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Growth Patterns: From Stable Curved Fronts to Fractal Structures

  • Chapter
Chaos, Order, and Patterns

Part of the book series: NATO ASI 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.

associé au C.N.R.S. et aux Universités Paris 6 et Paris 7

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References

  1. P. G. Saffman and G. I. Taylor, Proc. Roy. Soc. London, Ser. A, 245, 312 (1958)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. R. L. Chuoke, P. Van Meurs and C. Van der Pol, Tr. AIME 216, 188 (1959)

    Google Scholar 

  3. D. Bensimon, L. P. Kadanoff, S. Liang, B.I. Shraiman and Chao Tang, Rev. Mod. Phys. 58, 977 (1986)

    Article  ADS  Google Scholar 

  4. P. G. Saffman, J. Fluid Mech. 173, 73 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. G. M. Homsy, Ann. Rev. Fluid Mech. 19, 271 (1987)

    Article  ADS  Google Scholar 

  6. P. Pelcé, Dynamics of Curved Fronts (Academic Press, Orlando, 1988)

    MATH  Google Scholar 

  7. H. Thomé, M. Rabaud, V. Hakim and Y. Couder, Phys. Fluids A1, 224 (1989)

    ADS  Google Scholar 

  8. J. Bataille, Revue Inst. Pétrole 23, 1349 (1968)

    Google Scholar 

  9. L. Paterson, J. Fluid Mech. 113, 513 (1981)

    Article  ADS  Google Scholar 

  10. P. Tabeling, G. Zocchi and A. Libchaber, J. Fluid Mech. 177, 67 (1987)

    Article  ADS  Google Scholar 

  11. D.A. Reinelt, Phys. Fluids, 30, 2617 (1987)

    Article  ADS  MATH  Google Scholar 

  12. S. Tanveer, Preprint

    Google Scholar 

  13. J. W. Mc Lean and P. G. Saffman, J. Fluid Mech. 102, 455 (1981)

    Article  ADS  Google Scholar 

  14. J. M. Vanden-Broeck, Phys. of Fluids 26, 2033 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  16. D. C. Hong and J. Langer, Phys. Rev. Lett. 56, 2032 (1986)

    Article  ADS  Google Scholar 

  17. B. Shraiman, Phys. Rev. Lett. 56, 2028 (1986)

    Article  ADS  Google Scholar 

  18. (a) D. Bensimon, P. Pelcé, B.I. Shraiman J. Phys. 48, 2081 (1987) (b) S. Tanveer, Phys. of Fluids, 30, 2318 (1987)

    Article  Google Scholar 

  19. M. Ben Amar, To appear in Phys. Rev. A, in May 1991

    Google Scholar 

  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. E. Ben Jacob, R. Godbey, N. D. Goldenfeld, J. Koplik, H. Levine, T. Mueller and L.M. Sander, Phys. Rev. Lett. 55, 1315 (1985)

    Article  ADS  Google Scholar 

  22. Y. Couder, O. Cardoso, D. Dupuy, P. Tavernier and W. Thorn, Europhys.Lett. 2, 437 (1986)

    Article  ADS  Google Scholar 

  23. A. Buka, J. Kertesz and T. Vicsek, Nature 323, 424 (1986)

    Article  ADS  Google Scholar 

  24. S.K. Sarkar and D. Jasnow, Phys. Rev. A 39, 5299 (1989)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  26. G. Zocchi, B.E. Shaw, A. Libchaber and L.P. Kadanoff, Phys. Rev. A 36, 1894 (1987)

    Article  ADS  Google Scholar 

  27. A.R. Kopf-Sill and G.M. Homsy, Phys. Fluids 30, 2607 (1987)

    Article  ADS  Google Scholar 

  28. A. T. Dorsey and O. Martin, Phys. Rev. A 35, 3989, (1987)

    Article  MathSciNet  ADS  Google Scholar 

  29. H. Thomé, R. Combescot and Y. Couder, Phys. Rev. A 41, 5739 (1990)

    Article  ADS  Google Scholar 

  30. D. C. Hong and J. Langer, Phys. Rev. Lett. 56, 2032 (1986)

    Article  ADS  Google Scholar 

  31. D. C. Hong and J. Langer, Phys. Rev. A 36, 2325 (1987)

    Article  ADS  Google Scholar 

  32. R. Combescot and T. Dombre, Rev. A 39, 3525 (1989)

    Article  MathSciNet  Google Scholar 

  33. E. Ben Jacob, G. Deutscher, P. Garik, N. D. Goldenfeld and Y. Lereah, Phys. Rev. Lett. 57, 1903 (1986)

    Article  ADS  Google Scholar 

  34. S. N. Rauseo, P. D. Barnes and J. V. Maher, Phys. Rev. A 35, 1245 (1987)

    Article  ADS  Google Scholar 

  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. 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)

    Article  ADS  Google Scholar 

  37. A. Arneodo, Y. Couder, G. Grasseau, V. Hakim and M. Rabaud, Phys. Rev. Lett. 63, 984 (1989)

    Article  ADS  Google Scholar 

  38. W.W. Mullins and R.F. Sekerka, J. of App. Phys. 35, 444 (1964)

    Article  ADS  Google Scholar 

  39. J. S. Langer, Rev. of Mod. Phys. 52, 1 (1980)

    Article  ADS  Google Scholar 

  40. C. Huang and M. E. Glicksman, Acta Metall. 29, 701 (1982)

    Google Scholar 

  41. J. S. Langer, in Chance and Matter, edited by J. Souletie, J. Vanimenus and R. Stora (North-Holland, Amsterdam, 1987)

    Google Scholar 

  42. D. A. Kessler, J. Koplik and H. Levine, Advances in Physics 37, 255 (1988)

    Article  ADS  Google Scholar 

  43. G. P. Ivantsov, Dokl. Acad. Nauk. 58, 567 (1947)

    Google Scholar 

  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)

    Article  ADS  Google Scholar 

  45. B. Caroli, C. Caroli, C. Misbah and B. Roulet, J. de Phys.48, 547 (1987)

    Article  Google Scholar 

  46. M. Ben Amar, Phys. Rev. A. 41, 2080 (1990)

    Article  ADS  Google Scholar 

  47. A. Dougherty and J. P. Gollub, Phys. Rev. A. 38, 3043 (1988).

    Article  ADS  Google Scholar 

  48. J. Maurer, P. Bouissou, B. Perrin and P. Tabeling, Europhys. Lett. 6, 67 (1989)

    Article  ADS  Google Scholar 

  49. H. Honjo, S. Ohta and M. Matsushita, J. of the Phys. Soc. Japan 55, 2487 (1986)

    Article  ADS  Google Scholar 

  50. Y. Couder, F. Argoul, A. Arnéodo, J. Maurer and M. Rabaud, Phys. Rev. A 42, 3499 (1990)

    Article  ADS  Google Scholar 

  51. T. Witten and L. M. Sander, Phys. Rev. Lett. 47, 1400 (1981) and Phys. Rev. B 27, 5686 (1983)

    Article  ADS  Google Scholar 

  52. Random Fluctuations and Pattern Growth, edited by H.E. Stanley and N. Ostrowsky (Kluwer Academic Publisher, Dordrecht, 1988)

    Google Scholar 

  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. D. A. Kessler, J. Koplik and H. Levine, Advances in Physics 37, 255 (1988)

    Article  ADS  Google Scholar 

  55. T. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore, 1989)

    MATH  Google Scholar 

  56. L. Paterson, Phys. Rev. Lett. 52, 1621 (1984)

    Article  ADS  Google Scholar 

  57. T. Vicsek, Phys. Rev. Lett. 53, 2281 (1984)

    Article  ADS  Google Scholar 

  58. S. K. Sarkar, Phys. Rev. A 32, 3114 (1985)

    Article  ADS  Google Scholar 

  59. L.M. Sander, P. Ramanlal and E. Ben Jacob, Phys. Rev. A 32, 3160 (1985)

    Article  ADS  Google Scholar 

  60. L.P. Kadanoff, J. of Stat. Phys. 39, 267 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  61. S. Liang, Phys. Rev. A 33, 2663 (1986)

    Article  ADS  Google Scholar 

  62. J. Nittmann and H. E. Stanley, Nature 321, 663 (1986)

    Article  ADS  Google Scholar 

  63. J. Kertěsz and T. Vicsek, J. Phys. A 19, L257 (1986)

    Article  ADS  Google Scholar 

  64. P. Meakin, Phys. Rev. A 36, 332 (1987)

    Article  ADS  Google Scholar 

  65. P. Meakin, J. Kertesz and T. Viczek, J. Phys. A 21, 1271 (1988)

    Article  ADS  Google Scholar 

  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. R. Julien, M. Kolb and M. Botet, J. de Phys. 45, 395 (1984)

    Article  Google Scholar 

  68. J.P. Eckmann, P. Meakin, I. Procaccia and R. Zeitak, Phys. Rev.A 29, 3185 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  69. B. Shraiman and D. Bensimon, Phys. Rev.A 30, 2840 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  70. S. D. Howison, J. Fluid Mech. 167, 439 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  71. D. Bensimon and P. Pelcé, Phys. Rev. A 33, 4477 (1986)

    Article  ADS  Google Scholar 

  72. Ya. B. Zel’dovich, A. G. Istratov, N. I. Kidin and V. B. Librovich, Combust. Science Technol. 24, 1 (1980)

    Article  Google Scholar 

  73. P. Pelcé, Thése d’état, and P. Pelcé and P. Clavin, Europhys. Lett. 3, 907 (1987)

    Article  ADS  Google Scholar 

  74. B. Caroli, C. Caroli and B. Roulet, J. Phys. 48, 1423 (1987)

    Article  Google Scholar 

  75. D. Kessler and H. Levine, Europhys. Lett. 4, 215 (1987)

    Article  ADS  Google Scholar 

  76. P. Huerre and P.A. Monkewitz, J.Fluid Mech. 159, 151 (1985)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  77. X. W. Quian and H. Z. Cummins, Phys. Rev. Lett. 64, 3038 (1990)

    Article  ADS  Google Scholar 

  78. P. Bouissou, A. Chiffaudel, B. Perrin and P. Tabeling, Europhys. Lett. 13, 89 (1990)

    Article  ADS  Google Scholar 

  79. A. Arnéodo, F. Argoul, Y. Couder and M. Rabaud, Phys. Rev. Lett. 66, 2332 (1991)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  81. Jing-Den Chen and D. Wilkinson, Phys. Rev. Lett. 55, 1892 (1985)

    Article  ADS  Google Scholar 

  82. R. Lenormand, E. Touboul and Zarcone, J. Fluid Mech. 189, 165 (1988)

    Article  ADS  Google Scholar 

  83. H. Van Damme, E. Alsac, C. Laroche, and L. Gatineau, Europhys. Lett. 5, 25 (1988)

    Article  ADS  Google Scholar 

  84. V. Hakim, Private communication

    Google Scholar 

  85. E. Brener, H. Levine and Y. Tu, Phys. Rev. Lett. 66, 1878 (1991)

    Article  ADS  Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratoire de Physique Statistique, Ecole Normale Supérieure, 24 rue Lhomond, 75231, Paris Cedex 05, France

    Yves Couder

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  1. Yves Couder
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Editors and Affiliations

  1. University of Milan, Milan, Italy

    Roberto Artuso  & Giulio Casati  & 

  2. Niels Bohr Institute, Copenhagen, Denmark

    Predrag Cvitanović

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© 1991 Plenum Press, New York

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Couder, Y. (1991). Growth Patterns: From Stable Curved Fronts to Fractal Structures. In: Artuso, R., Cvitanović, P., Casati, G. (eds) Chaos, Order, and Patterns. NATO ASI Series, vol 280. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-0172-2_8

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  • DOI: https://doi.org/10.1007/978-1-4757-0172-2_8

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