Journal of Engineering Mathematics

, Volume 46, Issue 1, pp 1–32 | Cite as

The selection of Saffman-Taylor fingers by kinetic undercooling

  • S.J. Chapman
  • J.R. King
Article

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.

asymptotics beyond-all-orders finger selection Hele-Shaw kinetic undercooling 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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.Google Scholar
  2. 2.
    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.Google Scholar
  3. 3.
    R. Combescot, T. Dombre, V. Hakim and Y. Pomeau, Shape selection of Saffman-Taylor fingers. Phys. Rev. Lett. 56 (1986) 2036–2039.Google Scholar
  4. 4.
    R. Combescot, V. Hakim, T. Dombre, Y. Pomeau and A. Pumir, Analytic theory of the Saffman-Taylor fingers. Phys. Rev. A37 (1988) 1270–1283.Google Scholar
  5. 5.
    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.Google Scholar
  6. 6.
    B.I. Shraiman, Velocity selection and the Saffman-Taylor problem. Phys. Rev. Lett. 56 (1986) 2028–2031.Google Scholar
  7. 7.
    S. Tanveer, Surprises in viscous fingering. J. Fluid Mech. 409 (2000) 273–308.Google Scholar
  8. 8.
    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.Google Scholar
  9. 9.
    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.Google Scholar
  10. 10.
    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.Google Scholar
  11. 11.
    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.Google Scholar
  12. 12.
    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.Google Scholar
  13. 13.
    S.J. Chapman, Asymptotic analysis of the Ginzburg-Landau model of superconductivity-reduction to a freeboundary model. Quart. Appl. Math. 53 (1995) 601–627.Google Scholar
  14. 14.
    H.K. Kuiken, Edge effects in crystal growth under intermediate diffusive kinetic control. IMA J. Appl. Math. 35 (1985) 117–129.Google Scholar
  15. 15.
    N.B. Pleshchinskii and M. Reissig, Hele-Shaw flows with nonlinear kinetic undercooling regularization. Nonlin. Anal.-Theor. Meth. App. 50 (2002) 191–203.Google Scholar
  16. 16.
    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.Google Scholar
  17. 17.
    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.Google Scholar
  18. 18.
    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.Google Scholar
  19. 19.
    G.F. Carrier, M. Krook and C.E. Pearson, Functions of a Complex Variable. New York: McGraw-Hill (1966) 438 pp.Google Scholar
  20. 20.
    M.V. Berry, Waves near Stokes lines. Proc. R. Soc. London A427 (1990) 265–280.Google Scholar
  21. 21.
    J.R. King, Mathematical Aspects of Semiconductor Process Modelling. D Phil thesis, University of Oxford (1986) 408 pp.Google Scholar
  22. 22.
    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.Google Scholar
  23. 23.
    M.D. Kruskal and H. Segur, Asymptotics beyond all orders in a model of crystal growth. Stud. Appl. Math. 85 (1991) 129–181.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • S.J. Chapman
    • 1
  • J.R. King
    • 2
  1. 1.OxfordUnited Kingdom
  2. 2.Department of Theoretical MechanicsUniversity of NottinghamNottinghamUnited Kingdom

Personalised recommendations