The Printer’s Instability: The Dynamical Regimes of Directional Viscous Fingering

  • Y. Couder
  • S. Michalland
  • M. Rabaud
  • H. Thomé
Part of the NATO ASI Series book series (NSSB, volume 225)


Important experimental and theoretical efforts have been devoted recently to the dynamical behaviours of extended systems (reviews can be found in References 1 and 2). In this spirit, experiments were done in the Rayleigh-Bénard instability aimed at the study of the onset of chaos in periodic structures with a large number of cells3,4. On the other hand electroconvection of liquid crystals5,6, convection in binary fluid mixtures7,8 and the Faraday experiment9,10 show a rich dynamics due to the presence of two independent control parameters. In particular, propagative modes are observed, sometimes confined to limited domains. Propagation also exists in Taylor-Couette flows where it creates spiral structures11–13. For the sake of simplicity and comparison to theoretical models, one-dimensionality has been sought in extended systems. This is obtained by confinement in the above-mentioned convection experiments3,4 Another way to reach one-dimensionality is to study a boundary. This boundary can be the limit between two convective rolls affected by the oscillatory instability14 or an interface between two-dimensionally confined media. In the latter case, if a front instability generates a large number of cells, their dynamics is one dimensional to a good approximation. Such situations are obtained in directional phase transition of liquid crystal15, eutectic crystallisation16 and directional viscous fingering l7a,b. We presented previously17a results on the instability onset of the front of a viscous fluid in a widening gap. The present article reports the development of our work on the nonlinear regimes of this front instability.


Solitary Wave Directional Solidification Chaotic State Spatiotemporal Chaos Front Instability 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Propagation in Systems Far From Equilibrium, J.E. Wesfreid, H.R. Brand, P. Maneville, G. Albinet and N. Boccara editors (Springer, Berlin, 1984). Reviews on the dynamics of extended systems: A. C. Newell, p. 122, H. R. Brand, p. 206, P. Maneville, p. 265.Google Scholar
  2. 2.
    New trends in Nonlinear Dynamics and Pattern Forming Phenomena: The Geometry of Nonequilibrium, P. Coullet and P. Huerre editors, (Plenum Press, New York). To appear in 1990.MATHGoogle Scholar
  3. 3.
    S. Ciliberto and P. Bigazzi, Phys. Rev. Lett. 60, 286 (1988).ADSCrossRefGoogle Scholar
  4. 4.
    F. Daviaud, M. Dubois and P. Berge. Europhys. Lett. 9, 441 (1989).ADSCrossRefGoogle Scholar
  5. 5.
    A. Joëts and R. Ribotta, Phys. Rev. Lett. 60, 2164 (1988).ADSCrossRefGoogle Scholar
  6. 6.
    I. Rehberg, S. Rasenat, and V. Steinberg, Phys. Rev. Lett. 62, 756 (1989).ADSCrossRefGoogle Scholar
  7. 7.
    P. Kolodner and C. Surko, Phys. Rev. Lett. 61, 842 (1988).ADSCrossRefGoogle Scholar
  8. 8.
    E. Moses, J. Fineberg and V. Steinberg. Phys. Rev. Lett. 61, 838 (1988).ADSCrossRefGoogle Scholar
  9. 9.
    S. Douadi, S. Fauve and O. Thual, Europhys. Lett. 10, 309 (1989).ADSCrossRefGoogle Scholar
  10. 10.
    N.B. Turafillaro, N.B. Ramshankar and J.F. Gollub. Phys. Rev. Lett. 62, 422 (1989).ADSCrossRefGoogle Scholar
  11. 11.
    C. D. Andereck, S.S. Liu and H.L. Swinnev. J. Fluid Mech. 164, 155 (1986).ADSCrossRefGoogle Scholar
  12. 12.
    I. Mutabazi, J.J. Hegseth, C. D. Andereck and J. E. Weisfreid, Phys. Rev. A. 38, 4752 (1988).ADSCrossRefGoogle Scholar
  13. 13.
    Li Ning, G. Alhers and D.S. Cannell. Phys. Rev. Lett. 64, 1235 (1990).ADSCrossRefGoogle Scholar
  14. 14.
    B. Janiaud, E. Guyon, V. Croquette and D. Bensimon, in this volume.Google Scholar
  15. 15.
    A. Simon, J. Bechhoefer and A. Libchaber. Phys. Rev. Lett. 61, 2574 (1988)ADSCrossRefGoogle Scholar
  16. 15a.
    J. Bechhoefer, A. Simon, A. Libchaber and P. Oswald. Phys. Rev. A 40, 2042 (1989).ADSCrossRefGoogle Scholar
  17. 16.
    G. Faivre, S. de Cheveigné, C. Guthmann and P. Kurowski. Europhys. Lett. 9, 779 (1989).ADSCrossRefGoogle Scholar
  18. 17. (a)
    V. Hakim, M. Rabaud, H. Thomé and Y. Couder, in New trends in Nonlinear Dynamics and Pattern Forming Phenomena: The Geometry of Nonequilibrium, P. Coullet and P. Huerre editors, (Plenum Press, New York).Google Scholar
  19. 17. (b)
    M. Rabaud, S. Michalland and Y. Couder. Phys. Rev. Lett. 64, 184 (1990).ADSCrossRefGoogle Scholar
  20. 18.
    Y. Kuramoto and T. Tsuzuki, Prog. Theor. Phys. 55, 356 (1976),ADSCrossRefGoogle Scholar
  21. 18a.
    G.I. Sivashinsky, Acta Astronaut. 4, 1177 (1977).MathSciNetMATHCrossRefGoogle Scholar
  22. 19.
    A.N. Kolmogorov, in Seminar Notes edited by V.I. Arnold and L.D. Meshalkin, Uspekhi Mat. Naut. 15, 247 (1960).Google Scholar
  23. 20.
    H. Chaté and P. Maneville, Phys. Rev. Lett. 58, 112 (1987)ADSCrossRefGoogle Scholar
  24. 20a.
    H. Chaté and B. Nicolaenko, to appear in New trends in Nonlinear Dynamics and Pattern Forming Phenomena: The Geometry of Nonequilibrium, P. Coullet and P. Huerre editors, (Plenum Press, New York). 1987Google Scholar
  25. 21.
    G. I. Sivashinsky, Physica D 8. 243 (1983)MathSciNetADSCrossRefGoogle Scholar
  26. 21a.
    A. Novick-Cohen, Physica D. 23, 118 (1986)MathSciNetADSMATHCrossRefGoogle Scholar
  27. 21b.
    A. Novick-Cohen. Physica D 26. 403 (1987).ADSMATHCrossRefGoogle Scholar
  28. 22.
    P. Coullet, R. E. Goldstein and G.H. Gunaratne, Phys. Rev. Lett. 63, 1954 (1989).ADSCrossRefGoogle Scholar
  29. 23.
    S. Douady, S. Fauve and O. Thual. Europhys. Lett. 10, 309 (1989).ADSCrossRefGoogle Scholar
  30. 24.
    W. Van Saarloos and P.C. Hohenberg. Phys. Rev. Lett. 64, 749 (1990).ADSCrossRefGoogle Scholar
  31. 25.
    J. R. A. Pearson. J. Fluid Mech. 7, 481 (1960).MathSciNetADSMATHCrossRefGoogle Scholar
  32. 26.
    M. D. Savage, J. Fluid Mech. 80, 743 (1977);ADSMATHCrossRefGoogle Scholar
  33. 26a.
    M. D. Savage, J. Fluid Mech. 80. 757 (1977)ADSMATHCrossRefGoogle Scholar
  34. 26b.
    M. D. Savage, J. Fluid Mech. 117, 443 (1982).ADSMATHCrossRefGoogle Scholar
  35. 27.
    E. Pitts and J. Greiller, J. Fluid Mech. 11, 33 (1961).ADSMATHCrossRefGoogle Scholar
  36. 28.
    C. C. Mill, G. R. South. J. Fluid Mech. 28, 523 (1967).ADSCrossRefGoogle Scholar
  37. 29.
    J. Greener, T. Sullivan, B. Turner, S. Middleman, Chem. Eng. Commun. 5, 73 (1980).CrossRefGoogle Scholar
  38. 30.
    G. I. Taylor. J. Fluid Mech. 16, 595 (1963).ADSMATHCrossRefGoogle Scholar
  39. 31.
    A. D. McEwan and G. I. Taylor, J. Fluid Mech. 26, 1 (1966).ADSCrossRefGoogle Scholar
  40. 32.
    R. Trivedi and K. Somboonsuk. Acta Metall. 33, 1061 (1985).CrossRefGoogle Scholar
  41. 33.
    S. de Cheveigné, C. Guthmann and M. M. Lebrun, J. Physique. 47, 2095 (1986).CrossRefGoogle Scholar
  42. 34.
    These profiles were obtained numerically by M. Maashai (private communication).Google Scholar
  43. 35.
    T. Dombre and V. Hakim, Phys. Rev. A. 36, 2811 (1987).ADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • Y. Couder
    • 1
  • S. Michalland
    • 1
  • M. Rabaud
    • 1
  • H. Thomé
    • 1
  1. 1.Laboratoire de Physique StatistiqueEcole Normale SupérieureParis Cedex 05France

Personalised recommendations