Numerical solution to the Saffman--Taylor finger problem with kinetic undercooling regularisation

The Saffman--Taylor finger problem is to predict the shape and, in particular, width of a finger of fluid travelling in a Hele--Shaw cell filled with a different, more viscous fluid. In experiments the width is dependent on the speed of propagation of the finger, tending to half the total cell width as the speed increases. To predict this result mathematically, nonlinear effects on the fluid interface must be considered; usually surface tension is included for this purpose. This makes the mathematical problem sufficiently difficult that asymptotic or numerical methods must be used. We adapt numerical methods used to solve the Saffman--Taylor finger problem with surface tension to instead include the effect of kinetic undercooling, a regularisation effect important in Stefan melting-freezing problems, for which Hele--Shaw flow serves as a leading order approximation when the specific heat of a substance is much smaller than its latent heat. We find the existence of a solution branch where the finger width tends to zero as the propagation speed increases, disagreeing with some aspects of the asymptotic analysis of the same problem. We also find a second solution branch, supporting the idea of a countably infinite number of branches as for the surface tension problem. References P. G. Saffman and G. Taylor. The penetration of a fluid into a porous medium or Hele--Shaw cell containing a more viscous liquid. Proc. R. Soc. London, Ser. A, 245:312--329, 1958. doi:10.1098/rspa.1958.0085 J. W. McLean and P. G. Saffman. The effect of surface tension on the shape of fingers in a Hele--Shaw cell. J. Fluid. Mech., 102:455--469, 1981. doi:10.1017/S0022112081002735 J. M. Vanden-Broeck. Fingers in a Hele--Shaw cell with surface tension. Phys. Fluids, 26(8):2033--2034, 1983. doi:10.1063/1.864406 S. J. Chapman. On the role of Stokes lines in the selection of Saffman--Taylor fingers with small surface tension. Eur. J. Appl. Math., 10(06):513--534, 1999. doi:10.1017/S0956792599003848 R. Combescot, T. Dombre, V. Hakim, Y. Pomeau, and A. Pumir. Shape selection of Saffman--Taylor fingers. Phys. Rev. Lett., 56(19):2036--2039, May 1986. doi:10.1103/PhysRevLett.56.2036 R. Combescot, V. Hakim, T. Dombre, Y. Pomeau, and A. Pumir. Analytic theory of the Saffman--Taylor fingers. Phys. Rev. A, 37(4):1270--1283, February 1988. doi:10.1103/PhysRevA.37.1270 M. Reissig, S. 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(6):561--579, 1999. doi:10.1017/S0956792599003939 S. J. Chapman and J. R. King. The selection of Saffman--Taylor fingers by kinetic undercooling. J. Eng. Math., 46(1):1--32, 2003. doi:10.1023/A:1022860705459 D. Bensimon. Stability of viscous fingering. Phys. Rev. A, 33(2):1302--1309, 1986. doi:10.1103/PhysRevA.33.1302 D. A. Kessler and H. Levine. Theory of the Saffman--Taylor ``finger'' pattern. {I}. Phys. Rev. A, 33(4):2621--2633, 1986. doi:10.1103/PhysRevA.33.2621 D. A. Kessler and H. Levine. Theory of the Saffman--Taylor ``finger'' pattern. {II}. Phys. Rev. A, 33(4):2634--2639, 1986. doi:10.1103/PhysRevA.33.2634 A. J. Degregoria and L. W. Schwartz. A boundary integral method for two-phase displacement in Hele--Shaw cells. J. Fluid Mech., 164:383--400, 1986. doi:10.1017/S0022112086002604 T. Y. Hou, Z. Li, S. Osher, and H. Zhao. A hybrid method for moving interface problems with application to the Hele--Shaw flow. J. Comput. Phys., 134(2):236--252, 1997. doi:10.1006/jcph.1997.5689 G. Tryggvason and H. Aref. Numerical experiments on Hele--Shaw flow with a sharp interface. J. Fluid Mech., 136:1--30, 1983. doi:10.1017/S0022112083002037 N. Whitaker. Some numerical methods for the Hele--Shaw equations. J. Comput. Phys., 111(1):81--88, 1994. doi:10.1006/jcph.1994.104