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Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations

Published online by Cambridge University Press:  02 September 2019

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]  and
Scott W. McCue Open the ORCID record for Scott W. McCue [Opens in a new window]
Liam C. Morrow
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Brisbane QLD 4001, Australia
Timothy J. Moroney
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Brisbane QLD 4001, Australia
Scott W. McCue*
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Brisbane QLD 4001, Australia
*
†Email address for correspondence: scott.mccue@qut.edu.au
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Abstract

Viscous fingering experiments in Hele-Shaw cells lead to striking pattern formations which have been the subject of intense focus among the physics and applied mathematics community for many years. In recent times, much attention has been devoted to devising strategies for controlling such patterns and reducing the growth of the interfacial fingers. We continue this research by reporting on numerical simulations, based on the level set method, of a generalised Hele-Shaw model for which the geometry of the Hele-Shaw cell is altered. First, we investigate how imposing constant and time-dependent injection rates in a Hele-Shaw cell that is either standard, tapered or rotating can be used to reduce the development of viscous fingering when an inviscid fluid is injected into a viscous fluid over a finite time period. We perform a series of numerical experiments comparing the effectiveness of each strategy to determine how these non-standard Hele-Shaw configurations influence the morphological features of the inviscid–viscous fluid interface. Surprisingly, a converging or diverging taper of the plates leads to reduced metrics of viscous fingering at the final time when compared to the standard parallel configuration, especially with carefully chosen injection rates; for the rotating plate case, the effect is even more dramatic, with sufficiently large rotation rates completely stabilising the interface. Next, we illustrate how the number of non-splitting fingers can be controlled by injecting the inviscid fluid at a time-dependent rate while increasing the gap between the plates. Our simulations compare well with previous experimental results for various injection rates and geometric configurations. We demonstrate how the number of non-splitting fingers agrees with that predicted from linear stability theory up to some finger number; for larger values of our control parameter, the fully nonlinear dynamics of the problem leads to slightly fewer fingers than this linear prediction.

JFM classification

Interfacial Flows (free surface): Fingering instability Low-Reynolds-number flows: Hele-Shaw flows Mathematical Foundations: Computational methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 877 , 25 October 2019 , pp. 1063 - 1097
Copyright
© 2019 Cambridge University Press 

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References

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, 034502.10.1103/PhysRevLett.111.034502Google Scholar
Al-Housseiny, T. T. & Stone, H. A. 2013 Controlling viscous fingering in tapered Hele-Shaw cells. Phys. Fluids 25, 092102.10.1063/1.4819317Google Scholar
Al-Housseiny, T. T., Tsai, P. A. & Stone, H. A. 2012 Control of interfacial instabilities using flow geometry. Nat. Phys. 8, 747–750.10.1038/nphys2396Google Scholar
Alvarez-Lacalle, E., Ortın, J. & Casademunt, J. 2004 Low viscosity contrast fingering in a rotating Hele-Shaw cell. Phys. Fluids 16, 908–924.10.1063/1.1644149Google Scholar
Anjos, P. H. A., Alvarez, V. M. M., Dias, E. O. & Miranda, J. A. 2017 Rotating Hele-Shaw cell with a time-dependent angular velocity. Phys. Rev. Fluids 2, 124003.10.1103/PhysRevFluids.2.124003Google Scholar
Anjos, P. H. A., Dias, E. O. & Miranda, J. A. 2018 Fingering instability transition in radially tapered Hele-Shaw cells: insights at the onset of nonlinear effects. Phys. Rev. Fluids 3, 124004.10.1103/PhysRevFluids.3.124004Google Scholar
Ben Amar, M., Hakim, V., Mashaal, M. & Couder, Y. 1991 Self-dilating viscous fingers in wedge-shaped Hele-Shaw cells. Phys. Fluids A 3, 1687–1690.10.1063/1.858222Google Scholar
Ben-Jacob, E. & Garik, P. 1990 The formation of patterns in non-equilibrium growth. Nature 343, 523–530.10.1038/343523a0Google Scholar
Ben-Jacob, E., Shmueli, H., Shochet, O. & Tenenbaum, A. 1992 Adaptive self-organization during growth of bacterial colonies. Physica A 187, 378–424.10.1016/0378-4371(92)90002-8Google Scholar
Bongrand, G. & Tsai, P. A. 2018 Manipulation of viscous fingering in a radially tapered cell geometry. Phys. Rev. E 97, 061101.Google Scholar
Brener, E. A., Kessler, D. A., Levine, H. & Rappei, W. J. 1990 Selection of the viscous finger in the 90° geometry. Eur. Phys. Lett. 13, 161–166.10.1209/0295-5075/13/2/011Google Scholar
Carrillo, L., Magdaleno, F. X., Casademunt, J. & Ortn, J. 1996 Experiments in a rotating Hele-Shaw cell. Phys. Rev. E 54, 6260–6267.Google Scholar
Carrillo, L., Soriano, J. & Ortın, J. 1999 Radial displacement of a fluid annulus in a rotating Hele–Shaw cell. Phys. Fluids 11, 778–785.10.1063/1.869950Google Scholar
Chen, C. Y., Chen, C. H. & Miranda, J. A. 2005 Numerical study of miscible fingering in a time-dependent gap Hele-Shaw cell. Phys. Rev. E 71, 056304.Google Scholar
Chen, J.-D. 1987 Radial viscous fingering patterns in Hele-Shaw cells. Exp. Fluids 5, 363–371.10.1007/BF00264399Google Scholar
Chen, S., Merriman, B., Osher, S. & Smereka, P. 1997 A simple level set method for solving Stefan problems. J. Comput. Phys. 135, 8–29.10.1006/jcph.1997.5721Google Scholar
Combescot, R. & Ben Amar, M. 1991 Selection of Saffman–Taylor fingers in the sector geometry. Phys. Rev. Lett. 67, 453–456.10.1103/PhysRevLett.67.453Google Scholar
Dai, W.-S. & Shelley, M. J. 1993 A numerical study of the effect of surface tension and noise on an expanding Hele–Shaw bubble. Phys. Fluids A 5, 2131–2146.10.1063/1.858553Google Scholar
Dallaston, M. C. & McCue, S. W. 2013 Bubble extinction in Hele-Shaw flow with surface tension and kinetic undercooling regularization. Nonlinearity 26, 1639–1665.10.1088/0951-7715/26/6/1639Google Scholar
DeGregoria, A. J. & Schwartz, L. W. 1986 A boundary-integral method for two-phase displacement in Hele-Shaw cells. J. Fluid Mech. 164, 383–400.10.1017/S0022112086002604Google 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, 144502.10.1103/PhysRevLett.109.144502Google Scholar
Dias, E. O. & Miranda, J. 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
Dias, E. O., Parisio, F. & Miranda, J. A. 2010 Suppression of viscous fluid fingering: a piecewise-constant injection process. Phys. Rev. E 82, 067301.Google Scholar
Ducloué, L., Hazel, A. L., Pihler-Puzović, D. & Juel, A. 2017 Viscous fingering and dendritic growth under an elastic membrane. J. Fluid Mech. 826, R2.10.1017/jfm.2017.468Google Scholar
Enright, D., Fedkiw, R., Ferziger, J. & Mitchell, I. 2002 A hybrid particle level set method for improved interface capturing. J. Comput. Phys. 183, 83–116.10.1006/jcph.2002.7166Google 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, 117–142.10.1016/j.jcp.2003.08.034Google Scholar
Gadêlha, H. & Miranda, J. 2004 Finger competition dynamics in rotating Hele-Shaw cells. Phys. Rev. E 70, 066308.Google Scholar
Homsy, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271–311.10.1146/annurev.fl.19.010187.001415Google 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, 236–252.10.1006/jcph.1997.5689Google Scholar
Jackson, S. J., Power, H., Giddings, D. & Stevens, D. 2017 The stability of immiscible viscous fingering in Hele-Shaw cells with spatially varying permeability. Comput. Meth. Appl. Mech. Engng 320, 606–632.10.1016/j.cma.2017.03.030Google Scholar
Juel, A., Pihler-Puzović, D. & Heil, M. 2018 Instabilities in blistering. Annu. Rev. Fluid Mech. 50, 691–714.10.1146/annurev-fluid-122316-045106Google Scholar
Leshchiner, A., Thrasher, M., Mineev-Weinstein, M. B. & Swinney, H. L. 2010 Harmonic moment dynamics in Laplacian growth. Phys. Rev. E 81, 016206.Google Scholar
Li, S., Lowengrub, J. S., Leo, P. H. & Cristini, V. 2004 Nonlinear theory of self-similar crystal growth and melting. J. Cryst. Growth 267, 703–713.10.1016/j.jcrysgro.2004.04.002Google 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, 174501.10.1103/PhysRevLett.102.174501Google Scholar
Liang, S. 1986 Random-walk simulations of flow in Hele Shaw cells. Phys. Rev. A 33, 2663–2674.10.1103/PhysRevA.33.2663Google Scholar
Lindner, A., Derks, D. & Shelley, M. J. 2005 Stretch flow of thin layers of Newtonian liquids: fingering patterns and lifting forces. Phys. Fluids 17, 072107.10.1063/1.1939927Google Scholar
Lins, T. F. & Azaiez, J. 2017 Resonance-like dynamics in radial cyclic injection flows of immiscible fluids in homogeneous porous media. J. Fluid Mech. 819, 713–729.10.1017/jfm.2017.186Google Scholar
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, 154501.10.1103/PhysRevLett.111.154501Google Scholar
Lu, D., Municchi, F. & Christov, I. C.2018 Computational analysis of interfacial instability in angled Hele-Shaw cells. arXiv:1811.06960.Google Scholar
McCue, S. W. 2018 Short, flat-tipped, viscous fingers: novel interfacial patterns in a Hele-Shaw channel with an elastic boundary. J. Fluid Mech. 834, 1–4.10.1017/jfm.2017.692Google 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, 455–469.10.1017/S0022112081002735Google Scholar
Mirzadeh, M. & Bazant, M. Z. 2017 Electrokinetic control of viscous fingering. Phys. Rev. Lett. 119, 174501.10.1103/PhysRevLett.119.174501Google Scholar
Moroney, T. J., Lusmore, D. R., McCue, S. W. & McElwain, S. 2017 Extending fields in a level set method by solving a biharmonic equation. J. Comput. Phys. 343, 170–185.10.1016/j.jcp.2017.04.049Google Scholar
Mullins, W. W. & Sekerka, R. F. 1988 Stability of a planar interface during solidification of a dilute binary alloy. In Dynamics Curved Fronts, pp. 345–352. Elsevier.10.1016/B978-0-08-092523-3.50037-XGoogle Scholar
Nase, J., Derks, D. & Lindner, A. 2011 Dynamic evolution of fingering patterns in a lifted Hele–Shaw cell. Phys. Fluids 23, 123101.10.1063/1.3659140Google Scholar
Osher, S. & Sethian, J. A. 1988 Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49.10.1016/0021-9991(88)90002-2Google Scholar
Paterson, L. 1981 Radial fingering in a Hele Shaw cell. J. Fluid Mech. 113, 513–529.10.1017/S0022112081003613Google 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, 074502.10.1103/PhysRevLett.108.074502Google Scholar
Pihler-Puzović, D., Juel, A. & Heil, M. 2014 The interaction between viscous fingering and wrinkling in elastic-walled Hele-Shaw cells. Phys. Fluids 26, 022102.10.1063/1.4864188Google Scholar
Pihler-Puzović, D., Peng, G. G., Lister, J. R., Heil, M. & Juel, A. 2018 Viscous fingering in a radial elastic-walled Hele-Shaw cell. J. Fluid Mech. 849, 163–191.10.1017/jfm.2018.404Google Scholar
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, 162–183.10.1017/jfm.2013.375Google Scholar
Rabbani, H. S., Or, D., Liu, Y., Lai, C.-Y., Lu, N. B., Datta, S. S., Stone, H. A. & Shokri, N. 2018 Suppressing viscous fingering in structured porous media. Proc. Natl Acad. Sci. USA 115 (19), 4833–4838.10.1073/pnas.1800729115Google 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
Shelley, M. J., Tian, F. & Wlodarski, K. 1997 Hele-Shaw flow and pattern formation in a time-dependent gap. Nonlinearity 10, 1471–1495.10.1088/0951-7715/10/6/005Google Scholar
Stone, H. A. 2017 Seeking simplicity for the understanding of multiphase flows. Phys. Rev. Fluids 2, 100507.10.1103/PhysRevFluids.2.100507Google Scholar
Vaquero-Stainer, C., Heil, M., Juel, A. & Pihler-Puzović, D. 2019 Self-similar and disordered front propagation in a radial Hele-Shaw channel with time-varying cell depth. Phys. Rev. Fluids 4, 064002.10.1103/PhysRevFluids.4.064002Google Scholar
Witten, T. A. & Sander, L. M. 1983 Diffusion-limited aggregation. Phys. Rev. B 27, 5686–5697.10.1103/PhysRevB.27.5686Google Scholar
Zheng, Z., Kim, H. & Stone, H. A. 2015 Controlling viscous fingering using time-dependent strategies. Phys. Rev. Lett. 115, 174501.10.1103/PhysRevLett.115.174501Google Scholar