The Problem of Directional Solidification Revisited

  • M. Ben Amar
  • T. Dombre
  • V. Hakim
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 41)


The connection of directional solidification at low velocity with Saffman-Taylor fingering is discussed. We put forward some quantitative predictions concerning the shape of solidification cells, which can be gained from this analogy. We present also some numerical results without surface tension, which seem to go against the existence of wavelength selection by a solvability mechanism in this problem.


Directional Solidification Wavelength Selection Amplitude Steady State Solidification Cell Directional Solidification Experiment 
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  1. 1.
    K.A. Jackson, J.D. Hunt: Trans. Met. Soc. AIME 236, 1929 (1966)Google Scholar
  2. See also S. de Cheveigne, C. Guthmann, M.M. Lebrun: J. Cryst. Growth 75, 242 (1985)CrossRefGoogle Scholar
  3. 2.
    C. Normand, Y. Pomeau, M.G. Velarde: Rev. Mod. Phys. 49, 581 (1977)CrossRefADSMathSciNetGoogle Scholar
  4. 3.
    D.J. Wollkind, L.A. Segel: Philos. Trans. R. Soc. Lond. 51, 268 (1970)Google Scholar
  5. 4.
    A. Karma: Phys. Rev. Lett. 57, 858 (1986)CrossRefADSGoogle Scholar
  6. 5.
    T. Dombre, V. Hakim: Phys. Rev.A 36, 2811 (1987)CrossRefADSGoogle Scholar
  7. 6.
    J.S. Langer: Rev. Mod. Phys. 52, 1 (1980)CrossRefADSGoogle Scholar
  8. 7.
    G. Tryggvason, H. Aref: J. Fluid Mech. 136, 1 (1983)CrossRefzbMATHADSGoogle Scholar
  9. 8.
    P.G. Saffman, G.I. Taylor: Proc. Roy. Soc. Lond. Ser.A 245, 312 (1958)CrossRefzbMATHADSMathSciNetGoogle Scholar
  10. 9.
    In order to derive (22), we calculate ∫∫ dx dz div[z G(po,p)] as -∫ ds(f n.∇G(po,p)), where the surface integration area is limited by the two lines (x=±0.5) and by an arbitrary smooth profile called here f. Alternatively, we may estimate the first quantity by using the diffusion equation of the Green’s function.Google Scholar
  11. 10.
    We define dilog(z) as dilog(z) = ∫0 Z du log(1-u)/uGoogle Scholar
  12. 11.
    M. Ben Amar, B. Moussallam: Physica D 25, 155 (1987)CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • M. Ben Amar
    • 1
  • T. Dombre
    • 1
  • V. Hakim
    • 1
  1. 1.Groupe de Physique des SolidesEcole Normale SupérieureParis, Cedex 05France

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