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A REVIEW OF ONE-PHASE HELE-SHAW FLOWS AND A LEVEL-SET METHOD FOR NONSTANDARD CONFIGURATIONS

Part of: Partial differential equations, boundary value problems Incompressible viscous fluids Miscellaneous topics - Partial differential equations Geometric function theory

Published online by Cambridge University Press:  23 September 2021

LIAM C. MORROW Open the ORCID record for LIAM C. MORROW [Opens in a new window] ,
TIMOTHY J. MORONEY Open the ORCID record for TIMOTHY J. MORONEY [Opens in a new window] ,
MICHAEL C. DALLASTON Open the ORCID record for MICHAEL C. DALLASTON [Opens in a new window]  and
SCOTT W. MCCUE Open the ORCID record for SCOTT W. MCCUE [Opens in a new window]
LIAM C. MORROW
Affiliation:
Department of Engineering Science, University of Oxford, OxfordOX1 3PJ, UK; e-mail: [email protected] School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, 4001, Australia; e-mail: [email protected], [email protected]
TIMOTHY J. MORONEY
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, 4001, Australia; e-mail: [email protected], [email protected]
MICHAEL C. DALLASTON
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, 4001, Australia; e-mail: [email protected], [email protected]
SCOTT W. MCCUE*
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, 4001, Australia; e-mail: [email protected], [email protected]
*
e-mail: [email protected]
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Abstract

The classical model for studying one-phase Hele-Shaw flows is based on a highly nonlinear moving boundary problem with the fluid velocity related to pressure gradients via a Darcy-type law. In a standard configuration with the Hele-Shaw cell made up of two flat stationary plates, the pressure is harmonic. Therefore, conformal mapping techniques and boundary integral methods can be readily applied to study the key interfacial dynamics, including the Saffman–Taylor instability and viscous fingering patterns. As well as providing a brief review of these key issues, we present a flexible numerical scheme for studying both the standard and nonstandard Hele-Shaw flows. Our method consists of using a modified finite-difference stencil in conjunction with the level-set method to solve the governing equation for pressure on complicated domains and track the location of the moving boundary. Simulations show that our method is capable of reproducing the distinctive morphological features of the Saffman–Taylor instability on a uniform computational grid. By making straightforward adjustments, we show how our scheme can easily be adapted to solve for a wide variety of nonstandard configurations, including cases where the gap between the plates is linearly tapered, the plates are separated in time, and the entire Hele-Shaw cell is rotated at a given angular velocity.

Keywords

Hele-Shaw flowSaffman–Taylor instabilityviscous fingering patternsmoving boundary problemconformal mappinglevel-set method

MSC classification

Primary: 76D27: Other free-boundary flows; Hele-Shaw flows
Secondary: 35R37: Moving boundary problems 65N06: Finite difference methods 30C35: General theory of conformal mappings
Type
Research Article
Information
The ANZIAM Journal , Volume 63 , Issue 3 , July 2021 , pp. 269 - 307
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Copyright
© Australian Mathematical Society 2021

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Footnotes

*

This is a contribution to the series of invited papers by past Tuck medallists (Editorial, Issue 62(1)). Scott W. McCue was awarded the 2019 Tuck medal.

References

Al-Housseiny, T. T., Christov, I. C. and Stone, H. A., “Two-phase fluid displacement and interfacial instabilities under elastic membranes”, Phys. Rev. Lett. 111 (2013) 034502; doi:10.1103/PhysRevLett.111.034502.CrossRefGoogle ScholarPubMed
Al-Housseiny, T. T. and Stone, H. A., “Controlling viscous fingering in tapered Hele-Shaw cells”, Phys. Fluids 25 (2013) 092102; doi:10.1063/1.4819317.CrossRefGoogle Scholar
Al-Housseiny, T. T., Tsai, P. A. and Stone, H. A., “Control of interfacial instabilities using flow geometry”, Nat. Phys. 8 (2012) 747–750; doi:10.1038/nphys2396.CrossRefGoogle Scholar
Alvarez-Lacalle, E., Gadêlha, H. and Miranda, J. A., “Coriolis effects on fingering patterns under rotation”, Phys. Rev. E 78 (2008) 026305; doi:10.1103/PhysRevE.78.026305.CrossRefGoogle ScholarPubMed
Anjos, P. H. A., Alvarez, V. M. M., Dias, E. O. and Miranda, J. A., “Rotating Hele-Shaw cell with a time-dependent angular velocity”, Phys. Rev. Fluids 2 (2017) 124003; doi:10.1103/PhysRevFluids.2.124003.CrossRefGoogle Scholar
Anjos, P. H. A., Dias, E. O. and Miranda, J. A., “Kinetic undercooling in Hele-Shaw flows”, Phys. Rev. E 92 (2015) 043019; doi:10.1103/PhysRevE.92.043019.CrossRefGoogle ScholarPubMed
Anjos, P. H. A., Dias, E. O. and Miranda, J. A., “Fingering instability transition in radially tapered Hele-Shaw cells: insights at the onset of nonlinear effects”, Phys. Rev. Fluids 3 (2018) 124004; doi:10.1103/PhysRevFluids.3.124004.CrossRefGoogle Scholar
Anjos, P. H. A. and Miranda, J. A., “Radial viscous fingering: wetting film effects on pattern-forming mechanisms”, Phys. Rev. E 88 (2013) 053003; doi:10.1103/PhysRevE.88.053003.CrossRefGoogle ScholarPubMed
Anjos, P. H. A., Zhao, M., Lowengrub, J., Bao, W. and Li, S., “Controlling fingering instabilities in Hele-Shaw flows in the presence of wetting film effects”, Phys. Rev. E 103 (2021) 063105; doi:10.1103/PhysRevE.103.063105.CrossRefGoogle ScholarPubMed
Aronsson, G. and Janfalk, U., “On Hele-Shaw flow of power-law fluids”, European J. Appl. Math. 3 (1992) 343–366; doi:10.1017/S0956792500000905.CrossRefGoogle Scholar
Arun, R., Dawson, S. T. M., Schmid, P. J., Laskari, A. and McKeon, B. J., “Control of instability by injection rate oscillations in a radial Hele-Shaw cell”, Phys. Rev. Fluids 5 (2020) 123902; doi:10.1103/PhysRevFluids.5.123902.CrossRefGoogle Scholar
Beeson-Jones, T. H. and Woods, A. W., “Control of viscous instability by variation of injection rate in a fluid with time-dependent rheology”, J. Fluid Mech. 829 (2017) 214–235; doi:10.1017/jfm.2017.581.CrossRefGoogle Scholar
Ben-Jacob, E. and Garik, P., “The formation of patterns in non-equilibrium growth”, Nature 343 (1990) 523–530; doi:10.1038/343523a0.CrossRefGoogle Scholar
Bongrand, G. and Tsai, P. A., “Manipulation of viscous fingering in a radially tapered cell geometry”, Phys. Rev. E 97 (2018) 061101; doi:10.1103/PhysRevE.97.061101.CrossRefGoogle Scholar
Bookstein, F. L., “Principal warps: thin-plate splines and the decomposition of deformations”, IEEE Trans. Pattern Anal. Mach. Intell. 11 (1989) 567–585; doi:10.1109/34.24792.CrossRefGoogle Scholar
Bretherton, F. P., “The motion of long bubbles in tubes”, J. Fluid Mech. 10 (1961) 166–188; doi:10.1017/S0022112061000160.CrossRefGoogle Scholar
Carrillo, L., Magdaleno, F. X., Casademunt, J. and Ortín, J., “Experiments in a rotating Hele-Shaw cell”, Phys. Rev. E 54 (1996) 6260; doi:10.1103/PhysRevE.54.6260.CrossRefGoogle Scholar
Casademunt, J., “Viscous fingering as a paradigm of interfacial pattern formation: recent results and new challenges”, Chaos 14 (2004) 809–824; doi:10.1063/1.1784931.CrossRefGoogle ScholarPubMed
Ceniceros, H. D. and Hou, T. Y., “The singular perturbation of surface tension in Hele-Shaw flows”, J. Fluid Mech. 409 (2000) 251–272; doi:10.1017/S0022112099007703.CrossRefGoogle Scholar
Ceniceros, H. D., Hou, T. Y. and Si, H., “Numerical study of Hele-Shaw flow with suction”, Phys. Fluids 11 (1999) 2471–2486; doi:10.1063/1.870112.CrossRefGoogle Scholar
Chapman, S. J., “On the role of Stokes lines in the selection of Saffman–Taylor fingers with small surface tension”, European J. Appl. Math. 10 (1999) 513–534; doi:10.1017/S0956792599003848.CrossRefGoogle Scholar
Chapman, S. J. and King, J. R., “The selection of Saffman–Taylor fingers by kinetic undercooling”, J. Engrg. Math. 46 (2003) 1–32; doi:10.1023/A:1022860705459. CrossRefGoogle Scholar
Chen, C.-Y., Huang, Y.-S. and Miranda, J. A., “Radial Hele-Shaw flow with suction: fully nonlinear pattern formation”, Phys. Rev. E 89 (2014) 053006; doi:10.1103/PhysRevE.89.053006.CrossRefGoogle ScholarPubMed
Chen, J.-D., “Radial viscous fingering patterns in Hele-Shaw cells”, Exp. Fluids 5 (1987) 363–371; doi:10.1103/PhysRevE.78.016306.CrossRefGoogle Scholar
Chen, S., Merriman, B., Osher, S. and Smereka, P., “A simple level set method for solving Stefan problems”, J. Comput. Phys. 135 (1997) 8–29; doi:10.1006/jcph.1997.5721.CrossRefGoogle Scholar
Coco, A. and Russo, G., “Finite-difference ghost-point multigrid methods on Cartesian grids for elliptic problems in arbitrary domains”, J. Comput. Phys. 241 (2013) 464–501; doi:10.1016/j.jcp.2012.11.047.CrossRefGoogle Scholar
Combescot, R., Dombre, T., Hakim, V., Pomeau, Y. and Pumir, A., “Shape selection of Saffman–Taylor fingers”, Phys. Rev. Lett. 56 (1986) 2036; doi:10.1103/PhysRevLett.56.2036.CrossRefGoogle ScholarPubMed
Coutinho, Í. M. and Miranda, J. A., “Control of viscous fingering through variable injection rates and time-dependent viscosity fluids: beyond the linear regime”, Phys. Rev. E 102 (2020) 063102; doi:10.1103/PhysRevE.80.026303.CrossRefGoogle ScholarPubMed
Crowdy, D. and Tanveer, S., “The effect of finiteness in the Saffman–Taylor viscous fingering problem”, J. Comput. Phys. 114 (2004) 1501–1536; doi:10.1023/B:JOSS0000013962.78542.33.Google Scholar
Cummings, L. J. and King, J. R., “Hele-Shaw flow with a point sink: generic solution breakdown”, European J. Appl. Math. 15 (2004) 1–37; doi:10.1017/S095679250400539X.CrossRefGoogle Scholar
Cummings, L. M., Howison, S. D. and King, J. R., “Two-dimensional Stokes and Hele-Shaw flows with free surfaces”, Euro. J. Appl. Math. 10 (1999) 635–680; doi:10.1017/S0956792599003964.CrossRefGoogle Scholar
Cuttle, C., Pihler-Puzović, D. and Juel, A., “Dynamics of front propagation in a compliant channel”, J. Fluid Mech. 886 (2020) A20; doi:10.1017/jfm.2019.1037.CrossRefGoogle Scholar
Dai, W.-S. and Shelley, M. J., “A numerical study of the effect of surface tension and noise on an expanding Hele-Shaw bubble”, Phys. Fluids 5 (1993) 2131–2146; doi:10.1063/1.858553.CrossRefGoogle Scholar
Dallaston, M. C. and McCue, S. W., “New exact solutions for Hele-Shaw flow in doubly connected regions”, Phys. Fluids 24 (2012); doi:10.1063/1.4711274.CrossRefGoogle Scholar
Dallaston, M. C. and McCue, S. W., “Bubble extinction in Hele-Shaw flow with surface tension and kinetic undercooling regularization”, Nonlinearity 26 (2013) 1639–1665; doi:10.1088/0951-7715/26/6/1639.CrossRefGoogle Scholar
Dallaston, M. C. and McCue, S. W., “Corner and finger formation in Hele-Shaw flow with kinetic undercooling regularisation”, European J. Appl. Math. 25 (2014) 707–727; doi:10.1017/S0956792514000230P.CrossRefGoogle Scholar
DeGregoria, A. J. and Schwartz, L. W., “A boundary-integral method for two-phase displacement in Hele-Shaw cells”, J. Fluid Mech. 164 (1986) 383–400; doi:10.1017/S0022112086002604.CrossRefGoogle Scholar
Dias, E. O., Alvarez-Lacalle, E., Carvalho, M. S. and Miranda, J. A., “Minimization of viscous fluid fingering: a variational scheme for optimal flow rates”, Phys. Rev. Lett. 109 (2012) 144502; doi:10.1103/PhysRevLett.109.144502.CrossRefGoogle ScholarPubMed
Dias, E. O. and Miranda, J., “Control of radial fingering patterns: a weakly nonlinear approach”, Phys. Rev. E 81 (2010) 016312; doi:10.1103/PhysRevE.81.016312.CrossRefGoogle ScholarPubMed
Dias, E. O., Parisio, F. and Miranda, J. A., “Suppression of viscous fluid fingering: a piecewise-constant injection process”, Phys. Rev. E 82 (2010) 067301; doi:10.1103/PhysRevE.82.067301.CrossRefGoogle ScholarPubMed
Ebert, U., Meulenbroeck, B. and Schäfer, L., “Convective stabilization of a Laplacian moving boundary problem with kinetic undercooling”, SIAM J. Appl. Math. 68 (2007) 292–310; doi:10.1137/070683908.CrossRefGoogle Scholar
Enright, D., Fedkiw, R., Ferziger, J. and Mitchel, I., “A hybrid particle level set method for improved interface capturing”, J. Comput. Phys. 183 (2002) 83–116; doi:10.1006/jcph.2002.7166.CrossRefGoogle Scholar
Entov, V. M., Etingof, P. I. and Kleinbock, D. Y., “On nonlinear interface dynamics in Hele-Shaw flows”, European J. Appl. Math. 6 (1995) 399–420; doi:10.1017/S0956792500001959.CrossRefGoogle Scholar
Eslami, A., Basak, R. and Taghavi, S. M., “Multiphase viscoplastic flows in a nonuniform Hele-Shaw cell: a fluidic device to control interfacial patterns”, Ind. Eng. Chem. Res. 59 (2020) 4119–4133; doi:10.1021/acs.iecr.9b06064.CrossRefGoogle Scholar
Fast, P., Kondic, L., Shelley, M. J. and Palffy-Muhoray, P., “Pattern formation in non-Newtonian Hele-Shaw flow”, Phys. Fluids 13 (2001) 1191; doi:10.1063/1.1359417.CrossRefGoogle Scholar
Folch, R., Alvarez-Lacalle, E., Or´tın, J. and Casademunt, J., “Pattern formation and interface pinch-off in rotating Hele-Shaw flows: a phase-field approach”, Phys. Rev. E 80 (2009) 056305; doi:10.1103/PhysRevE.80.056305.CrossRefGoogle ScholarPubMed
Fontana, J. V., Dias, E. O. and Miranda, J. A., “Controlling and minimizing fingering instabilities in non-Newtonian fluids”, Phys. Rev. E 89 (2014) 013016; doi:10.1103/PhysRevE.89.013016.CrossRefGoogle ScholarPubMed
Fontana, J. V., Juel, A., Bergemann, N., Heil, M. and Hazel, A. L., “Modelling finger propagation in elasto-rigid channels”, J. Fluid Mech. 916 (2021) A27; doi:10.1017/jfm.2021.219.CrossRefGoogle Scholar
Franco-Gómez, A., Thompson, A. B., Hazel, A. L. and Juel, A., “Sensitivity of Saffman–Taylor fingers to channel-depth perturbations”, J. Fluid Mech. 794 (2016) 343–368; doi:10.1017/jfm.2016.131.CrossRefGoogle Scholar
Gao, T., Mirzadeh, M., Bai, P., Conforti, K. M. and Bazant, M. Z., “Active control of viscous fingering using electric fields”, Nat. Commun. 10 (2019) 1–8; doi:10.1038/s41467-019-11939-7.CrossRefGoogle ScholarPubMed
Gardiner, B. P. J., McCue, S. W., Dallaston, M. C. and Moroney, T. J., “Saffman–Taylor fingers with kinetic undercooling”, Phys. Rev. E 91 (2015) 023016; doi:10.1103/PhysRevE.91.023016.CrossRefGoogle ScholarPubMed
Gardiner, B. P. J., McCue, S. W. and Moroney, T. J., “Discrete families of Saffman–Taylor fingers with exotic shapes”, Results Phys. 5 (2015) 103–104; doi:10.1016/j.rinp.2015.04.002.CrossRefGoogle Scholar
Gibou, F., Fedkiw, E. P., Cheng, L.-T. and Kang, M., “A second-order-accurate symmetric discretization of the Poisson equation on irregular domains”, J. Comput. Phys. 176 (2002) 205–227; doi:10.1006/jcph.2001.6977.CrossRefGoogle Scholar
Gibou, F., Fedkiw, R. and Osher, S., “A level set approach for the numerical simulation of dendritic growth”, J. Sci. Comput. 19 (2003) 183–199; doi:10.1023/A:1025399807998.CrossRefGoogle Scholar
Gibou, F., Fedkiw, R. and Osher, S., “A review of level-set methods and some recent applications”, J. Comput. Phys. 353 (2018) 82–109; doi:10.1016/j.jcp.2017.10.006.CrossRefGoogle Scholar
Gin, C. and Daripa, P., “Stability results for multi-layer radial Hele-Shaw and porous media flows”, Phys. Fluids 27 (2015) 012101; doi:10.1063/1.4904983.CrossRefGoogle Scholar
Gin, C. and Daripa, P., “Time-dependent injection strategies for multilayer Hele-Shaw and porous media flows”, Phys. Rev. Fluids 6 (2021) 033901; doi:10.1103/PhysRevFluids.6.033901.CrossRefGoogle Scholar
Givoli, D., Numerical methods for problems in infinite domains, Volume 33 of Stud. Appl. Mech. (Elsevier, Amsterdam, 2013).Google Scholar
Green, C. C., Lustri, C. J. and McCue, S. W., “The effect of surface tension on steadily translating bubbles in an unbounded Hele-Shaw cell”, Proc. R. Soc. Lond. A 473 (2017) 20170050; doi:10.1098/rspa.2017.0050.Google Scholar
Gustafsson, B. and Vasiĺev, A., Conformal and potential analysis in Hele-Shaw cells, Adv. in Math. Fluid Mech. (eds Galdi, G. P., Heywood, J. G. and Rannacher, R.), (Birkhäuser, Basel, 2006).Google Scholar
Hohlov, Y. E. and Howison, S. D., “On the classification of solutions to the zero-surface-tension model for Hele-Shaw free-boundary flows”, Q. Appl. Math. 51 (1993) 777–789; doi:10.1090/qam/1247441.CrossRefGoogle Scholar
Hohlov, Y. E., Howison, S. D., Huntingford, C., Ockendon, J. R. and Lacey, A. A., “A model for nonsmooth free boundaries in Hele-Shaw flows”, Q. J. Mech. Appl. Math. 47 (1994) 107–128; doi:10.1093/qjmam/47.1.107.CrossRefGoogle Scholar
Homsy, G. M., “Viscous fingering in porous media”, Annu. Rev. Fluid Mech. 19 (1987) 271–311; doi:10.1146/annurev.fl.19.010187.001415.CrossRefGoogle Scholar
Hong, D. C. and Family, F., “Bubbles in the Hele-Shaw cell—pattern selection and tip perturbations”, Phys. Rev. A 38 (1988) 5253–5259; doi:10.1103/PhysRevA.38.5253.CrossRefGoogle ScholarPubMed
Hong, D. C. and Langer, J. S., “Analytic theory of the selection mechanism in the Saffman–Taylor problem”, Phys. Rev. Lett. 56 (1986) 2032; doi:10.1103/PhysRevLett.56.2032.CrossRefGoogle ScholarPubMed
Hou, T. Y., Li, Z., Osher, S. and Zhao, H., “A hybrid method for moving interface problems with application to the Hele-Shaw flow”, J. Comput. Phys. 134 (1997) 236–252; doi:10.1006/jcph.1997.5689.CrossRefGoogle Scholar
Hou, T. Y., Lowengrub, J. S. and Shelley, M. J., “Removing the stiffness from interfacial flow with surface tension”, J. Comput. Phys. 114 (1994) 312–338; doi:10.1006/jcph.1994.1170.CrossRefGoogle Scholar
Hou, T. Y., Lowengrub, J. S. and Shelley, M. J., “Boundary integral methods for multicomponent fluids and multiphase materials”, J. Comput. Phys. 169 (2001) 302–362; doi:10.1006/jcph.2000.6626.CrossRefGoogle Scholar
Howison, S. D., “Bubble growth in porous media and Hele-Shaw cells”, Proc. Roy. Soc. Edinburgh Sect. A 102 (1986) 141–148; doi:10.1017/S0308210500014554.CrossRefGoogle Scholar
Howison, S. D., “Cusp development in Hele-Shaw flow with a free surface”, SIAM J. Appl. Math. 46 (1986) 20–26; doi:10.1137/0146003.CrossRefGoogle Scholar
Howison, S. D., “Fingering in Hele-Shaw cells”, J. Fluid Mech. 167 (1986) 439–453; doi:10.1017/S0022112086002902.CrossRefGoogle Scholar
Howison, S. D., Ockendon, J. R. and Lacey, A. A., “Singularity development in moving boundary problems”, Q. J. Mech. Appl. Math. 38 (1985) 343–354; doi:10.1093/qjmam/38.3.343.CrossRefGoogle Scholar
Jackson, S. J., Power, H., Giddings, D. and Stevens, D., “The stability of immiscible viscous fingering in Hele-Shaw cells with spatially varying permeability”, Comput. Methods Appl. Mech. Eng. 320 (2017) 606–632; doi:10.1002/fld.4028.CrossRefGoogle Scholar
Jackson, S. J., Stevens, D., Power, H. and Giddings, D., “A boundary element method for the solution of finite mobility ratio immiscible displacement in a Hele-Shaw cell”, Int. J. Numer. Methods Fluids 78 (2015) 521–551; doi:10.1002/fld.4028.CrossRefGoogle Scholar
Kelly, E. D. and Hinch, E. J., “Numerical simulations of sink flow in the Hele-Shaw cell with small surface tension”, European J. Appl. Math. 8 (1997) 533–550; doi:10.1017/S0956792597003203.CrossRefGoogle Scholar
Kondic, L., Shelley, M. J. and Palffy-Muhoray, P., “Non-Newtonian Hele-Shaw flow and the Saffman–Taylor instability”, Phys. Rev. Lett. 80 (1998) 1433–1436; doi:10.1103/PhysRevLett.80.1433.CrossRefGoogle Scholar
Lacey, A. A., “Moving boundary problems in the flow of liquid through porous media”, ANZIAM J. 24 (1982) 171–193; doi:10.1017/S0334270000003660.Google Scholar
Leshchiner, A., Thrasher, M., Mineev-Weinstein, M. B. and Swinney, H. L., “Harmonic moment dynamics in Laplacian growth”, Phys. Rev. E 81 (2010) 016206; doi:10.1103/PhysRevE.81.016206.CrossRefGoogle ScholarPubMed
Li, S., Lowengrub, J. S., Fontana, J. and Palffy-Muhoray, P., “Control of viscous fingering patterns in a radial Hele-Shaw cell”, Phys. Rev. Lett. 102 (2009) 174501; doi:10.1103/PhysRevLett.102.174501.CrossRefGoogle Scholar
Li, S., Lowengrub, J. S., H. Leo, P. and Cristini, V., “Nonlinear theory of self-similar crystal growth and melting”, J. Cryst. Growth 267 (2004) 703–713; doi:10.1016/j.jcrysgro.2004.04.002.CrossRefGoogle Scholar
Li, S., Lowengrub, J. S. and Leo, P. H., “A rescaling scheme with application to the long-time simulation of viscous fingering in a Hele-Shaw cell”, J. Comput. Phys. 228 (2007) 554–567; doi:10.1016/j.jcp.2006.12.023.CrossRefGoogle Scholar
Liang, S., “Random-walk simulations of flow in Hele-Shaw cells”, Phys. Rev. A 33 (1986) 2663; doi:10.1103/PhysRevA.33.2663.CrossRefGoogle ScholarPubMed
Lindner, A., Derks, D. and Shelley, M. J., “Stretch flow of thin layers of Newtonian liquids: fingering patterns and lifting forces”, Phys. Fluids 17 (2005) 072107; doi:10.1063/1.1939927.CrossRefGoogle Scholar
Lins, T. F. and Azaiez, J., “Resonance-like dynamics in radial cyclic injection flows of immiscible fluids in homogeneous porous media”, J. Fluid Mech. 819 (2017) 713–729; doi:10.1017/jfm.2017.186.CrossRefGoogle Scholar
Lister, J. R., Peng, G. G. and Neufeld, J. A., “Viscous control of peeling an elastic sheet by bending and pulling”, Phys. Rev. Lett. 111 (2013) 154501; doi:10.1103/PhysRevLett.111.154501.CrossRefGoogle Scholar
Lu, D., Municchi, F. and Christov, I. C., “Computational analysis of interfacial dynamics in angled Hele-Shaw cells: instability regimes”, Transp. Porous Media 131 (2020) 907–934; doi:10.1007/s11242-019-01371-2.CrossRefGoogle Scholar
Lustri, C. J., Green, C. C. and McCue, S. W., “Selection of a Hele-Shaw bubble via exponential asymptotics”, SIAM J. Appl. Math. 80 (2020) 289–311; doi:10.1137/18M1220868.CrossRefGoogle Scholar
McCloud, K. V. and Maher, J. V., “Experimental perturbations to Saffman–Taylor flow”, Phys. Rep. 260 (1995) 139–185; doi:10.1016/0370-1573(95)91133-U.CrossRefGoogle Scholar
McCue, S. W., “Short, flat-tipped, viscous fingers: novel interfacial patterns in a Hele-Shaw channel with an elastic boundary”, J. Fluid Mech. 834 (2018) 1–4; doi:10.1017/jfm.2017.692.CrossRefGoogle Scholar
McCue, S. W. and King, J. R., “Contracting bubbles in Hele-Shaw cells with a power-law fluid”, Nonlinearity 24 (2011) 613–641; doi:10.1088/0951-7715/24/2/009.CrossRefGoogle Scholar
McLean, J. W. and Saffman, P. G., “The effect of surface tension on the shape of fingers in a Hele-Shaw cell”, J. Fluid Mech. 102 (1981) 455–469; doi:10.1016/B978-0-08-092523-3.50018-6.CrossRefGoogle Scholar
Mineev-Weinstein, M., “Selection of the Saffman–Taylor finger width in the absence of surface tension: an exact result”, Phys. Rev. Lett. 80 (1998) 2113–2116; doi:10.1103/PhysRevLett.81.5950.CrossRefGoogle Scholar
Mineev-Weinstein, M., Wiegmann, P. B. and Zabrodin, A., “Integrable structure of interface dynamics”, Phys. Rev. Lett. 84 (2000) 5106–5109; doi:10.1103/PhysRevLett.84.5106.CrossRefGoogle ScholarPubMed
Miranda, J. and Widom, M., “Radial fingering in a Hele-Shaw cell: a weakly nonlinear analysis”, Phys. D 120 (1998) 315–328; doi:10.1016/S0167-2789(98)00097-9.CrossRefGoogle Scholar
Mirzadeh, M. and Bazant, M. Z., “Electrokinetic control of viscous fingering”, Phys. Rev. Lett. 119 (2017) 174501; doi:10.1103/PhysRevLett.119.174501.CrossRefGoogle ScholarPubMed
Moroney, T. J., Lusmore, D. R., McCue, S. W. and McElwain, D. L. S., “Extending fields in a level set method by solving a biharmonic equation”, J. Comput. Phys. 343 (2017); doi:10.1016/j.jcp.2017.04.049.CrossRefGoogle Scholar
Morrow, L. C., Dallaston, M. C. and McCue, S. W., “Interfacial dynamics and pinch-off singularities for axially symmetric Darcy flow”, Phys. Rev. E 100 (2019) 053109; doi:10.1103/PhysRevE.100.053109.CrossRefGoogle ScholarPubMed
Morrow, L. C., King, J. R., Moroney, T. J. and McCue, S. W.., “Moving boundary problems for quasi-steady conduction limited melting”, SIAM J. Appl. Math. 79 (2019) 2107–2131; doi:10.1137/18M123445X.CrossRefGoogle Scholar
Morrow, L. C., Moroney, T. J. and McCue, S. W., “Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations”, J. Fluid Mech. 877 (2019) 1063–1097; doi:10.1017/jfm.2019.623.CrossRefGoogle Scholar
Nase, J., Derks, D. and Lindner, A., “Dynamic evolution of fingering patterns in a lifted Hele-Shaw cell”, Phys. Fluids 23 (2011) 123101; doi:10.1063/1.3659140.CrossRefGoogle Scholar
Nie, Q. and Tian, F. R., “Singularities in Hele-Shaw flows”, SIAM J. Appl. Math. 58 (1998) 34–54; doi:10.1137/S0036139996297924.Google Scholar
Osher, S. and Fedkiw, R., Level set methods and dynamic implicit surfaces, Volume 153 of Appl. Math. Sci. (Springer, New York, 2003); doi:10.1115/1.1760520.CrossRefGoogle Scholar
Osher, S. and Fedkiw, R. P., “Level set methods: an overview and some recent results”, J. Comput. Phys. 169 (2001) 463–502; doi:10.1006/jcph.2000.6636.CrossRefGoogle Scholar
Osher, S. and Sethian, J. A., “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations”, J. Comput. Phys. 79 (1988) 12–49; doi:10.1016/0021-9991(88)90002-2.CrossRefGoogle Scholar
Paiva, A. S. S., Lira, S. H. A. and Andrade, R. F. S., “Non-linear effects in a closed rotating radial Hele-Shaw cell”, AIP Adv. 9 (2019) 025121; doi:10.1063/1.5086525.CrossRefGoogle Scholar
Park, C.-W. and Homsy, G. M., “Two-phase displacement in Hele-Shaw cells: theory”, J. Fluid Mech. 139 (1984) 291–308; doi:10.1017/S0022112084000367.CrossRefGoogle Scholar
Paterson, L., “Radial fingering in a Hele-Shaw cell”, J. Fluid Mech. 113 (1981) 513–529; doi:10.1017/S0022112081003613.CrossRefGoogle Scholar
Pihler-Puzović, D., Illien, P., Heil, M. and Juel, A., “Suppression of complex fingerlike patterns at the interface between air and a viscous fluid by elastic membranes”, Phys. Rev. Lett. 108 (2012) 074502; doi:10.1103/PhysRevLett.108.074502.CrossRefGoogle Scholar
Pihler-Puzović, D., Juel, A. and Heil, M., “The interaction between viscous fingering and wrinkling in elastic-walled Hele-Shaw cells”, Phys. Fluids 26 (2014) 022102; doi:10.1063/1.4864188.CrossRefGoogle Scholar
Pihler-Puzović, D., Peng, G. G., Lister, J. R., Heil, M. and Juel, A., “Viscous fingering in a radial elastic-walled Hele-Shaw cell”, J. Fluid Mech. 849 (2018) 163–191; doi:10.1017/jfm.2018.404.CrossRefGoogle Scholar
Pihler-Puzović, D., Périllat, R., Russell, M., Juel, A. and Heil, M., “Modelling the suppression of viscous fingering in elastic-walled Hele-Shaw cells”, J. Fluid Mech. 731 (2013) 162–183; doi:10.1017/jfm.2013.375.CrossRefGoogle Scholar
Pleshchinskii, N. B. and Reissig, M., “Hele-Shaw flows with nonlinear kinetic undercooling regularization”, Nonlinear Anal. 50 (2002) 191–203; doi:10.1016/S0362-546X(01)00745-3.CrossRefGoogle Scholar
Polubarinova-Kochina, P. Y., “On motion of the contour of an oil layer”, Dokl. Akad. Nauk SSSR 47 (1945) 254–257.Google Scholar
Power, H., Stevens, D., Cliffe, K. A. and Golin, A., “A boundary element study of the effect of surface dissolution on the evolution of immiscible viscous fingering within a Hele-Shaw cell”, Eng. Anal. Bound. Elem. 37 (2013) 1318–1330; doi:10.1016/j.enganabound.2013.04.010.CrossRefGoogle Scholar
Richardson, S., “Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel”, J. Fluid Mech. 56 (1972) 609–618; doi:10.1017/S0022112072002551.CrossRefGoogle Scholar
Sader, J. E., Chan, D. Y. C. and Hughes, B. D., “Non-Newtonian effects on immiscible viscous fingering in a radial Hele-Shaw cell”, Phys. Rev. E 49 (1994) 420–432; doi:10.1103/PhysRevE.49.420.CrossRefGoogle Scholar
Saffman, P. G., “Viscous fingering in Hele-Shaw cells”, J. Fluid Mech. 173 (1986) 73–94; doi:10.1017/S0022112086001088.CrossRefGoogle Scholar
Saffman, P. G. and Taylor, G. I., “The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid”, Proc. R. Soc. Lond. A 245 (1958) 312–329; doi:10.1098/rspa.1958.0085.Google Scholar
Schwartz, L. W., “Instability and fingering in a rotating Hele-Shaw cell or porous medium”, Phys. Fluids 1 (1989) 167–169; doi:10.1063/1.857543.CrossRefGoogle Scholar
Sethian, J. A., Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, Volume 3 (Cambridge University Press, Cambridge, 1993).Google Scholar
Sethian, J. A. and Smereka, P., “Level set methods for fluid interfaces”, Annu. Rev. Fluid Mech. 35 (2003) 341–372; doi:10.1146/annurev.fluid.35.101101.161105.CrossRefGoogle Scholar
Shelley, M. J., Tian, F. and Wlodarski, K., “Hele-Shaw flow and pattern formation in a time-dependent gap”, Nonlinearity 10 (1997) 1471; doi:10.1088/0951-7715/10/6/005.CrossRefGoogle Scholar
Shraiman, B. I., “Velocity selection and the Saffman–Taylor problem”, Phys. Rev. Lett. 56 (1986) 2028–2031; doi:10.1103/PhysRevLett.56.2028.CrossRefGoogle ScholarPubMed
Tabeling, P., Zocchi, G. and Libchaber, A., “An experimental study of the Saffman–Taylor instability”, J. Fluid Mech. 177 (1987) 67–82; doi:10.1017/S0022112087000867.CrossRefGoogle Scholar
Tanveer, S., “New solutions for steady bubbles in a Hele-Shaw cell”, Phys. Fluids 30 (1987) 651–658; doi:10.1063/1.866369.CrossRefGoogle Scholar
Tanveer, S., “Analytic theory for the selection of a symmetric Saffman–Taylor finger in a Hele-Shaw cell”, Phys. Fluids 30 (1987) 1589–1605; doi:10.1063/1.866225.CrossRefGoogle Scholar
Tanveer, S., “Analytic theory for the determination of velocity and stability of bubbles in a Hele-Shaw cell”, Theor. Comput. Fluid Dyn. 1 (1989) 135–163; doi:10.1007/BF00417918.CrossRefGoogle Scholar
Tanveer, S., “Surprises in viscous fingering”, J. Fluid Mech. 409 (2000) 273–308; doi:10.1017/S0022112099007788.CrossRefGoogle Scholar
Tanveer, S. and Saffman, P. G., “Stability of bubbles in a Hele-Shaw cell”, Phys. Fluids 30 (1987) 2624–2635; doi:10.1063/1.866106.CrossRefGoogle Scholar
Thomé, H., Rabaud, M., Hakim, V. and Couder, Y., “The Saffman–Taylor instability: from the linear to the circular geometry”, Phys. Fluids 1 (1989) 224–240; doi:10.1063/1.857493.CrossRefGoogle Scholar
Thompson, A. B., Juel, A. and Hazel, A. L., “Multiple finger propagation modes in Hele-Shaw channels of variable depth”, J. Fluid Mech. 746 (2014) 123–164; doi:10.1063/1.857493.CrossRefGoogle Scholar
Vanden-Broeck, J.-M., “Fingers in a Hele-Shaw cell with surface tension”, Phys. Fluids 26 (1983) 2033; doi:10.1063/1.864406.CrossRefGoogle Scholar
Vaquero-Stainer, C., Heil, M., Juel, A. and Pihler-Puzović, D., “Self-similar and disordered front propagation in a radial Hele-Shaw channel with time-varying cell depth”, Phys. Rev. Fluids 4 (2019) 064002; doi:10.1103/PhysRevFluids.4.064002.CrossRefGoogle Scholar
Vasconcelos, G., “Exact solutions for steady bubbles in a Hele-Shaw cell with rectangular geometry”, J. Fluid Mech. 444 (2001)175–198; doi:10.1017/S0022112001005365.CrossRefGoogle Scholar
Vasconcelos, G. L., “Multiple bubbles in a Hele-Shaw cell”, Phys. Rev. E 50 (1994) R3306–R3309; doi:10.1103/PhysRevE.50.R3306.CrossRefGoogle Scholar
Vasiĺev, A., “From the Hele-Shaw experiment to integrable systems: a historical overview”, Complex Anal. Oper. Theory 3 (2009) 551–585; doi:10.1007/s11785-008-0104-8.CrossRefGoogle Scholar
Waters, S. L. and Cummings, L. J., “Coriolis effects in a rotating Hele-Shaw cell”, Phys. Fluids 17 (2005) 048101; doi:10.1063/1.1861752.CrossRefGoogle Scholar
Xie, X., “Rigorous results in existence and selection of Saffman–Taylor fingers by kinetic undercooling”, European J. Appl. Math. 30 (2019) 63–116; doi:10.1017/S0956792517000390.CrossRefGoogle Scholar
Xie, X., “Analytic solution to an interfacial flow with kinetic undercooling in a time-dependent gap Hele-Shaw cell”, Discrete Contin. Dyn. Syst. Ser. B 26 (2021) 4663–4680; doi:10.3934/dcdsb.2020307.Google Scholar
Zhao, H., Casademunt, J., Yeung, C. and Maher, J. V., “Perturbing Hele-Shaw flow with a small gap gradient”, Phys. Rev. A 45 (1992) 2455; doi:10.1103/PhysRevA.45.2455.CrossRefGoogle ScholarPubMed
Zhao, M., Li, X., Ying, W., Belmonte, A., Lowengrub, J. and Li, S., “Computation of a shrinking interface in a Hele-Shaw cell”, SIAM J. Sci. Comput. 40 (2018) B1206–B1228; doi:10.1017/jfm.2020.983.CrossRefGoogle Scholar
Zhao, M., Niroobakhsh, Z., Lowengrub, J. and Li, S., “Nonlinear limiting dynamics of a shrinking interface in a Hele-Shaw cell”, J. Fluid Mech. 910 (2021) A41; doi:10.1017/jfm.2020.983.CrossRefGoogle Scholar
Zheng, Z., Kim, H. and Stone, H. A., “Controlling viscous fingering using time-dependent strategies”, Phys. Rev. Lett. 115 (2015) 174501; doi:10.1103/PhysRevLett.115.174501.CrossRefGoogle ScholarPubMed