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Corner and finger formation in Hele-Shaw flow with kinetic undercooling regularisation



We examine the effect of a kinetic undercooling condition on the evolution of a free boundary in Hele-Shaw flow, in both bubble and channel geometries. We present analytical and numerical evidence that the bubble boundary is unstable and may develop one or more corners in finite time, for both expansion and contraction cases. This loss of regularity is interesting because it occurs regardless of whether the less viscous fluid is displacing the more viscous fluid, or vice versa. We show that small contracting bubbles are described to leading order by a well-studied geometric flow rule. Exact solutions to this asymptotic problem continue past the corner formation until the bubble contracts to a point as a slit in the limit. Lastly, we consider the evolving boundary with kinetic undercooling in a Saffman-Taylor channel geometry. The boundary may either form corners in finite time, or evolve to a single long finger travelling at constant speed, depending on the strength of kinetic undercooling. We demonstrate these two different behaviours numerically. For the travelling finger, we present results of a numerical solution method similar to that used to demonstrate the selection of discrete fingers by surface tension. With kinetic undercooling, a continuum of corner-free travelling fingers exists for any finger width above a critical value, which goes to zero as the kinetic undercooling vanishes. We have not been able to compute the discrete family of analytic solutions, predicted by previous asymptotic analysis, because the numerical scheme cannot distinguish between solutions characterised by analytic fingers and those which are corner-free but non-analytic.



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Angenent, S. B. & Aronson, D. G. (1995) The focusing problem for the radially symmetric porous medium equation. Comm. Part. Diff. Eq. 20, 12171240.
Angenent, S. B. & Aronson, D. G. (2004) The focusing problem for the eikonal equation. In: Nonlinear Evolution Equations and Related Topics, Springer, pp. 137151.
Back, J. M., McCue, S. W., Hsieh, M. H.-N. & Moroney, T. J. (2014) The effect of surface tension and kinetic undercooling on a radially-symmetric melting problem. Appl. Math. Comp. 229, 4152.
Bensimon, D. (1986) Stability of viscous fingering. Phys. Rev. A 33, 13021308.
Ceniceros, H. D. & Hou, T. Y. (1998) Convergence of a non-stiff boundary integral method for interfacial flows with surface tension. Math. Comp. 67, 137182.
Chapman, S. J. (1999) On the rôle of Stokes lines in the selection of Saffman–Taylor fingers with small surface tension. Eur. J. Appl. Math. 10, 513534.
Chapman, S. J. & King, J. R. (2003) The selection of Saffman–Taylor fingers by kinetic undercooling. J. Eng. Math. 46, 132.
Cohen, D. S. & Erneux, T. (1988) Free boundary problems in controlled release pharmaceuticals. I: Diffusion in glassy polymers. SIAM J. Appl. Math. 48, 14511465.
Crowdy, D. G. (2002) A theory of exact solutions for the evolution of a fluid annulus in a rotating Hele–Shaw cell. Quart. Appl. Math. 60, 1136.
Dallaston, M. C. (2013) Mathematical Models of Bubble Evolution in a Hele–Shaw Cell. PhD thesis, Queensland University of Technology.
Dallaston, M. C. & McCue, S. W. (2011) Numerical solution to the Saffman–Taylor finger problem with kinetic undercooling regularisation. In: Proceedings of the 15th Biennial Computational Techniques and Applications Conference, CTAC–2010, volume 52 of ANZIAM J., pp. C124–C138.
Dallaston, M. C. & McCue, S. W. (2012) New exact solutions for Hele–Shaw flow in doubly connected regions. Phys. Fluids 24, 052101.
Dallaston, M. C. & McCue, S. W. (2013) An accurate numerical scheme for the contraction of a bubble in a Hele–Shaw cell. In: Proceedings of the 16th Biennial Computational Techniques and Applications Conference, CTAC–2012, volume 54 of ANZIAM J., pp. C309–C326.
Dallaston, M. C. & McCue, S. W. (2013) Bubble extinction in Hele–Shaw flow with surface tension and kinetic undercooling regularisation. Nonlinearity 26, 16391665.
Degregoria, A. J. & Schwartz, L. W. (1986) A boundary-integral method for 2-phase displacement in Hele–Shaw cells. J. Fluid Mech. 164, 383400.
Dias, E. O., Alvarez-Lacalle, E., Carvalho, M. S. & Miranda, J. S. (2012) Minimization of viscous fluid fingering: A variational scheme for optimal flow rates. Phys. Rev. Lett. 109, 144502.
Dias, E. O. & Miranda, J. A. (2010) Control of radial fingering patterns: A weakly nonlinear approach. Phys. Rev. E 81, 016312 (1–7).
Ebert, U., Brau, F., Derks, G., Hundsdorfer, W., Kao, C-Y., Li, C., Luque, A., Meulenbroek, B., Nijdam, S., Ratushnaya, V., Schäfer, L. & Tanveer, S. (2011) Multiple scales in streamer discharges, with an emphasis on moving boundary approximations. Nonlinearity 24, C1C26.
Ebert, U., Meulenbroek, B. J. & Schäfer, L. (2007) Convective stabilization of a Laplacian moving boundary problem with kinetic undercooling. SIAM J. Appl. Math. 68, 292310.
Entov, V. M. & Etingof, P. I. (1991) Bubble contraction in Hele–Shaw cells. Q. J. Mech. Appl. Math. 44, 507535.
Entov, V. M. & Etingof, P. I. (2011) On the breakup of air bubbles in a Hele–Shaw cell. Eur. J. Appl. Math. 22, 125149.
Evans, J. D. & King, J. R. (2000) Asymptotic results for the Stefan problem with kinetic undercooling. Q. J. Mech. Appl. Math. 53, 449473.
Fasano, A., Meyer, G. H. & Primicerio, M. (1986) On a problem in the polymer industry: theoretical and numerical investigation of swelling. SIAM J. Math. Anal. 17, 945960.
Fox, L. & Parker, I. B. (1968) Chebyshev Polynomials in Numerical Analysis, Vol. 29, Oxford University Press, London.
Gage, M. & Hamilton, R.The shrinking of convex plane curves by heat equation. J. Diff. Geom. 23, 6996, 1986.
Galin, L. A. (1945) Unsteady filtration with a free surface. Dokl. Akad. Nauk. S. S. S. R. 47, 246249.
Grayson, M. (1987) The heat equation shrinks embedded curves to points. J. Diff. Geom. 26, 285314.
Günther, M. & Prokert, G. (2009) On travelling-wave solutions for a moving boundary problem of Hele–Shaw type. IMA J. Appl. Math. 74, 107127.
Hou, T. Y., Li, Z. L., Osher, S. & Zhao, H. K. (1997) A hybrid method for moving interface problems with application to the Hele–Shaw flow. J. Comp. Phys. 134, 236252.
Hou, T. Y., Lowengrub, J. S. & Shelley, M. J. (1994) Removing the stiffness from interfacial flow with surface tension. J. Comp. Phys. 114, 312338.
Howison, S. D. (1986) Bubble growth in porous media and Hele–Shaw cells. Proc. Roy. Soc. Edin. A 102, 141148.
Howison, S. D. (1986) Cusp development in Hele–Shaw flow with a free surface. SIAM J. Appl. Math. 46, 2026.
Howison, S. D. (1986) Fingering in Hele–Shaw cells. J. Fluid Mech. 167, 439453.
Howison, S. D. (1992) Complex variable methods in Hele–Shaw moving boundary problems. Eur. J. Appl. Math. 3, 209224.
Howison, S. D. (August 1998) Bibliography of free and moving boundary problems in Hele–shaw and Stokes flow.
Kao, C.-Y., Brau, F., Ebert, U., Schäfer, L. & Tanveer, S. (2010) A moving boundary model motivated by electric breakdown: II. initial value problem. Physica D 239, 15421559.
Kessler, D. A., Koplik, J. & Levine, H. (1988) Pattern selection in fingered growth phenomena. Adv. Phys. 37, 255339.
King, J. R. & Evans, J. D. (2005) Regularization by kinetic undercooling of blow-up in the ill-posed Stefan problem. SIAM J. Appl. Math. 65, 16771707.
King, J. R., Lacey, A. A. & Vazquez, J. L. (1995) Persistence of corners in free boundaries in Hele–Shaw. Eur. J. Appl. Math. 6, 455490.
King, J. R. & McCue, S. W. (2009) Quadrature domains and p-Laplacian growth. Compl. Anal. Oper. Th. 3, 453469.
Lee, S-Y., Bettelheim, E. & Weigmann, P. (2006) Bubble break-off in Hele–Shaw flows–-singularities and integrable structures. Physica D 219, 2234.
Luque, A., Brau, F. & Ebert, U. (2008) Saffman–Taylor streamers: Mutual finger interaction in spark formation. Phys. Rev. E 78, 016206.
Martyushev, L. M. & Birzina, A. I. (2008) Specific features of the loss of stability during radial displacement of fluid in the Hele–Shaw cell. J. Phys.: Condens. Matter 20, 045201.
McCue, S. W., Hsieh, M., Moroney, T. J. & Nelson, M. I. (2011) Asymptotic and numerical results for a model of solvent-dependent drug diffusion through polymeric spheres. SIAM J. Appl. Math. 71, 22872311.
McCue, S. W. & King, J. R. (2011) Contracting bubbles in Hele–Shaw cells with a power-law fluid. Nonlinearity 24, 613641.
McCue, S. W., King, J. R. & Riley, D. S. (2003) Extinction behaviour of contracting bubbles in porous media. Q. J. Mech. Appl. Math. 56, 455482.
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, 455469.
Meulenbroeck, B., Ebert, U. & Schäfer, L. (2005) Regularization of moving boundaries in a Laplacian field by a mixed Dirichlet–Neumann boundary condition: Exact results. Phys. Rev. Lett. 95, 195004.
Mitchell, S. L. & O'Brien, S. B. G. (2012) Asymptotic, numerical and approximate techiques for a free boundary problem arising in the diffusion of glassy polymers. Appl. Math. Comp. 219, 376388.
Paterson, L. (1981) Radial fingering in a Hele–Shaw cell. J. Fluid Mech. 113, 513529.
Pleshchinskii, N. B. & Reissig, M. (2002) Hele–Shaw flows with nonlinear kinetic undercooling regularization. Nonlinear Anal. 50, 191203.
Polubarinova-Kochina, P. Ya. (1945) On the motion of the oil contour. Dokl. Akad. Nauk. S. S. S. R. 47, 254257.
Reissig, M., Rogosin, D. V. & Hübner, F. (1999) Analytical and numerical treatment of a complex model for Hele–Shaw moving boundary value problems with kinetic undercooling regularization. Eur. J. Appl. Math. 10, 561579.
Rocha, F. M. & Miranda, J. A. (2013) Manipulation of the Saffman–Taylor instability: A curvature-dependent surface tension approach. Phys. Rev. E 87, 013017.
Romero, L. A. (1981) The Fingering Problem in a Hele–Shaw Cell. PhD thesis, California Institute of Technology.
Sethian, J. A. (1999) Level Set Methods and Fast Marching Methods, Cambridge University Press.
Tanveer, S. (1987) New solutions for steady bubbles in a Hele–Shaw cell. Phys. Fluids 30, 651658.
Tanveer, S., Schäfer, L., Brau, F. & Ebert, U. (2009) A moving boundary problem motivated by electric breakdown, I: Spectrum of linear perturbations. Physica D 238, 888901.
Tanveer, S. & Xie, X. (2003) Analyticity and nonexistence of classical steady Hele–Shaw fingers. Comm. Pure Appl. Math. 56, 353402.
Tryggvason, G. & Aref, H. (1983) Numerical experiments on Hele–Shaw flow with a sharp interface. J. Fluid Mech. 136, 130.
Vanden-Broeck, J-M. (1983) Fingers in a Hele–Shaw cell with surface tension. Phys. Fluids 26, 20332034.
Xie, X. & Tanveer, S. (2003) Rigorous results in steady finger selection in viscous fingering. Arch. Rational Mech. Anal. 166, 219286.
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European Journal of Applied Mathematics
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