Advertisement

Growth Patterns: From Stable Curved Fronts to Fractal Structures

  • Yves Couder
Part of the NATO ASI Series book series (NSSB, volume 280)

Abstract

The present lectures intend to review some recent experiments on the properties of patterns obtained in growth processes. This covers a very large range of situations and I will limit myself here to three systems which can be considered as archetypes demonstrating some of the effects at work. The three systems, Saffman Taylor viscous fingering (1–37), dendritic crystal growth (38–50), and diffusion limited aggregation (51–68) have been widely studied and reviews on these three problems can be found respectively in references 3–6, 39–42, and 52–55. Though they are related to still other pattern forming systems these have been chosen here for their (relative) simplicity. All three are close to an ideal problem which is intrinsically mathematical.

Keywords

Fractal Structure Isotropic Case Stefan Problem Diffusion Limited Aggregation Curve Front 
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).
    P. G. Saffman and G. I. Taylor, Proc. Roy. Soc. London, Ser. A, 245, 312 (1958)MathSciNetADSMATHCrossRefGoogle Scholar
  2. (2).
    R. L. Chuoke, P. Van Meurs and C. Van der Pol, Tr. AIME 216, 188 (1959)Google Scholar
  3. (3).
    D. Bensimon, L. P. Kadanoff, S. Liang, B.I. Shraiman and Chao Tang, Rev. Mod. Phys. 58, 977 (1986)ADSCrossRefGoogle Scholar
  4. (4).
    P. G. Saffman, J. Fluid Mech. 173, 73 (1986)MathSciNetADSMATHCrossRefGoogle Scholar
  5. (5).
    G. M. Homsy, Ann. Rev. Fluid Mech. 19, 271 (1987)ADSCrossRefGoogle Scholar
  6. (6).
    P. Pelcé, Dynamics of Curved Fronts (Academic Press, Orlando, 1988)MATHGoogle Scholar
  7. (7).
    H. Thomé, M. Rabaud, V. Hakim and Y. Couder, Phys. Fluids A1, 224 (1989)ADSGoogle Scholar
  8. (8).
    J. Bataille, Revue Inst. Pétrole 23, 1349 (1968)Google Scholar
  9. (9).
    L. Paterson, J. Fluid Mech. 113, 513 (1981)ADSCrossRefGoogle Scholar
  10. (10).
    P. Tabeling, G. Zocchi and A. Libchaber, J. Fluid Mech. 177, 67 (1987)ADSCrossRefGoogle Scholar
  11. (11).
    D.A. Reinelt, Phys. Fluids, 30, 2617 (1987)ADSMATHCrossRefGoogle Scholar
  12. (12).
    S. Tanveer, PreprintGoogle Scholar
  13. (13).
    J. W. Mc Lean and P. G. Saffman, J. Fluid Mech. 102, 455 (1981)ADSCrossRefGoogle Scholar
  14. (14).
    J. M. Vanden-Broeck, Phys. of Fluids 26, 2033 (1983)MathSciNetADSCrossRefGoogle Scholar
  15. (15).
    (a) R. Combescot, T. Dombre, V. Hakim, Y. Pomeau and A. Pumir, Phys. Rev. Lett. 58, 2036 (1986) and (b) Phys. Rev. A 37, 1270 (1988)ADSCrossRefGoogle Scholar
  16. (16).
    D. C. Hong and J. Langer, Phys. Rev. Lett. 56, 2032 (1986)ADSCrossRefGoogle Scholar
  17. (17).
    B. Shraiman, Phys. Rev. Lett. 56, 2028 (1986)ADSCrossRefGoogle Scholar
  18. (18).
    (a) D. Bensimon, P. Pelcé, B.I. Shraiman J. Phys. 48, 2081 (1987) (b) S. Tanveer, Phys. of Fluids, 30, 2318 (1987)CrossRefGoogle Scholar
  19. (19).
    M. Ben Amar, To appear in Phys. Rev. A, in May 1991Google Scholar
  20. (20).
    (a) M. Ben Amar, V. Hakim, M. Mashaal and Y. Couder, to appear in Phys. of Fluids, July 1991 (b) R. Combescot and M. Ben Amar, To appear in Phys. Rev. Lett.Google Scholar
  21. (21).
    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
  22. (22).
    Y. Couder, O. Cardoso, D. Dupuy, P. Tavernier and W. Thorn, Europhys.Lett. 2, 437 (1986)ADSCrossRefGoogle Scholar
  23. (23).
    A. Buka, J. Kertesz and T. Vicsek, Nature 323, 424 (1986)ADSCrossRefGoogle Scholar
  24. (24).
    S.K. Sarkar and D. Jasnow, Phys. Rev. A 39, 5299 (1989)ADSCrossRefGoogle Scholar
  25. (25).
    (a) Y. Couder, N. Gerard and M. Rabaud, Phys. Rev. A 34, 5175 (1986) (b) M. Rabaud, Y.Couder and N. Gerard, Phys. Rev. A 37, 935 (1988)ADSCrossRefGoogle Scholar
  26. (26).
    G. Zocchi, B.E. Shaw, A. Libchaber and L.P. Kadanoff, Phys. Rev. A 36, 1894 (1987)ADSCrossRefGoogle Scholar
  27. (27).
    A.R. Kopf-Sill and G.M. Homsy, Phys. Fluids 30, 2607 (1987)ADSCrossRefGoogle Scholar
  28. (28).
    A. T. Dorsey and O. Martin, Phys. Rev. A 35, 3989, (1987)MathSciNetADSCrossRefGoogle Scholar
  29. (29).
    H. Thomé, R. Combescot and Y. Couder, Phys. Rev. A 41, 5739 (1990)ADSCrossRefGoogle Scholar
  30. (30).
    D. C. Hong and J. Langer, Phys. Rev. Lett. 56, 2032 (1986)ADSCrossRefGoogle Scholar
  31. (31).
    D. C. Hong and J. Langer, Phys. Rev. A 36, 2325 (1987)ADSCrossRefGoogle Scholar
  32. (32).
    R. Combescot and T. Dombre, Rev. A 39, 3525 (1989)MathSciNetCrossRefGoogle Scholar
  33. (33).
    E. Ben Jacob, G. Deutscher, P. Garik, N. D. Goldenfeld and Y. Lereah, Phys. Rev. Lett. 57, 1903 (1986)ADSCrossRefGoogle Scholar
  34. (34).
    S. N. Rauseo, P. D. Barnes and J. V. Maher, Phys. Rev. A 35, 1245 (1987)ADSCrossRefGoogle Scholar
  35. (35).
    Y. Couder, in Random Fluctuations and Pattern Growth, edited by H.E. Stanley and N. Ostrowsky (Kluwer Academic Publisher, Dordrecht, 1988)Google Scholar
  36. (36).
    A.R. Kopf-Sill and G.M. Homsy, Phys. Fluids 31, 242 (1988) and E. Meiburg and G.M. Homsy, Phys. Fluids 31, 429 (1988)ADSCrossRefGoogle Scholar
  37. (37).
    A. Arneodo, Y. Couder, G. Grasseau, V. Hakim and M. Rabaud, Phys. Rev. Lett. 63, 984 (1989)ADSCrossRefGoogle Scholar
  38. (38).
    W.W. Mullins and R.F. Sekerka, J. of App. Phys. 35, 444 (1964)ADSCrossRefGoogle Scholar
  39. (39).
    J. S. Langer, Rev. of Mod. Phys. 52, 1 (1980)ADSCrossRefGoogle Scholar
  40. (40).
    C. Huang and M. E. Glicksman, Acta Metall. 29, 701 (1982)Google Scholar
  41. (41).
    J. S. Langer, in Chance and Matter, edited by J. Souletie, J. Vanimenus and R. Stora (North-Holland, Amsterdam, 1987)Google Scholar
  42. (42).
    D. A. Kessler, J. Koplik and H. Levine, Advances in Physics 37, 255 (1988)ADSCrossRefGoogle Scholar
  43. (43).
    G. P. Ivantsov, Dokl. Acad. Nauk. 58, 567 (1947)Google Scholar
  44. (44).
    (a) D. Kessler, J. Koplik and H. Levine Phys. Rev. A 33, 3352 (1986) (b) D. Kessler, J. Koplik and H. Levine, Phys. Rev. A 34, 4980 (1986)ADSCrossRefGoogle Scholar
  45. (45).
    B. Caroli, C. Caroli, C. Misbah and B. Roulet, J. de Phys.48, 547 (1987)CrossRefGoogle Scholar
  46. (46).
    M. Ben Amar, Phys. Rev. A. 41, 2080 (1990)ADSCrossRefGoogle Scholar
  47. (47).
    A. Dougherty and J. P. Gollub, Phys. Rev. A. 38, 3043 (1988).ADSCrossRefGoogle Scholar
  48. (48).
    J. Maurer, P. Bouissou, B. Perrin and P. Tabeling, Europhys. Lett. 6, 67 (1989)ADSCrossRefGoogle Scholar
  49. (49).
    H. Honjo, S. Ohta and M. Matsushita, J. of the Phys. Soc. Japan 55, 2487 (1986)ADSCrossRefGoogle Scholar
  50. (50).
    Y. Couder, F. Argoul, A. Arnéodo, J. Maurer and M. Rabaud, Phys. Rev. A 42, 3499 (1990)ADSCrossRefGoogle Scholar
  51. (51).
    T. Witten and L. M. Sander, Phys. Rev. Lett. 47, 1400 (1981) and Phys. Rev. B 27, 5686 (1983)ADSCrossRefGoogle Scholar
  52. (52).
    Random Fluctuations and Pattern Growth, edited by H.E. Stanley and N. Ostrowsky (Kluwer Academic Publisher, Dordrecht, 1988)Google Scholar
  53. (53).
    P. Meakin, in Phase Transitions and Critical Phenomena, Vol. 12, Edited by C. Domb and J.L. Lebowitz (Academic Press, Orlando, 1988)Google Scholar
  54. (54).
    D. A. Kessler, J. Koplik and H. Levine, Advances in Physics 37, 255 (1988)ADSCrossRefGoogle Scholar
  55. (55).
    T. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore, 1989)MATHGoogle Scholar
  56. (56).
    L. Paterson, Phys. Rev. Lett. 52, 1621 (1984)ADSCrossRefGoogle Scholar
  57. (57).
    T. Vicsek, Phys. Rev. Lett. 53, 2281 (1984)ADSCrossRefGoogle Scholar
  58. (58).
    S. K. Sarkar, Phys. Rev. A 32, 3114 (1985)ADSCrossRefGoogle Scholar
  59. (59).
    L.M. Sander, P. Ramanlal and E. Ben Jacob, Phys. Rev. A 32, 3160 (1985)ADSCrossRefGoogle Scholar
  60. (60).
    L.P. Kadanoff, J. of Stat. Phys. 39, 267 (1985)MathSciNetADSCrossRefGoogle Scholar
  61. (61).
    S. Liang, Phys. Rev. A 33, 2663 (1986)ADSCrossRefGoogle Scholar
  62. (62).
    J. Nittmann and H. E. Stanley, Nature 321, 663 (1986)ADSCrossRefGoogle Scholar
  63. (63).
    J. Kertěsz and T. Vicsek, J. Phys. A 19, L257 (1986)ADSCrossRefGoogle Scholar
  64. (64).
    P. Meakin, Phys. Rev. A 36, 332 (1987)ADSCrossRefGoogle Scholar
  65. (65).
    P. Meakin, J. Kertesz and T. Viczek, J. Phys. A 21, 1271 (1988)ADSCrossRefGoogle Scholar
  66. (66).
    H. Kondo, M. Matsushita, and S. Ohnishi. Proceedings of the first symposium for science on form, Editor S. Ishizaka, (KTK Scientific publishers, Tokyo, 1986)Google Scholar
  67. (67).
    R. Julien, M. Kolb and M. Botet, J. de Phys. 45, 395 (1984)CrossRefGoogle Scholar
  68. (68).
    J.P. Eckmann, P. Meakin, I. Procaccia and R. Zeitak, Phys. Rev.A 29, 3185 (1989)MathSciNetADSCrossRefGoogle Scholar
  69. (69).
    B. Shraiman and D. Bensimon, Phys. Rev.A 30, 2840 (1984)MathSciNetADSCrossRefGoogle Scholar
  70. (70).
    S. D. Howison, J. Fluid Mech. 167, 439 (1986)MathSciNetADSMATHCrossRefGoogle Scholar
  71. (71).
    D. Bensimon and P. Pelcé, Phys. Rev. A 33, 4477 (1986)ADSCrossRefGoogle Scholar
  72. (72).
    Ya. B. Zel’dovich, A. G. Istratov, N. I. Kidin and V. B. Librovich, Combust. Science Technol. 24, 1 (1980)CrossRefGoogle Scholar
  73. (73).
    P. Pelcé, Thése d’état, and P. Pelcé and P. Clavin, Europhys. Lett. 3, 907 (1987)ADSCrossRefGoogle Scholar
  74. (74).
    B. Caroli, C. Caroli and B. Roulet, J. Phys. 48, 1423 (1987)CrossRefGoogle Scholar
  75. (75).
    D. Kessler and H. Levine, Europhys. Lett. 4, 215 (1987)ADSCrossRefGoogle Scholar
  76. (76).
    P. Huerre and P.A. Monkewitz, J.Fluid Mech. 159, 151 (1985)MathSciNetADSMATHCrossRefGoogle Scholar
  77. (77).
    X. W. Quian and H. Z. Cummins, Phys. Rev. Lett. 64, 3038 (1990)ADSCrossRefGoogle Scholar
  78. (78).
    P. Bouissou, A. Chiffaudel, B. Perrin and P. Tabeling, Europhys. Lett. 13, 89 (1990)ADSCrossRefGoogle Scholar
  79. (79).
    A. Arnéodo, F. Argoul, Y. Couder and M. Rabaud, Phys. Rev. Lett. 66, 2332 (1991)ADSCrossRefGoogle Scholar
  80. (80).
    J. Nittmann, G. Daccord and H. E. Stanley, Nature 314, 141 (1985), and G. Daccord, J. Nittmann and H. E. Stanley, Phys. Rev. Lett. 56, 336 (1986)ADSCrossRefGoogle Scholar
  81. (81).
    Jing-Den Chen and D. Wilkinson, Phys. Rev. Lett. 55, 1892 (1985)ADSCrossRefGoogle Scholar
  82. (82).
    R. Lenormand, E. Touboul and Zarcone, J. Fluid Mech. 189, 165 (1988)ADSCrossRefGoogle Scholar
  83. (83).
    H. Van Damme, E. Alsac, C. Laroche, and L. Gatineau, Europhys. Lett. 5, 25 (1988)ADSCrossRefGoogle Scholar
  84. (84).
    V. Hakim, Private communicationGoogle Scholar
  85. (85).
    E. Brener, H. Levine and Y. Tu, Phys. Rev. Lett. 66, 1878 (1991)ADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • Yves Couder
    • 1
  1. 1.Laboratoire de Physique StatistiqueEcole Normale SupérieureParis Cedex 05France

Personalised recommendations