An accurate numerical scheme for the contraction of a bubble in a Hele--Shaw cell

We report on an accurate numerical scheme for the evolution of an inviscid bubble in radial Hele--Shaw flow, where the nonlinear boundary effects of surface tension and kinetic undercooling are included on the bubble-fluid interface. As well as demonstrating the onset of the Saffman--Taylor instability for growing bubbles, the numerical method is used to show the effect of the boundary conditions on the separation (pinch off) of a contracting bubble into multiple bubbles, and the existence of multiple possible asymptotic bubble shapes in the extinction limit. The numerical scheme also allows for the accurate computation of bubbles which pinch off very close to the theoretical extinction time, raising the possibility of computing solutions for the evolution of bubbles with non-generic extinction behaviour. References M. C. Dallaston and S. W. McCue. New exact solutions for Hele--Shaw flow in doubly connected regions. Phys. Fluids, 24:052101, 2012. doi:10.1063/1.4711274. M. C. Dallaston and S. W. McCue. Bubble extinction in Hele--Shaw flow with surface tension and kinetic undercooling regularisation. Nonlinearity, 26:1639--1665, 2013. doi:10.1088/0951-7715/26/6/1639. E. O. Dias and J. A. Miranda. Control of radial fingering patterns: A weakly nonlinear approach. Phys. Rev. E, 81(1):016312 (1--7), 2010. doi:10.1103/PhysRevE.81.016312. E. O. Dias, E. Alvarez--Lacalle, M. S. Carvalho, and J. S. Miranda. Minimization of viscous fluid fingering: a variational scheme for optimal flow rates. Phys. Rev. Lett., 109:144502, 2012. doi:10.1103/PhysRevLett.109.144502. V. Entov and P. Etingof. On the breakup of air bubbles in a Hele--Shaw cell. Eur. J. Appl. Math., 22:125--149, 2011. doi:10.1017/S095679251000032X. V. M. Entov and P. I. Etingof. Bubble contraction in Hele--Shaw cells. Q. J. Mech. Appl. Math., 44:507--535, 1991. doi:10.1093/qjmam/44.4.507. L. K. Forbes. A cylindrical Rayleigh--Taylor instability: radial outflow from pipes or stars. J. Eng. Math., 70:205--224, 2011. doi:10.1007/s10665-010-9374-z. T. Y. Hou, Z. Li, S. Osher, and H. Zhao. A hybrid method for moving interface problems with application to the Hele--Shaw flow. Journal of Computational Physics, 134(2):236--252, 1997. doi:10.1006/jcph.1997.5689. S. D. Howison. Complex variable methods in Hele--Shaw moving boundary problems. Eur. J. Appl. Math., 3:209--224, 1992. doi:10.1017/S0956792500000802. J. R. King and S. W. McCue. Quadrature domains and p-Laplacian growth. Complex Anal. Oper. Th., 3:453--469, 2009. doi:10.1007/s11785-008-0103-9. S.-Y. Lee, E. Bettelheim, and P. Weigmann. Bubble break-off in Hele--Shaw flows---singularities and integrable structures. Physica D, 219:22--34, 2006. doi:10.1016/j.physd.2006.05.010. S. Li, J. S. Lowengrub, and P. H. Leo. A rescaling scheme with application to the long-time simulation of viscous fingering in a Hele--Shaw cell. J. Comp. Phys., 225(1):554--567, 2007. doi:10.1016/j.jcp.2006.12.023. S. Li, J. S. Lowengrub, J. Fontana, and P. Palffy--Muhoray. Control of viscous fingering patterns in a radial Hele--Shaw cell. Phys. Rev. Lett., 102(17):174501, 2009. doi:10.1103/PhysRevLett.102.174501. S. W. McCue and J. R. King. Contracting bubbles in Hele--Shaw cells with a power-law fluid. Nonlinearity, 24:613--641, 2011. doi:10.1088/0951-7715/24/2/009. S. W. McCue, J. R. King, and D. S. Riley. Extinction behaviour of contracting bubbles in porous media. Q. J. Mech. Appl. Math., 56:455--482, 2003. doi:10.1093/qjmam/56.3.455. S. W. McCue, J. R. King, and D. S. Riley. Extinction behavior for two-dimensional inward-solidification problems. Proc. R. Soc. Lond. A, 459:977--999, 2003. doi:10.1098/rspa.2002.1059. S. W. McCue, J. W. King, and D. S. Riley. The extinction problem for three-dimensional inward solidification. J. Eng. Math., 52:389--409, 2005. doi:10.1007/s10665-005-3501-2. M. Reissig, D. V. Rogosin, and F. Hubner. Analytical and numerical treatment of a complex model for Hele--Shaw moving boundary value problems with kinetic undercooling regularization. Eur. J. Appl. Math., 10:561--579, 1999. http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=44021&fulltextType=RA&fileId=S0956792599003939 P. G. Saffman and G. I. Taylor. 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, 1958. doi:10.1098/rspa.1958.0085