The selection of SaffmanTaylor fingers by kinetic undercooling
 S.J. Chapman,
 J.R. King
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The selection of SaffmanTaylor fingers by surface tension has been widely studied. Here their selection is analysed by another regularisation widely adopted in studying otherwise illposed Stefan problems, namely kinetic undercooling. An asymptoticbeyondallorders 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 HeleShaw 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 HeleShaw problem with kinetic undercooling.
 P.G. Saffman and G.I. Taylor, The penetration of a fluid into a porous medium or HeleShaw cell. Proc. R. Soc. London A245 (1958) 312–329.
 S.J. Chapman, On the rôle of Stokes lines in the selection of SaffmanTaylor 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 SaffmanTaylor fingers. Phys. Rev. Lett. 56 (1986) 2036–2039.
 R. Combescot, V. Hakim, T. Dombre, Y. Pomeau and A. Pumir, Analytic theory of the SaffmanTaylor fingers. Phys. Rev. A37 (1988) 1270–1283.
 D.C. Hong and J.S. Langer, Analytic theory of the selection mechanism in the SaffmanTaylor problem. Phys. Rev. Lett. 56 (1986) 2032–2035.
 B.I. Shraiman, Velocity selection and the SaffmanTaylor 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 SaffmanTaylor finger problem and the TaylorSaffman 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 SaffmanTaylor finger and TaylorSaffman 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 GinzburgLandau model of superconductivityreduction 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, HeleShaw 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 HeleShaw 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 HeleShaw cell. J. Fluid Mech. 102 (1981) 445–469.
 G.F. Carrier, M. Krook and C.E. Pearson, Functions of a Complex Variable. New York: McGrawHill (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.
 Title
 The selection of SaffmanTaylor fingers by kinetic undercooling
 Journal

Journal of Engineering Mathematics
Volume 46, Issue 1 , pp 132
 Cover Date
 20030501
 DOI
 10.1023/A:1022860705459
 Print ISSN
 00220833
 Online ISSN
 15732703
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 asymptotics
 beyondallorders
 finger selection
 HeleShaw
 kinetic undercooling
 Industry Sectors
 Authors

 S.J. Chapman ^{(1)}
 J.R. King ^{(2)}
 Author Affiliations

 1. 24–29 St. Giles', Oxford, OX1 3LB, United Kingdom
 2. Department of Theoretical Mechanics, University of Nottingham, Nottingham, NG7 2RD, United Kingdom