefGoogle Scholar
Fan, Y. & Pop, I. S. 2011 A class of pseudo-parabolic equations: existence, uniqueness of weak solutions, and error estimates for the Euler-implicit discretization. Math. Meth. Appl. Sci. 34, 2329–2339.CrossRefGoogle 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.CrossRefGoogle Scholar
Fast, P. & Shelley, M. J. 2004 A moving overset grid method for interface dynamics applied to non-Newtonian Hele-Shaw flow. J. Comput. Phys. 195, 117–142.CrossRefGoogle Scholar
Folch, R., Casademunt, J., Hernández-Machado, A. & Ramirez-Piscina, L. 1999a Phase-field models for Hele-Shaw flows with arbitrary viscosity contrast. I. Theoretical approach. Phys. Rev. E 60, 1724–1733.CrossRefGoogle ScholarPubMed
Folch, R., Casademunt, J., Hernández-Machado, A. & Ramirez-Piscina, L. 1999b Phase-field models for Hele-Shaw flows with arbitrary viscosity contrast. II. Numerical study. Phys. Rev. E 60, 1734–1740.CrossRefGoogle ScholarPubMed
Fourar, M. & Lenormand, R.1998 A viscous coupling model for relative permeabilities in fractures. In SPE Annual Technical Conference and Exhibition, New Orleans, 27–30 September. Paper SPE-49006-MS.CrossRefGoogle Scholar
de Gennes, P. G. 1980 Dynamics of fluctuations and spinodal decomposition in polymer blends. J. Chem. Phys. 72, 4756–4763.CrossRefGoogle Scholar
Glasner, K. 2003 A diffuse-interface approach to Hele-Shaw flow. Nonlinearity 16, 49–66.CrossRefGoogle Scholar
Glass, R. J., Rajaram, H., Nicholl, M. J. & Detwiler, R. L. 2001 The interaction of two fluid phases in fractured media. Curr. Opin. Colloid Interface Sci. 6, 223–235.CrossRefGoogle Scholar
Gurtin, M. E. 1994 Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance. Physica D 92, 178–192.CrossRefGoogle Scholar
Hassanizadeh, S. M. & Gray, W. G. 1993 Thermodynamic basis of capillary pressure in porous media. Water Resour. Res. 29, 3389–3405.CrossRefGoogle Scholar
Hernández-Machado, A., Lacasta, M., Mayoral, E. & Corvera-Poiré, E. 2003 Phase-field model of Hele-Shaw flows in the high-viscosity contrast regime. Phys. Rev. E 68, 046310.CrossRefGoogle ScholarPubMed
Homsy, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271–311.CrossRefGoogle Scholar
Honda, T., Honjo, H. & Katsuragi, H. 2006 Experimental study on the morphology in a large Hele-Shaw cell. J. Phys. Soc. Japan 75, 034005.CrossRefGoogle Scholar
Hou, T. Y., Li, Z., Osher, S. & Zhao, H. 1997 A hybrid method for moving interface problems with application to the Hele-Shaw flow. J. Comput. Phys. 134, 236–252.CrossRefGoogle Scholar
Hou, T. Y., Lowengrub, J. S. & Shelley, M. J. 1994 Removing the stiffness from interfacial flows with surface tension. J. Comput. Phys. 114, 312–338.CrossRefGoogle Scholar
Hou, T. Y., Lowengrub, J. S. & Shelley, M. J. 2001 Boundary integral methods for multicomponent fluids and multiphase materials. J. Comput. Phys. 169, 302–362.CrossRefGoogle Scholar
Howison, S. D. 1986 Fingering in Hele-Shaw cells. J. Fluid Mech. 167, 439–453.CrossRefGoogle Scholar
Jacqmin, D. 1999 Calculation of two-phase Navier–Stokes flows using phase-field modeling. J. Comput. Phys. 155, 96–127.CrossRefGoogle Scholar
Jasnow, D. & Viñals, J. 1996 Coarse-grained description of thermo-capillary flow. Phys. Fluids 8, 660–669.CrossRefGoogle Scholar
Kawaguchi, M., Hibino, Y. & Kato, T. 2001 Anisotropy effects of Hele-Shaw cells on viscous fingering instability in dilute polymer solutions. Phys. Rev. E 64, 051806.CrossRefGoogle ScholarPubMed
Kawaguchi, M., Shimomoto, K., Shibata, A. & Kato, T. 1999 Effect of anisotropy on viscous fingering patterns of polymer solutions in linear Hele-Shaw cells. Chaos 9, 323–328.CrossRefGoogle ScholarPubMed
Kawaguchi, M., Yamazaki, S., Yonekura, K. & Kato, T. 2004 Viscous fingering instabilities in an oil in water emulsion. Phys. Fluids 16, 1908–1914.CrossRefGoogle Scholar
Kim, J. 2005 A continuous surface tension force formulation for diffuse-interface models. J. Comput. Phys. 204, 784–804.CrossRefGoogle Scholar
Kondic, L., Palffy-Muhoray, P. & Shelley, M. J. 1996 Models of non-Newtonian Hele-Shaw flow. Phys. Rev. E 54, R4536.CrossRefGoogle ScholarPubMed
Kondic, L., Palffy-Muhoray, P. & Shelley, M. J. 1998 Non-Newtonian Hele-Shaw flow and the Saffman–Taylor instability. Phys. Rev. Lett. 80, 1433–1436.CrossRefGoogle Scholar
Kopf-Sill, A. R. & Homsy, G. M. 1988a Bubble motion in a Hele-Shaw cell. Phys. Fluids 31, 18–26.CrossRefGoogle Scholar
Kopf-Sill, A. R. & Homsy, G. M. 1988b Nonlinear unstable viscous fingers in Hele-Shaw flows. I. Experiments. Phys. Fluids 31, 242–249.CrossRefGoogle Scholar
Lajeunesse, E. & Couder, Y. 2000 On the tip-splitting instability of viscous fingers. J. Fluid Mech. 419, 125–149.CrossRefGoogle Scholar
Lamorguese, A. G. & Mauri, R. 2008 Diffuse-interface modeling of phase segregation in liquid mixtures. Intl J. Multiphase Flow 34, 987–995.CrossRefGoogle Scholar
Lax, P. D. 1957 Hyperbolic systems of conservation laws II. Commun. Pure Appl. Maths 10, 537–566.CrossRefGoogle Scholar
Lee, H.-G., Lowengrub, J. S. & Goodman, J. 2002a Modeling pinchoff and reconnection in a Hele-Shaw cell. I. The models and their calibration. Phys. Fluids 14, 492–513.CrossRefGoogle Scholar
Lee, H.-G., Lowengrub, J. S. & Goodman, J. 2002b Modeling pinchoff and reconnection in a Hele-Shaw cell. II. Analysis and simulation in the nonlinear regime. Phys. Fluids 14, 514–545.CrossRefGoogle Scholar
Lemaire, E., Levitz, P., Daccords, G. & Damme, H. V. 1991 From viscous fingering to viscoelastic fracturing in colloidal fluids. Phys. Rev. Lett. 67, 2009–2012.CrossRefGoogle ScholarPubMed
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
Li, S., Lowengrub, J. S. & Leo, P. H. 2007 A rescaling scheme with application to the long-time simulation of viscous fingering in a Hele-Shaw cell. J. Comput. Phys. 225, 554–567.CrossRefGoogle Scholar
Lindner, A., Bonn, D., Corvera Poiré, E., Ben Amar, M. & Meunier, J. 2002 Viscous fingering in non-Newtonian fluids. J. Fluid Mech. 469, 237–256.CrossRefGoogle Scholar
Lindner, A., Bonn, D. & Meunier, J. 2000a Viscous fingering in a shear-thinning fluid. Phys. Fluids 12, 256–261.CrossRefGoogle Scholar
Lindner, A., Coussot, P. & Bonn, D. 2000b Viscous fingering in a yield stress fluid. Phys. Rev. Lett. 85, 314–317.CrossRefGoogle Scholar
Lowengrub, J. S. & Truskinovsky, L. 1998 Quasi-incompressible Cahn–Hilliard fluids and topological transitions. Proc. R. Soc. Lond. A 454, 2617–2654.CrossRefGoogle Scholar
Lu, H.-W., Glasner, K., Bertozzi, A. L. & Kim, C.-J. 2007 A diffuse-interface model for electrowetting drops in a Hele-Shaw cell. J. Fluid Mech. 590, 411–435.CrossRefGoogle Scholar
Maher, J. V. 1985 Development of viscous fingering patterns. Phys. Rev. Lett. 54, 1498–1501.CrossRefGoogle ScholarPubMed
Maruvada, S. R. & Park, C. W. 1996 Retarded motion of bubbles in Hele-Shaw cells. Phys. Fluids 8, 3229–3233.CrossRefGoogle Scholar
Maxworthy, T. 1986 Bubble formation, motion and interaction in a Hele-Shaw cell. J. Fluid Mech. 173, 95–114.CrossRefGoogle Scholar
McCloud, K. V. & Maher, J. V. 1995a Experimental perturbations to Hele-Shaw flow. Phys. Rep. 260, 139–185.CrossRefGoogle Scholar
McCloud, K. V. & Maher, J. V. 1995b Pattern selection in an anisotropic Hele-Shaw cell. Phys. Rev. E 51, 1184–1190.CrossRefGoogle 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
Meiburg, E. 1989 Bubbles in a Hele-Shaw cell: numerical simulation and three-dimensional effects. Phys. Fluids A 1, 938–946.CrossRefGoogle Scholar
Meiburg, E. & Homsy, G. M. 1988 Nonlinear unstable viscous fingers in Hele-Shaw flows. II. Numerical simulation. Phys. Fluids 31, 429–439.CrossRefGoogle Scholar
Miranda, J. A. & Alvarez-Lacalle, E. 2005 Viscosity contrast effects on fingering formation in rotating Hele-Shaw flows. Phys. Rev. E 72, 026306.CrossRefGoogle ScholarPubMed
Moore, M. G., Juel, A., Burgess, J. M., McCormick, W. D. & Swinney, H. L. 2002 Fluctuations in viscous fingering. Phys. Rev. E 65, 030601(R).CrossRefGoogle ScholarPubMed
Nagatsu, Y., Matsuda, K., Kato, Y. & Tada, Y. 2007 Experimental study on miscible viscous fingering involving viscosity changes induced by variations in chemical species concentrations due to chemical reactions. J. Fluid Mech. 571, 475–493.CrossRefGoogle Scholar
Nagel, M. & Gallaire, F. 2013 A new prediction of wavelength selection in radial viscous fingering involving normal and tangential stresses. Phys. Fluids 25, 124107.CrossRefGoogle Scholar
Nguyen, S., Folch, R., Verma, V. K., Henry, H. & Plapp, M. 2010 Phase-field simulations of viscous fingering in shear-thinning fluids. Phys. Fluids 22, 103102.CrossRefGoogle Scholar
Otto, F. & E, W. 1997 Thermodynamically driven incompressible fluid mixtures. J. Chem. Phys. 107, 10177–10184.CrossRefGoogle Scholar
Park, S. S. & Durian, D. J. 1994 Viscous and elastic fingering instabilities in foam. Phys. Rev. Lett. 72, 3347–3350.CrossRefGoogle ScholarPubMed
Park, C.-W., Gorell, S. & Homsy, G. M. 1984 Two-phase displacement in Hele-Shaw cells: experiments. J. Fluid Mech. 141, 257–287.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, 1583–1585.CrossRefGoogle Scholar
Paterson, L. 1981 Radial fingering in a Hele-Shaw cell. J. Fluid Mech. 113, 513–529.CrossRefGoogle Scholar
Persoff, P. & Pruess, K. 1995 Two-phase flow visualization and relative permeability measurement in natural rough-walled rock fractures. Water Resour. Res. 31, 1175–1186.CrossRefGoogle Scholar
Saffman, P. G. & Taylor, G. 1958a 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
Saffman, P. G. & Taylor, G. I. 1958b 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
Sarkar, S. K. 1990 Scaling dynamics of immiscible radial viscous fingering. Phys. Rev. Lett. 65, 2680–2683.CrossRefGoogle ScholarPubMed
Sarkar, S. K. & Jasnow, D. 1989 Viscous fingering in an anisotropic Hele-Shaw cell. Phys. Rev. A 39, 5299–5307.CrossRefGoogle Scholar
Shelley, M., Tian, F.-R. & Wlodarski, K. 1997 Hele-Shaw flow and pattern formation in a time-dependent gap. Nonlinearity 10, 1471–1495.CrossRefGoogle Scholar
Siegel, M. & Tanveer, S. 1996 Singular perturbation of smoothly evolving Hele-Shaw solutions. Phys. Rev. Lett. 76, 419–422.CrossRefGoogle ScholarPubMed
Siegel, M., Tanveer, S. & Dai, W.-S. 1996 Singular effects of surface tension in evolving Hele-Shaw flows. J. Fluid Mech. 323, 201–236.CrossRefGoogle Scholar
Somfai, E., Sander, L. M. & Ball, R. C. 1999 Scaling and crossovers in diffusion-limited aggregation. Phys. Rev. Lett. 83, 5523–5526.CrossRefGoogle Scholar
Strait, M., Shearer, M., Levy, R., Cueto-Felgueroso, L. & Juanes, R.2013 Two fluid flow in a capillary tube. Topics from the 9th Annual UNCG Regional Mathematics and Statistics Conference, Springer Proceedings in Mathematics & Statistics (in Press).CrossRefGoogle Scholar
Sun, Y. & Beckermann, C. 2004 Diffuse interface modeling of two-phase flows based on averaging: mass and momentum equations. Physica D 198, 281–308.CrossRefGoogle Scholar
Sun, Y. & Beckermann, C. 2007 Sharp interface tracking using the phase-field equation. J. Comput. Phys. 220, 626–653.CrossRefGoogle Scholar
Sun, Y. & Beckermann, C. 2008 A two-phase diffuse-interface model for Hele-Shaw flows with large property contrasts. Physica D 237, 3089–3098.CrossRefGoogle 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. 1986 The effect of surface tension on the shape of a Hele-Shaw cell bubble. Phys. Fluids 29, 3537–3548.CrossRefGoogle Scholar
Tanveer, S. 2000 Surprises in viscous fingering. J. Fluid Mech. 409, 273–308.CrossRefGoogle Scholar
Tryggvason, G. & Aref, H. 1983 Numerical experiments on Hele-Shaw flow with a sharp interface. J. Fluid Mech. 136, 1–30.CrossRefGoogle Scholar
Vlad, D. H. & Maher, J. V. 2000 Tip-splitting instabilities in the channel Saffman–Taylor flow of constant viscosity elastic fluids. Phys. Rev. E 61, 5439–5444.CrossRefGoogle ScholarPubMed
Weinstein, S. J., Dussan, E. B. & Ungar, L. H. 1990 A theoretical study of two-phase flow through a narrow gap with a moving contact line: viscous fingering in a Hele-Shaw cell. J. Fluid Mech. 221, 53–76.CrossRefGoogle Scholar
Whitaker, N. 1994 Some numerical methods for the Hele-Shaw equations. J. Comput. Phys. 111, 81–88.CrossRefGoogle Scholar
Witelski, T. P. 1998 Equilibrium interface solutions of a degenerate singular Cahn–Hilliard equation. Appl. Maths Lett. 11, 127–133.CrossRefGoogle Scholar
Yamamoto, T., Kamikawa, H., Tanaka, H., Nakamura, K. & Mori, N. 2001 Viscous fingering of non-Newtonian fluids in a rectangular Hele-Shaw cell. Nihon Reoroji Gakkaishi 29, 81–87.CrossRefGoogle Scholar
Yue, P., Feng, J. J., Liu, C. & Shen, J. 2004 A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293–317.CrossRefGoogle Scholar
Zhao, H., Casademunt, J., Yeung, C. & Maher, J. V. 1992 Perturbing Hele-Shaw flow with a small gap gradient. Phys. Rev. A 45, 2455–2460.CrossRefGoogle ScholarPubMed
Zhao, H., Casademunt, J., Yeung, C. & Maher, J. V. 1993 Erratum: Perturbing Hele-Shaw flow with a small gap gradient. Phys. Rev. E 48, 1601.CrossRefGoogle ScholarPubMed
Zhao, H. & Maher, J. V. 1993 Associating-polymer effects in a Hele-Shaw experiment. Phys. Rev. E 47, 4278–4283.CrossRefGoogle Scholar

Cueto-Felgueroso supplementary movie

Numerical simulation of viscous fingering in a Hele-Shaw cell (corresponds to Figs. 13c and 14 in the manuscript)

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