Advertisement

Directional Growth in Viscous Fingering

  • V. Hakim
  • M. Rabaud
  • H. Thomé
  • Y. Couder
Part of the NATO ASI Series book series (NSSB, volume 237)

Abstract

During the past few years there has been a renewed interest in the Saffman-Taylor instability (Saffman and Taylor (1958)), as this instability between strongly viscosity-contrasted fluids gives rise to shape selection processes. Its dynamics is very similar to that of another pattern forming instability, the Mullins-Sekerka instability, occurring during the free growth of a crystal in an undercooled liquid (Mullins and Sekerka (1964)). The theoretical analogy between the growth of viscous fingers in Hele-Shaw cells and the growth of crystalline dendrites is now well-known (e.g. Langer (1988); Kessler and Levine (1988 a) for a review) and the similarity between the two phenomena has been demonstrated experimentally (Rabaud et al. (1988)). Both processes are free growths in which the non linear evolution of the pattern defines the field in which it keeps growing.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. W. H. Bank and C. C. Mill (1954), Proc. Roy. Soc. London, A 223, 414.CrossRefGoogle Scholar
  2. J. Bechhoefer, A. Simon and A. Libchaber (1988), in these proceedings.Google Scholar
  3. E. Ben-Jacob, R. Godbey, N. D. Goldenfeld, J. Koplik, H. Levine, T. Mueller and L. M. Sander (1985), Phys.Rev. Lett. 55, 1315.CrossRefGoogle Scholar
  4. B. Billia (1987), J. of Crystal Growth 82, 747 and these proceedings.CrossRefGoogle Scholar
  5. F. P. Bretherton (1961), J. Fluid Mech. 10, 166.MathSciNetCrossRefGoogle Scholar
  6. S. de Cheveigné, C. Guthmann and M. M. Lebrun (1986), J. Physique, 47, 2095 and these proceedings.CrossRefGoogle Scholar
  7. T. Dombre and V. Hakim (1987), Phys.Rev. A, 36, 2811.CrossRefGoogle Scholar
  8. D. A. Kessler and H. Levine (1988 a), Adv. in Physics 37, 255.Google Scholar
  9. D. A. Kessler and H. Levine (1988 b) Preprint.Google Scholar
  10. K. A. Jackson and J. D. Hunt (1966) Trans. Met. Soc. of AIME 236, 1129.Google Scholar
  11. L. D. Landau and B. Levich (1942), Acta Phys. Chim. URSS, 17, 42.Google Scholar
  12. J. S. Langer (1980), Rev. Mod. Phys. 52, 1.CrossRefGoogle Scholar
  13. J. S. Langer (1988), preprint.Google Scholar
  14. H. Martin (1916), Engineering 102, 119.Google Scholar
  15. A. D. McEwan and G. I. Taylor (1966), J. Fluid Mech. 26, 1.CrossRefGoogle Scholar
  16. H. Muller Krumbhaar, these proceedings.Google Scholar
  17. W. W. Mullins and R. F. Sekerka (1964), J. Appl. Phys., 35, 444.CrossRefGoogle Scholar
  18. C. W. Park and G. M. Homsy (1985), Phys. Fluids 28, 1583.CrossRefGoogle Scholar
  19. J. R. A. Pearson (1960), J. Fluid Mech. 7, 481.MathSciNetCrossRefGoogle Scholar
  20. E. Pitts and J. Greiller (1961), J. Fluid Mech. 11, 33.CrossRefGoogle Scholar
  21. M. Rabaud, Y. Couder and N. Gerard (1988) Phys. Rev. A, 37, 935.CrossRefGoogle Scholar
  22. D. A. Reinelt and P. G. Saffman (1985), Journal Sci. Stat. Comp. 6, 542.CrossRefGoogle Scholar
  23. P. G. Saffman and G.I. Taylor (1958), Proc. Roy. Soc. London, A 245, 312.CrossRefGoogle Scholar
  24. P. Tabeling, G. Zocchi and A. Libchaber (1987), J. Fluid Mech. 177, 67.CrossRefGoogle Scholar
  25. G. I. Taylor (1963), J. Fluid Mech. 16, 595.CrossRefGoogle Scholar
  26. R. Trivedi (1984), Metal. Trans. A 15, 977 and Part I, p. 967.Google Scholar
  27. L. H. Ungar and R. A. Brown (1985), Phys. Rev. B 31, 5931.CrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • V. Hakim
    • 1
  • M. Rabaud
    • 2
  • H. Thomé
    • 2
  • Y. Couder
    • 2
  1. 1.Laboratoire de Physique Statistique de l’Ecole Normale SupérieureParis Cedex 05France
  2. 2.Groupe de Physique des Solides de l’Ecole Normale SupérieureParis Cedex 05France

Personalised recommendations