Elastic properties of a cellular dissipative structure

Drift and oscillations in a 1-D pattern
Article

Abstract.

Transition towards spatio-temporal chaos in one-dimensional interfacial patterns often involves two degrees of freedom: drift and out-of-phase oscillations of cells, respectively associated to parity breaking and vacillating-breathing secondary bifurcations. In this paper, the interaction between these two modes is investigated in the case of a single domain propagating along a circular array of liquid jets. As observed by Michalland and Rabaud for the printer’s instability [1], the velocity V g of a constant width domain is linked to the angular frequency \(\omega\) of oscillations and to the spacing between columns \(\lambda_0\) by the relationship \(V_g = \alpha \lambda_0 \omega\). We show by a simple geometrical argument that \(\alpha\) should be close to \(1/ \pi\) instead of the initial value \(\alpha = 1/2\) deduced from their analogy with phonons. This fact is in quantitative agreement with our data, with a slight deviation increasing with flow rate.

Keywords

Elastic Property Angular Frequency Velocity Versus Single Domain Quantitative Agreement 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Michalland, M. Rabaud, Physica D 61, 197 (1991)MATHGoogle Scholar
  2. 2.
    T. Bohr, M.H. Jensen, G. Paladin, A. Vulpiani, Dynamical systems approach to turbulence (Cambridge University Press, 1998)Google Scholar
  3. 3.
    P. Coullet, G. Iooss, Phys. Rev. Lett. 64, 866 (1990)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    F. Daviaud, M. Dubois, P. Bergé, Europhys. Lett. 9, 441 (1989)Google Scholar
  5. 5.
    J.-M. Flesselles, A.J. Simon, A.J. Libchaber, Adv. Phys. 40, 1 (1991)Google Scholar
  6. 6.
    G. Faivre, J. Mergy, Phys. Rev. A 45, 7320 (1992)CrossRefGoogle Scholar
  7. 7.
    J.T. Gleeson, P.L. Finn, P.E. Cladis, Phys. Rev. Lett. 66, 236 (1991)CrossRefGoogle Scholar
  8. 8.
    I. Mutabazi, C.D. Andereck, Phys. Rev. Lett. 70, 1429 (1993)CrossRefGoogle Scholar
  9. 9.
    R. Wiener, D.F. MacAlister, Phys. Rev. Lett. 69, 2915 (1992)CrossRefGoogle Scholar
  10. 10.
    L. Pan, J.R. de Bruyn, Phys. Rev. Lett. 70, 1791 (1993)CrossRefGoogle Scholar
  11. 11.
    H.Z. Cummins, L. Fourtune, M. Rabaud, Phys. Rev. E 47, 1727 (1993)CrossRefGoogle Scholar
  12. 12.
    C. Misbah, A. Valance, Phys. Rev. E 49, 166 (1994)CrossRefMathSciNetGoogle Scholar
  13. 13.
    C. Counillon, L. Daudet, T. Podgorski, L. Limat, Phys. Rev. Lett. 80, 2117 (1998)CrossRefGoogle Scholar
  14. 14.
    P. Brunet, J.-M. Flesselles, L. Limat, Europhys. Lett. 56, 221 (2001)Google Scholar
  15. 15.
    F. Giorgiutti, L. Limat, Physica D 103, 590 (1997)CrossRefGoogle Scholar
  16. 16.
    S. Fauve, S. Douady, O. Thual, J. Phys. II France 1, 311 (1991)CrossRefGoogle Scholar
  17. 17.
    B. Caroli, C. Caroli, S. Fauve, J. Phys. I France 2, 281 (1992)CrossRefGoogle Scholar
  18. 18.
    R.E. Goldstein, G.H. Gunaratne, L. Gil, P. Coullet, Phys. Rev. A 43, 6700 (1991)CrossRefGoogle Scholar
  19. 19.
    L. Gil, Europhys. Lett. 48, 156 (1999)Google Scholar
  20. 20.
    R. Alvarez, M. van Hecke, W. van Saarloos, Phys. Rev. E 56, R1306 (1997). See also P. Habdas, M.J. Case, J.R. de Bruyn, Phys. Rev. E 63, 066305 (2001)Google Scholar
  21. 21.
    B.I. Shraiman, Phys. Rev. Lett. 57, 325 (1986)CrossRefGoogle Scholar
  22. 22.
    P. Coullet, S. Fauve, Phys. Rev. Lett. 55, 2857 (1985)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin/Heidelberg 2003

Authors and Affiliations

  1. 1.Laboratoire PMMH-ESPCIParisFrance

Personalised recommendations