Abstract
The selection of Saffman-Taylor fingers by surface tension has been widely studied. Here their selection is analysed by another regularisation widely adopted in studying otherwise ill-posed Stefan problems, namely kinetic undercooling. An asymptotic-beyond-all-orders analysis (which forms the core of the paper) reveals for small kinetic undercooling how a discrete family of fingers is selected; while these are similar to those arising for surface tension, the asymptotic analysis exhibits a number of additional subtleties. In Appendix 1 a description of some general features of the Hele-Shaw problem with kinetic undercooling and an analysis of the converse limit in which kinetic undercooling effects are large are included, while Appendix 2 studies the role of exponentially small terms in a simple linear problem which clarifies the rather curious behaviour at the origin of Stokes lines in the Hele-Shaw problem with kinetic undercooling.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P.G. Saffman and G.I. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell. Proc. R. Soc. London A245 (1958) 312–329.
S.J. Chapman, On the rôle of Stokes lines in the selection of Saffman-Taylor fingers with small surface tension. Eur. J. Appl. Math. 10 (1999) 513–534.
R. Combescot, T. Dombre, V. Hakim and Y. Pomeau, Shape selection of Saffman-Taylor fingers. Phys. Rev. Lett. 56 (1986) 2036–2039.
R. Combescot, V. Hakim, T. Dombre, Y. Pomeau and A. Pumir, Analytic theory of the Saffman-Taylor fingers. Phys. Rev. A37 (1988) 1270–1283.
D.C. Hong and J.S. Langer, Analytic theory of the selection mechanism in the Saffman-Taylor problem. Phys. Rev. Lett. 56 (1986) 2032–2035.
B.I. Shraiman, Velocity selection and the Saffman-Taylor problem. Phys. Rev. Lett. 56 (1986) 2028–2031.
S. Tanveer, Surprises in viscous fingering. J. Fluid Mech. 409 (2000) 273–308.
A.P. Aldushin and B.J. Matkowsky, Selection in the Saffman-Taylor finger problem and the Taylor-Saffman bubble problem without surface tension. Appl. Math. Lett. 11 (1998) 57–62.
A.P. Aldushin and B.J. Matkowsky, Extremum principles for selection in the Saffman-Taylor finger and Taylor-Saffman bubble problems. Phys. Fluids 11 (1999) 1287–1296.
C. Charach, B. Zaltzman and I.G. Gotz, Interfacial kinetic effect in planar solidification problems without initial undercooling. Math. Model Method Appl. Sci. 4 (1994) 331–354.
J.D. Evans and J.R. King, Asymptotic results for the Stefan problem with kinetic undercooling. Q. J. Mech. Appl. Math. 53 (2000) 449–473.
R.C. Kerr, A.W. Woods, M.G. Worster and H.E. Huppert, Solidification of an alloy cooled from above. 2. Nonequilibrium interfacial kinetics. J. Fluid Mech. 217 (1990) 331–348.
S.J. Chapman, Asymptotic analysis of the Ginzburg-Landau model of superconductivity-reduction to a freeboundary model. Quart. Appl. Math. 53 (1995) 601–627.
H.K. Kuiken, Edge effects in crystal growth under intermediate diffusive kinetic control. IMA J. Appl. Math. 35 (1985) 117–129.
N.B. Pleshchinskii and M. Reissig, Hele-Shaw flows with nonlinear kinetic undercooling regularization. Nonlin. Anal.-Theor. Meth. App. 50 (2002) 191–203.
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 (1999) 561–579.
S.J. Chapman, J.R. King and K.L. Adams, Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations. Proc. R. Soc. London A454 (1998) 2733–2755.
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 (1981) 445–469.
G.F. Carrier, M. Krook and C.E. Pearson, Functions of a Complex Variable. New York: McGraw-Hill (1966) 438 pp.
M.V. Berry, Waves near Stokes lines. Proc. R. Soc. London A427 (1990) 265–280.
J.R. King, Mathematical Aspects of Semiconductor Process Modelling. D Phil thesis, University of Oxford (1986) 408 pp.
J.R. King, Interacting Stokes lines. In: C.J. Howls, T. Kawai and Y. Takei (eds.), Towards the Exact WKB Analysis of Differential Equations, Linear or Nonlinear. Kyoto University Press (2000) pp. 165–178.
M.D. Kruskal and H. Segur, Asymptotics beyond all orders in a model of crystal growth. Stud. Appl. Math. 85 (1991) 129–181.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chapman, S., King, J. The selection of Saffman-Taylor fingers by kinetic undercooling. Journal of Engineering Mathematics 46, 1–32 (2003). https://doi.org/10.1023/A:1022860705459
Issue Date:
DOI: https://doi.org/10.1023/A:1022860705459