Advertisement

Surface Tension Gradient-Driven Wave Motion in Shallow Liquid Layers

  • M. G. Velarde
  • H. Linde
  • A. Ye. Rednikov
  • Yu. S. Ryazantsev
  • A. A. Nepomnyashchy
  • V. N. Kurdyumov
Part of the Fluid Mechanics and Its Applications book series (FMIA, volume 34)

Abstract

During the last decade, the nonstationary regimes of Rayleigh-Bénard, buoyancy driven convection in single and binary fluid layers heated from below or above were subject of numerous investigations. The experiments performed first in rectangular geometries showed the existence of different kinds of motions, including counterpropagating waves,’blinking’ waves, etc. The situation was clarified when experiments were performed in narrow rectagular cells, and especially in annular cylindrical geometries. The properties observed in nonstationary patterns agreed qualitatively with theoretical predictions based on the complex Ginzburgh-Landau equation or its extension including the mass concentration field. Essential to the phenomena observed was the maintenance of a suitable level of external constraint which in appropriate balance with dissipation, due to viscosity and heat or mass diffusion, sustains the stationary or nonstationary structure

Keywords

Solitary Wave Wave Train Solitary Wave Solution Wave Crest Annular Channel 
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.

A few references in historical perspective

  1. Russell, J.S. (1845) On Waves. In Report of 14th Meeting (1944) of the British Association for the Advancement of Science, 311–390, York.Google Scholar
  2. Boussinesq, J.V. (1872) Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl., 17, 55 – 108Google Scholar
  3. Lord Rayleigh (1876) On waves. Phil. Mag., 1, 257 – 279Google Scholar
  4. Korteweg, D.J. & De Vries, G. (1895) On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil. Mag., ser. 5, 39, 422 – 443CrossRefGoogle Scholar
  5. Zabusky, N.J. & Kruskal, M.D. (1965) Interaction of’solitons’ in collisionless plasma and the recurrence of initial states. Phys. Rev. Lett., 15, 57 – 62CrossRefGoogle Scholar
  6. Zabusky, N.J. (1969) Nonlinear lattice dynamics and energy sharing. J. Phys. Soc. Japan, 26, 196 – 202Google Scholar
  7. Gjevik, B. (1970) Occurrence of finite-amplitude surface waves on falling liquid films, Phys. Fluids, 13, 1918 – 1925zbMATHCrossRefGoogle Scholar
  8. Johnson, R.S. (1972) Shallow water waves on a viscous fluid. The undular bore. Phys. Fluids, 15, 1639 – 1699CrossRefGoogle Scholar
  9. Whithams G.B. (1974) Linear and nonlinear waves. J. Wiley, NY.Google Scholar
  10. Homsy, G.M. (1974) Model equations for wavy viscous film flow, Lect. in Appl. Math, 15, 191 – 194MathSciNetGoogle Scholar
  11. Nepomnyashchy, A.A. (1976) Wave motions in a layer of viscous flowing down an inclined plane. Trans. Perm Univ. 362, pp. 114–119. (In Russian)Google Scholar
  12. Sivashinsky, G.I. & Michelson, D.M. (1980) On irregular wavy flow of a liquid film down a vertical plane, Progr. Theor. Phys., 63, 2112 – 2114Google Scholar
  13. Benjamin, T.B. (1982) The solitary wave with surface tension, Quart. Appl. Math., 40, 231 – 234MathSciNetzbMATHGoogle Scholar
  14. Kawahara, T. & Toh, S. (1985) Nonlinear dispersive periodic waves in the presence of instability and damping. Phys. Fluids, 28, 1636 – 1638CrossRefGoogle Scholar
  15. Kawahara, T. & Toh, S. (1988) Pulse interactions in an unstable dissipative-dispersive nonlinear system. Phys. Fluids, 31, 2103 – 2111MathSciNetCrossRefGoogle Scholar
  16. Segur, H. (1991) Who cares about integrability? Physica, D, 51, 343 – 359MathSciNetzbMATHCrossRefGoogle Scholar
  17. Chu, X.-L. & Velarde, M.G. (1991) Korteweg-de Vries solution excitation in Bénard-Marangoni convection, Phys. Rev., A43, 1094 – 1096Google Scholar
  18. Velarde, M.G., Chu, X.-L. & Garazo, A.N. (1991) Onset of possible solutions in surface tension-driven convection, Phys. Scripta, T35, 71 - 74CrossRefGoogle Scholar
  19. Elphick, C, Ierley, G.R., Regev, O. & Spiegel, E.A. (1991) Interacting localized structures with Galilean invariance, Phys. Rev., A 44, 1110 – 1122CrossRefGoogle Scholar
  20. Garazo, A.N. & Velarde, M.G. (1991) Dissipative Korteweg-de Vries description of Marangoni-Bénard oscillatory convection. Phys. Fluids, A 3, 2295 – 2300.zbMATHCrossRefGoogle Scholar
  21. Janiaud, B., Pumir, A., Bensimon, D., Croquette, V., Richter, H. & Kramer L. (1992) The Eckhaus instability for travelling waves, Physica, D 55, 269 – 289MathSciNetzbMATHCrossRefGoogle Scholar
  22. Weidman, P.D., Linde, H. & Velarde, M.G. (1992) Evidence for solitary wave behavior in Marangoni-Bénard convection. Phys. Fluids, A 4, 921 – 926CrossRefGoogle Scholar
  23. Chang, H.-C, Demekhin, A. & Kopelevich, D.I. (1993) Laminarizing effect of dispersion in an active.dissipative nonlinear medium, Physica, D 63, 299 – 320zbMATHCrossRefGoogle Scholar
  24. Linde, H., Chu, X.-L. & Velarde, M.G. (1993a) Oblique and head-on collisions of solitary waves in Marangoni-Bénard convection. Phys. Fluids A 5, 1068 – 1070CrossRefGoogle Scholar
  25. Linde, H., Chu, X.-L., Velarde, M.G. & Waldhelm, W. (1993b) Wall reflections of solitary waves in Marangoni-Bénard convection. Phys. Fluids, A 5, 3162 – 3166CrossRefGoogle Scholar
  26. Christov, C.I. & Velarde, M.G. (1994) Inelastic interaction of Boussinesq soli-tons, Int. J. Bif. Chaos 4, 1095 – 1112zbMATHCrossRefGoogle Scholar
  27. Nekorkin, V.l. & Velarde, M.G. (1994) Solitary waves and soliton bound states of a dissipative Korteweg-de Vries equation describing Marangoni-Bénard convection and other thermoconvective instabilities. Int. J. Bif. Chaos, 4, 1135 – 1146MathSciNetzbMATHCrossRefGoogle Scholar
  28. Nepomnyashchy, A.A. & Velarde, M.G. (1994) A Three-dimensional description of solitary waves and their interaction in Marangoni-Bénard layers. Phys. Fluids, 6, 187 – 198MathSciNetzbMATHCrossRefGoogle Scholar
  29. Christov, C.I. & Velarde, M.G. (1995) Dissipative solitons. Physica, D (to appear)Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • M. G. Velarde
    • 1
  • H. Linde
    • 1
    • 2
  • A. Ye. Rednikov
    • 1
  • Yu. S. Ryazantsev
    • 1
    • 2
  • A. A. Nepomnyashchy
    • 4
  • V. N. Kurdyumov
    • 1
    • 3
  1. 1.Instituto PluridisciplinarUniversidad ComplutenseMadridSpain
  2. 2.Institut für Organische und Bioorganische chemie, Fachbereich ChemieHumboldt UniversitätBerlinGermany
  3. 3.Department of MathematicsTechnionHaifaIsrael
  4. 4.E.T.S.I.AeronáuticosUniversidad Politécnica de MadridMadridSpain

Personalised recommendations