Advertisement

Interfacial Pattern Formation: Dynamical Effects

  • D. A. Kessler
  • H. Levine
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 37)

Abstract

In this talk, we focus on the dynamical aspects of pattern forming systems. These include the formation of dendritic sidebranches as well as the tip-splitting instability. We study these phenomena through first computing the steady-state pattern (via the “solvability” mechanism) and then solving the associated linear stability problem. The results to date suggest that both of the aforementioned effects must be treated by including the effects of finite amplitude perturbations on the deterministic growth equations.

Keywords

Capillary Number Needle Crystal Spectral Flow Velocity Selection Capillary Pressure Drop 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    For a recent review, see D. Kessler, J. Koplik, and H. Levine, Pattern Selection in Fingered Growth Phenomena, to appear in Advances in Physics Google Scholar
  2. 2.
    D.P. Woodruff, The Solid-Liquid Interface, Cambridge University Press, 1973Google Scholar
  3. 3.
    H.J.S. Hele-Shaw, Nature, 58, 334 (1898)CrossRefGoogle Scholar
  4. 4.
    W.W. Mullins and R.F. Sekerka, J.Appl. Phys., 34, 323 (1963)ADSCrossRefGoogle Scholar
  5. 5.
    P.G. Saffman and G.I. Taylor, Proc. Roy. Soc, A 245, 312 (1958)MathSciNetADSGoogle Scholar
  6. 6.
    J.W. McLean and P.G. Saffman, J. Fluid Mech., 102, 455 (1981)ADSzbMATHCrossRefGoogle Scholar
  7. 7.
    G. Trygvasson and H. Aref, J. Fluid Mech., 136, 1 (1983)ADSCrossRefGoogle Scholar
  8. 7a.
    L.M. Schwartz and A.J. DeGregorio, J. Fluid Mech., 164, 383 (1986)MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 8.
    G.P. Ivantsov, Dokl. Akad. Nauk SSSR, 58, 567 (1947)Google Scholar
  10. 9.
    For a review of early attempts at resolving this paradox, see J.S. Langer, Rev. Mod. Phys., 52, 1 (1980)Google Scholar
  11. 10.
    For a discussion of local models of pattern formation, see D.A. Kessler, J. Koplik, and H. Levine, Physiochem. Hydro., 6, 507 (1985)Google Scholar
  12. 11.
    D.A. Kessler and H. Levine, Phys. Rev., A 33, 2634 (1986)ADSGoogle Scholar
  13. 12.
    D.A. Kessler and H. Levine, Phys. Rev., B 33, 7687 (1986)Google Scholar
  14. 12a.
    D. Meiron, Phys. Rev., A 33, 2704 (1986)ADSGoogle Scholar
  15. 12b.
    D.A. Kessler, J. Koplik, and H. Levine, Phys. Rev., A 33, 3352 (1986)ADSGoogle Scholar
  16. 12c.
    M. Ben-Amar and B. Moussalam, to appearGoogle Scholar
  17. 13.
    B.I. Shraiman, Phys. Rev. Lett., 56, 2028 (1986)ADSCrossRefGoogle Scholar
  18. 13a.
    D.C. Hong and J.S. Langer, Phys. Rev. Lett., 56, 2032 (1986)ADSCrossRefGoogle Scholar
  19. 13b.
    R. Combescot, T. Dombre, V. Hakim, Y. Pomeau, and A. Pumir, Phys. Rev. Lett., 56, 2036 (1986)ADSCrossRefGoogle Scholar
  20. 14.
    D.A. Kessler, J. Koplik, and H. Levine, in Proc. NATO A.R.W. Patterns, Defects and Microstructure, (1986, to appear)Google Scholar
  21. 14a.
    A. Barbieri, D.C. Hong, and J.S. Langer, to appearGoogle Scholar
  22. 14b.
    M. Ben-Amar and Y. Pomeau, Europhys. Lett., 2, 307 (1986)ADSCrossRefGoogle Scholar
  23. 15.
    For more details, see D. Kessler and H. Levine, Phys. Rev. Lett., 57, 3069 (1986)ADSCrossRefGoogle Scholar
  24. 16.
    D. Kessler and H. Levine, Discrete Set Selection of Saffman-Taylor Fingers, submitted to Phys. FluidsGoogle Scholar
  25. 17.
    For a simple WKB approach to this result, see D. Bensimon, L. Kadanoff, S. Liang, B.I. Shraiman, and C. Tang, Rev. Mod. Phys., 58, 977 (1986)ADSCrossRefGoogle Scholar
  26. 17a.
    P. Pelce, Dynamique des Fronts Courbés, Thesis Université de Provence (1986)Google Scholar
  27. 18.
    D.A. Kessler and H. Levine, Determining the Wavelength of Dendritic Sidebranches, in preparationGoogle Scholar
  28. 19.
    D. Bensimon, Phys. Rev., A 33, 1302 (1986)ADSGoogle Scholar
  29. 20.
    E. Ben-Jacob, R. Godbey, N.D. Goldenfeld, J. Koplik, H. Levine, T. Mueller, and L.M. Sander, Phys. Rev. Lett., 55, 1315 (1985)ADSCrossRefGoogle Scholar
  30. 21.
    A. Buka, J. Kertesz, and T. Vicsek, Nature, 323, 424 (1986)ADSCrossRefGoogle Scholar
  31. 22.
    H. Honjo, S. Ohta and M. Matsushita, Irregular Fractal-like Growth of Ammonium Chloride, submitted to J. Phys. Soc. Japan (1986)Google Scholar
  32. 23.
    T.A. Witten and L.M. Sander, Phys. Rev., B 27, 5696 (1983)MathSciNetADSGoogle Scholar
  33. 24.
    C. Tang, Phys. Rev., A 31, 1977 (1985)ADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • D. A. Kessler
    • 1
  • H. Levine
    • 2
  1. 1.Department of PhysicsUniversity of MichiganAnn ArborUSA
  2. 2.Schlumberger-Doll ResearchRidgefieldUSA

Personalised recommendations