Skip to main content
SpringerLink
Log in
Book cover

Propagation in Systems Far from Equilibrium pp 35–43Cite as

The Problem of Directional Solidification Revisited

  • Conference paper

  • 249 Accesses

Part of the book series: Springer Series in Synergetics ((SSSYN,volume 41))

Abstract

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.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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)

    Article  Google Scholar 

  3. C. Normand, Y. Pomeau, M.G. Velarde: Rev. Mod. Phys. 49, 581 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  4. D.J. Wollkind, L.A. Segel: Philos. Trans. R. Soc. Lond. 51, 268 (1970)

    Google Scholar 

  5. A. Karma: Phys. Rev. Lett. 57, 858 (1986)

    Article  ADS  Google Scholar 

  6. T. Dombre, V. Hakim: Phys. Rev.A 36, 2811 (1987)

    Article  ADS  Google Scholar 

  7. J.S. Langer: Rev. Mod. Phys. 52, 1 (1980)

    Article  ADS  Google Scholar 

  8. G. Tryggvason, H. Aref: J. Fluid Mech. 136, 1 (1983)

    Article  MATH  ADS  Google Scholar 

  9. P.G. Saffman, G.I. Taylor: Proc. Roy. Soc. Lond. Ser.A 245, 312 (1958)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. 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. We define dilog(z) as dilog(z) = ∫0 Z du log(1-u)/u

    Google Scholar 

  12. M. Ben Amar, B. Moussallam: Physica D 25, 155 (1987)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Groupe de Physique des Solides, Ecole Normale Supérieure, F-75231, Paris, Cedex 05, France

    M. Ben Amar, T. Dombre & V. Hakim

Authors
  1. M. Ben Amar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Dombre
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. Hakim
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Laboratoire d’Hydrodynamique et Mécanique Physique, E.S.P.C.I. Paris, 10, rue Vauquelin, F-75231, Paris Cedex 05, France

    José E. Wesfreid

  2. Fachbereich 7 (Physik), Universität Essen, D-4300, Essen 1, Fed. Rep. of Germany

    Helmut R. Brand

  3. IRF-DPh-G/PSRM, CEN-Saclay, F-91191, Gif-sur-Yvette, France

    Paul Manneville

  4. Lab. de Recherches en Combustion, Université de Provence, Centre St. Jérôme, F-13397, Marseille Cedex 13, France

    Gilbert Albinet

  5. Centre de Physique Théorique, F-74310, Les Houches, France

    Nino Boccara

Rights and permissions

Reprints and permissions

Copyright information

© 1988 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Amar, M.B., Dombre, T., Hakim, V. (1988). The Problem of Directional Solidification Revisited. In: Wesfreid, J.E., Brand, H.R., Manneville, P., Albinet, G., Boccara, N. (eds) Propagation in Systems Far from Equilibrium. Springer Series in Synergetics, vol 41. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-73861-6_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-73861-6_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-73863-0

  • Online ISBN: 978-3-642-73861-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation