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Numerical analysis of the Eckhaus instability in travelling-wave convection in binary mixtures

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Abstract.

The Eckhaus stability boundaries of travelling periodic roll patterns arising in binary fluid convection is analysed using high-resolution numerical methods. We present results corresponding to three different values of the separation ratio used in experiments. Our results show that the subcritical branches of travelling waves bifurcating at the onset of convection suffer sideband instabilities that are restabilised further away in the branch. If this restabilisation is produced after the turning point of the travelling-wave branch, these waves do not become stable in a saddle node bifurcation as would have been the case in a smaller domain. In the regions of instability of the uniform travelling waves we expect to find either transitions between states of different wave number or modulated travelling waves arising in these bifurcations.

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References

  1. M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65, 998 (1993).

    Article  Google Scholar 

  2. M. Lücke, W. Barten, P. Büchel, C. Fütterer, St. Hollinger, Ch. Jung, Evolution of Spontaneous Structures in Dissipative Continuous Systems, edited by F.H. Busse, S.C. Müller (Springer-Verlag, Berlin, Heidelberg, 1998) pp. 127-196.

  3. D.R. Ohlsen, S.Y. Yamamoto, C.M. Surko, P. Kolodner, Phys. Rev. Lett. 65, 1431 (1990).

    Article  Google Scholar 

  4. P. Kolodner, J.A. Glazier, H. Williams Phys. Rev. Lett. 65, 1579 (1990).

    Article  Google Scholar 

  5. W. Barten, M. Lücke, M. Kamps, R. Schmitz, Phys. Rev. E 51, 5636 (1995).

    Article  Google Scholar 

  6. A.C. Or, F.H. Busse, J. Fluid Mech. 174, 313 (1987).

    MATH  Google Scholar 

  7. A.C. Or, F.H. Busse, J. Fluid Mech. 245, 155 (1992).

    Google Scholar 

  8. D. Pino, M. Net, J. Sánchez, I. Mercader, Phys. Rev. E 63, 056312 1 (2001).

    Article  Google Scholar 

  9. B. Janiaud, A. Pumir, D. Bensimon, V. Croquette, H. Richter, L. Kramer, Physica D 55, 269 (1992).

    MathSciNet  MATH  Google Scholar 

  10. A. Spina, J. Toomre, E. Knobloch, Phys. Rev. E 57, 524 (1998).

    Article  MathSciNet  Google Scholar 

  11. P. Kolodner, Phys. Rev. A 46, 1739 (1992).

    Article  Google Scholar 

  12. G.W. Baxter, K.D. Eaton, C.M. Surko, Phys. Rev. A 46, 1735 (1992).

    Article  Google Scholar 

  13. B. Huke, M. Lücke, P. Büchel, Ch. Jung, J. Fluid Mech. 408, 121 (2000).

    Article  MATH  Google Scholar 

  14. P. Büchel, M. Lücke, Entropie 218, 22 (1999).

    Google Scholar 

  15. J.K. Platten, J.C. Legros, Convection in Liquids (Springer-Verlag, Berlin, Heidelberg, 1984).

  16. J.D. Crawford, E. Knobloch, Annu. Rev. Fluid Mech. 23, 341 (1991).

    MATH  Google Scholar 

  17. J. Prat, J.M. Massaguer, I. Mercader, Phys. Fluids 7, 121 (1995).

    Article  MATH  Google Scholar 

  18. C.K. Mamun, L. Tuckerman, Phys. Fluids 7, 80 (1995).

    Article  MATH  Google Scholar 

  19. A. Bergeon, D. Henry, H. BenHadid, L. Tuckerman, J. Fluid Mech. 375, 143 (1998).

    Article  MATH  Google Scholar 

  20. V. Frayssé, L. Giraud, S. Gratton, J. Langou, Technical Report TR/PA/03/3, CERFACS (2003). Public domain software available at www.cerfacs/algor/Softs.

  21. A. Drissi, M. Net, I. Mercader, Phys. Rev. E 60, 1781 (1999).

    Article  Google Scholar 

  22. J. Prat, I. Mercader, E. Knobloch, Phys. Rev. E 58, 3145 (1998).

    Article  Google Scholar 

  23. S. Hugues, A. Randriamampianina, Int. J. Numer. Methods Fluids, 28, 501 (1998).

  24. O. Batiste, M. Net, I. Mercader, E. Knobloch, Phys. Rev. Lett. 86, 2309 (2001).

    Article  Google Scholar 

  25. A. Alonso, O. Batiste Theoret. Comput. Fluid Dyn. 18, 239 (2004).

    Article  Google Scholar 

  26. G. Dangelmayr, J. Hettel, E. Knobloch, Nonlinearity 10, 1093 (1997).

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Departament de Fısica Aplicada, Universitat Politècnica de Catalunya, Campus Nord, Mòdul B4, 08034, Barcelona, Spain

    I. Mercader, A. Alonso & O. Batiste

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  1. I. Mercader
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  2. A. Alonso
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  3. O. Batiste
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Correspondence to I. Mercader.

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Mercader, I., Alonso, A. & Batiste, O. Numerical analysis of the Eckhaus instability in travelling-wave convection in binary mixtures. Eur. Phys. J. E 15, 311–318 (2004). https://doi.org/10.1140/epje/i2004-10071-7

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  • DOI: https://doi.org/10.1140/epje/i2004-10071-7

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