Advertisement

Pattern Selection in Convective Instabilities with Axial Anisotropy

  • L. Kramer
  • W. Zimmermann
  • E. Bodenschatz
  • W. Pesch
Part of the Springer Series in Synergetics book series (SSSYN, volume 41)

Abstract

We examine the possible stable patterns slightly above threshold of quasi-two-dimensional pattern-forming systems with an axial anisotropy. Special attention is given to the electrohydrodynamic instability in nematic liquid crystals, where extensive calculations have been done, and to systems where the anisotropy can be turned on continuously.

Keywords

Nematic Liquid Crystal Trial Function Free Boundary Condition Amplitude Equation Normal Roll 
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.
    F. H. Busse: Rep. Prog. Phys. 41, 28 (1978); and in Hydrodynamic Instabilites and the Transition to Turbulence, 2nd ed., H. L. Swinney and J. P. Gollub, eds. (Springer, Berlin 1986).CrossRefGoogle Scholar
  2. 2.
    M. C. Cross and A. C. Newell: Physica 10D, 299 (1984).ADSMathSciNetGoogle Scholar
  3. 3.
    M. S. Heutmaker, P. N. Fraenkel and J. P. Gollub: Phys. Rev. Lett. 54, 1369 (1985)CrossRefADSGoogle Scholar
  4. 3.
    G. Ahlers, D. S. Cannell and V. Steinberg: Phys. Rev. Lett. 54, 1373 (1985)CrossRefADSGoogle Scholar
  5. 3.
    P. Le Gal, A. Pocheau and V. Croquette: Phys. Rev. Lett. 54, 2501 (1985) and 55, 1094 (1985).CrossRefADSGoogle Scholar
  6. 4.
    See e.g. W. J. A. Goossens: Advances in Liquid Crystals, edited by G. H. Brown (Academic, New York 1978), Vol.3, p.1.Google Scholar
  7. 5.
    A. Joets and R. Ribotta: in Cellular Structures in Instabilities, J. E. Wesfreid and S. Zaleski, eds. (Springer, Berlin 1984) p.294; J. Physique 47, 595 (1986)CrossRefGoogle Scholar
  8. 5.
    R. Ribotta, A. Joets and Lin Lei: Phys. Rev. Lett. 56, 1595 (1986).CrossRefADSGoogle Scholar
  9. 6.
    M. Lowe and J. P. Gollub: Phys. Rev. Lett. 55, 2575 (1985)CrossRefADSGoogle Scholar
  10. 6.
    Y. Tsuchiya and S. Horie: J. Phys. Soc. Japan 55, 2945 (1986)CrossRefADSGoogle Scholar
  11. 6.
    H. Yamazaki, S. Kai and K. Hirakawa: J. Phys. Soc. Jap. 56, 1 (1987).CrossRefADSGoogle Scholar
  12. 7.
    W. Zimmermann and L. Kramer: Phys. Rev. Lett. 55, 402 (1985).CrossRefADSGoogle Scholar
  13. 8.
    E. Bodenschatz, W. Zimmermann and L. Kramer: preprint (1988).Google Scholar
  14. 9.
    E. Dubois-Violette: C. R. hebd. seanc. Acad. Sci. 21, 275, 923 (1971).Google Scholar
  15. 10.
    E. Dubois-Violette: Solid State Comm. 14, 767 (1974).CrossRefADSGoogle Scholar
  16. 11.
    E. Dubois-Violette, E. Guyon and P. Pieranski: Mol. Cryst. Liq. Cryst. 26, 193 (1976).CrossRefGoogle Scholar
  17. 12.
    A. Joets, L. Kramer, R. Ribotta, J. Salan and W. Zimmermann: to be published.Google Scholar
  18. 13.
    W. Pesch and L. Kramer: Z. Phys. B63, 121 (1986).CrossRefADSGoogle Scholar
  19. 14.
    L. Kramer, E. Bodenschatz, W. Pesch and W. Zimmermann: In The Physics of Structure Formation, ed. by W. Güttinger and G. Dangelmayr, Springer Ser. Syn., Vol. 37 (Springer, Berlin, Heidelberg 1987).Google Scholar
  20. 15.
    S. Chandrasekhar: Hydrodynamic and Hydromagnetic Stability (Clarendon Press, Oxford, 1961) p.186.zbMATHGoogle Scholar
  21. 16.
    B. Lehnert and N. C. Little: Tellus 9, 97 (1957).CrossRefADSGoogle Scholar
  22. 16.
    For recent experiments on the influence of a horizontal magnetic field on Rayleigh-Bénard convection see S. Fauve, C. Laroche and A. Libschaber: J. Physique Lett. 42, L-455 (1981); 45, L-101 (1984).CrossRefGoogle Scholar
  23. 17.
    J. Salan and E. Guyon: J. Fluid Mech. 126, 13 (1983).CrossRefADSGoogle Scholar
  24. 18.
    P. Pieranski and E. Guyon: Phys. Rev. Lett. 39, 1281 (1977)CrossRefADSGoogle Scholar
  25. 18.
    E. Guazelli and E. Guyon: J. Physique 43, 985 (1982)CrossRefGoogle Scholar
  26. 18.
    E. Guazelli and A.-J. Koch: J. Physique 46, 673 (1985).CrossRefGoogle Scholar
  27. 19.
    E. Dubois-Violette and F. Rothen: J. Physique 10, 1039 (1978)Google Scholar
  28. 19.
    J. Sadik, F. Rothen, W. Bestgen and E. Dubois-Violette: J. Physique 42, 915 (1981)CrossRefGoogle Scholar
  29. 19.
    A.-J. Koch, F. Rothen, J. Sadik and J. Prost: Phil. Mag. A48, 727 (1983).Google Scholar
  30. 20.
    E. Guazelli, G. Dewel, P. Borckmann and W. Walgraef: to be published.Google Scholar
  31. 21.
    P. A. Penz and G. W. Ford: Phys. Rev. A6, 414 (1972).ADSGoogle Scholar
  32. 22.
    Experimental values of the material constants for MBBA at 25°C: Elastic constants \(\text k^{11} - 6.66 \times 10^{-12}\text{N}\), \(\text k^{22} - 4.2 \times 10^{-12}\text{N}\) and \(\text k^{33} - 8.61 \times 10^{-12}\text{N}\) are taken from H. W. de Jeu et al.: Mol. Cryst. & Liq. Cryst. 37, 269 (1976). Viscosities \(\alpha_1 = - 18.1 \times 10^{-3}\text{kg m}^{-1}\,\text{s}^{-1}\), \(\alpha_2 = - 110.4 \times 10^{-3}\text{kg m}^{-1}\,\text{s}^{-1}\), \(\alpha_3 = - 1.1 \times 10^{-3}\text{kg m}^{-1}\,\text{s}^{-1}\), \(\alpha_4 = 82.6 \times 10^{-3}\text{kg m}^{-1}\,\text{s}^{-1}\), \(\alpha_5 = 77.9 \times 10^{-3}\text{kg m}^{-1}\,\text{s}^{-1}\), \(\alpha_6 = - 33.6 \times 10^{-3}\text{kg m}^{-1}\,\text{s}^{-1}\) are taken from H. Kneppe et al.: J. Chem. Phys. 77, 3203 (1982). We have chosen \(3\bar\in = (2\in_{\,_\bot} + \in_{\,_{\text{,,}}}) = 15.22\), \(\in_a = - 0.53\), \(\sigma_{\,_\bot} = 10^{-8} \Omega^{-1} \text{ m}^{-1}\) and \(\sigma_{||}/\sigma_{\,_\bot} = 1.5\) unless stated otherwise.Google Scholar
  33. 23.
    A. Joets and R. Ribotta: this volume.Google Scholar
  34. 24.
    N. V. Madhusudana, V. A. Raghunathan and K. R. Sumathy, Pramäna J. Phys. 28, L 311 (1987).CrossRefADSGoogle Scholar
  35. 25.
    W. Thom, W. Zimmermann and L. Kramer: preprint (1988).Google Scholar
  36. 26.
    Y. P. Bobylev and S. A. Pikin: Zh. Eksp. Theor. Fiz. 72, 369 (1977) [English transl: Sov. Phys.-JETP 45, 195 (1977)]Google Scholar
  37. 26.
    Y. P. Bobylev, V. G. Chigrinov and S. A. Pikin: J. Physique Coll. 40, C3–331 (1979)Google Scholar
  38. 26.
    M. I. Barnik, L. M. Blinov, A. N. Trufanov and B. A. Umanskii: Zh. Eksp. Theor. Fiz. 73, 1936 (1977) [English transl.: Sov. Phys.-JETP 46, 1016 (1977)]; J. Physique 39, 417 (1978).Google Scholar
  39. 27.
    F. Lonberg and R. B. Meyer: Phys. Rev. Lett. 55, 718 (1985)CrossRefADSGoogle Scholar
  40. 27.
    E. Miraldi, C. Oldano and A. Strigazzl: Phys. Rev. A34, 4348 (1986)ADSGoogle Scholar
  41. 27.
    W. Zimmermann and L. Kramer: Phys. Rev. Lett. 56, 2655 (1986)CrossRefADSGoogle Scholar
  42. 27.
    U. D. Kini: J. Physique 47, 693 (1986); 47, 1829 (1986).CrossRefGoogle Scholar
  43. 28.
    R. M. Hornreich, M. Luban and S. Shtrikam: Phys. Rev. Lett. 35, 1678 (1975); Physica 86A, 465 (1977).CrossRefADSGoogle Scholar
  44. 29.
    See e.g. V. Steinberg, this conference. E. Ben-Jacob, H. Brand, G. Dee, L. Kramer and J. S. Langer: Physica 14D, 348 (1985).ADSMathSciNetGoogle Scholar
  45. 30.
    E. D. Siggia and A. Zippelius: Phys. Rev. A24, 1036 (1981)ADSGoogle Scholar
  46. 30.
    Y. Pomeau, S. Zaleski and P. Manneville: Phys. Rev. A27, 2710 (1983).ADSMathSciNetGoogle Scholar
  47. 30.
    G. Tesauro and M. C. Cross: Phys. Rev. A34, 1363 (1986).ADSGoogle Scholar
  48. 31.
    E. Bodenschatz, W. Pesch and L. Kramer: preprint (1988).Google Scholar
  49. 32.
    P. Coullet: this volume; P. Coullet, C. Elphick and D. Repaux: Phys. Rev. Lett. 58, 431 (1987).CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • L. Kramer
    • 1
  • W. Zimmermann
    • 1
  • E. Bodenschatz
    • 1
  • W. Pesch
    • 1
  1. 1.Physikalisches InstitutUniversität BayreuthBayreuthFed. Rep. of Germany

Personalised recommendations