Interfacial Pattern Formation: A Progress Report

  • H. Levine
Conference paper
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 69)


Interfacial patterns such as those which arise during dendritic solidification, viscous fingering and electrochemical deposition have long been of scientific and technological interest. Until recently, however, even the simplest steady-state structures have eluded theoretical treatment. In this talk, I review the progress made over the past several years towards remedying this situation. Specifically, I explain how the paradigm of microscopic solvability can be used to explain unique steady-state patterns and why secondary instabilities such as tip-splitting or sidebranching are inevitable. I focus on the Saffman-Taylor finger in a channel to illustrate the general framework.


Porous Medium Electrochemical Deposition Secondary Instability Dendritic Solidification Taylor Bubble 
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.


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  1. 1.
    H.J.S. Hele-Shaw: Nature (London) 58, 34 (1898)CrossRefADSGoogle Scholar
  2. 2.
    P. Tabeling, G. Zocchi, A. Libchaber: “An Experimental Study of the Saffman-Taylor Instability”, Chicago preprint (1986)Google Scholar
  3. 3.
    P.G. Saffman, G.I. Taylor: Proc. Roy. Soc A245, 312 (1958)ADSMathSciNetGoogle Scholar
  4. 4.
    See for example A. Scheidigger: Physics of Flow Through Porous Media, ( University of Toronto Press, Toronto, 1974 )Google Scholar
  5. 5.
    C.W. Park, G.M. Homsy: J. Fluid. Mech 139, 291 (1984)CrossRefMATHADSGoogle Scholar
  6. 6.
    W.W. Mullins, R.F. Sekerka: J. Appl. Phys. 34, 323 (1968), ibid. 35, 444 (1964)CrossRefADSGoogle Scholar
  7. 7.
    P.G. Saffman, G.I. Taylor: ref. 3. For later developments see E. Pitts: J. Fluid Mech. 97, 53 (1980)Google Scholar
  8. 8.
    J.W. McLean, P.G. Saffman: J. Fluid Mech. 102, 455 (1981)CrossRefMATHADSGoogle Scholar
  9. 9.
    G. Trygvasson, H. Aref: J. Fluid. Mech 136, 1 (1983)CrossRefADSGoogle Scholar
  10. 10.
    J.M. Vanden-Broeck: Phys. Fluids 26, 2033 (1983)CrossRefADSMathSciNetGoogle Scholar
  11. 11.
    D. Kessler, J. Koplik, H. Levine: “Dendritic Growth in a Channel”, Schlumberger preprint (1986)Google Scholar
  12. 12.
    D. Kessler, H. Levine: Phys. Rev. A33 2621, 2634 (1986)ADSGoogle Scholar
  13. 13.
    B.I. Shraiman, D. Bensimon: Phys. Rev. A 30, 2840 (1984), S.D. Howison: to appearGoogle Scholar
  14. 14.
    B.I. Shraiman: Phys. Rev. Lett. 56, 2028 (1986);CrossRefADSGoogle Scholar
  15. T. Pombre, V. Hakim, Y. Pomeau: C. R. Acad. Sci 302, 803 (1986);Google Scholar
  16. D.C. Hong, J.S. Langer: Phys. Rev. Lett. 56, 2032 (1986)CrossRefADSGoogle Scholar
  17. 15.
    R. Combescot, T. Dombre, V. Hakim, Y. Pomeau, A. Pumir: Phys. Rev. Lett. 56, 2036 (1986)CrossRefADSGoogle Scholar
  18. 16.
    M. Kruskal, H. Segur: “Asymptotics Beyond All Orders in a Model of Dendritic Crystal Growth”, A.R.A.P. Tech. Memo 85–25; For a different approach, see R. Dashen, D. Kessler, H. Levine, R. Savit: “The Geometric Model of Dendritic Growth; the Small Velocity Limit”, Physica D to appearGoogle Scholar
  19. 17.
    D. Kessler, H. Levine: Phys. Rev. A32, 1930 (1985)ADSGoogle Scholar
  20. 18.
    D. Bensimon: Phys. Rev. A33, 1302 (1986)ADSGoogle Scholar
  21. 19.
    A.J. DeGregoria, L.W. Schwartz: “A Boundary Integral Method for Two-Phase Displacement in Hele-shaw Cells”, Exxon preprints (1985)Google Scholar
  22. 20.
    D. Bensimon, L. Kadanoff, S. Liang, B. Shraiman, C. Tang: “Viscous Flows in Two Dimensions”, Chicago Preprint 1985Google Scholar
  23. 21.
    Ya.B. Zeldovich, A.G. Istratov, N.I. Kidin, V.B. Librovitch: Comb. Sci. and Tech. 24, 1 (1980)CrossRefGoogle Scholar
  24. 22.
    See ref. 2 and references thereinGoogle Scholar
  25. 23.
    C.W. Park and G.M. Homsy, Phys. Fluids 28, 1583 (1985)CrossRefADSGoogle Scholar
  26. 24.
    L. Patterson: J. Fluid Mech 113, 513 (1981)CrossRefADSGoogle Scholar
  27. 25.
    D. Kessler, J. Koplik, H. Levine: Phys. Rev. A30, 2820 (1984);ADSGoogle Scholar
  28. L. Sander, P. Ramanlal, E. Ben-Jacob: Phys. Rev. A32, 3160 (1985)ADSGoogle Scholar
  29. 26.
    J.D. Chen, D. Wilkinson: Phys. Rev. Lett. 55, 1982 (1985);Google Scholar
  30. K.J. Maloy, J. Feder, J. Jossang: Phys. Rev. Lett. 55, 2681 (1985)CrossRefGoogle Scholar
  31. 27.
    J. Nittman, G. Daccord, H.E. Stanley: Nature 314, 141 (1985)CrossRefADSGoogle Scholar
  32. 28.
    T.A. Witten, L.M. Sander: Phys. Rev. Lett. 47, 1400 (1981)CrossRefADSGoogle Scholar
  33. 29.
    L. Patterson: Phys. Rev. Lett. 52, 1621 (1984);CrossRefADSGoogle Scholar
  34. C. Tang: Phys. Rev. A31. 1977 (1985);ADSGoogle Scholar
  35. L. Kadanoff: J. Stat. Phys. 39, 267 (1985)CrossRefADSMathSciNetGoogle Scholar
  36. 30.
    E. Ben-Jacob, R. Godbey, N. Goldenfeld, J. Koplik, H. Levine, T. Mueller, L. Sander: Phys. Rev. Lett. 55, 1315 (1985)CrossRefADSGoogle Scholar
  37. 31.
    For a recent review, see D. Kessler, J. Koplik, H. Levine: “Pattern Formation Far From Equilibrium - The Free Space Dendrite”, NATO A.R.W. on “Patterns, Defects and Microstructures”, to appearGoogle Scholar
  38. 32.
    D. Kessler, H. Levine: Phys. Rev. B, to appear;Google Scholar
  39. D. Meiron: Phys. Rev A33, 2704 (1986);ADSGoogle Scholar
  40. D. Kessler, J. Koplik, H. Levine: Phys. Rev A33, 3352 (1986)ADSGoogle Scholar
  41. 33.
    See ref. 31 and also, A. Barbieri, D. Hong, J. Langer: “Velocity Selection in the Symmetric Model of Dendritic Crystal Growth”, ITP 86–65 preprint; M. Ben-Amar, Y. Pomeau: “Theory of Dendritic Growth in a Weakly Undercooled Melt”, Saclay preprint (1986)Google Scholar
  42. 34.
    P. Pelce: “Dynamique des Fronts Courbes”, Thesis (May 1986)Google Scholar
  43. 35.
    A. Karma: Caltech preprints (1986)Google Scholar
  44. 36.
    Y. Sawada, A. Dougherty, J. Gollub: Phys. Rev. Lett. 56, 1260 (1986);CrossRefADSGoogle Scholar
  45. D. Grier, E. Ben-Jacob, R. Clarke, L. Sander: Phys. Rev. Lett. 56, 1264 (1986)CrossRefADSGoogle Scholar
  46. 37.
    J.M. Vanden-Broeck: Phys. Fluids 27, 2604 (1984)CrossRefMATHADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • H. Levine
    • 1
  1. 1.Schlumberger-Doll ResearchRidgefieldUSA

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