RSS feeds
РусскийEnglish
Issue 11, 2024
Volume year page
Issues Authors PACS Subscription For authors

Issues

 / 

1998

 / 

March

←  →

Reviews of topical problems


The Benard-Marangoni thermocapillary-instability problem


Universite des Sciences et Technologies de Lille1, U.F.R.de Mathematiques Pures et Appliquees, Department de Mecanique Fondamentale, Batiment M 3, Villeneuve d’Ascq Cedex, 59655, France

Physically, there are two main mechanisms responsible for driving the instability in the coupled buoyancy (Benard) and thermocapillary (Marangoni) convection problem for a weakly expansible viscous liquid layer bounded from below by a heated solid surface and on the top by a free surface subject to a temperature-dependent surface tension. The first mechanism is density variation generated by the thermal expansion of the liquid; the second results from the surface-tension gradients due to temperature fluctuations along the upper free-surface. In the present paper we consider only the second effect as in the Benard experiments [the so-called Benard-Marangoni (BM) problem]. Indeed, for a thin layer we show that it is not consistent to consider both effects simultaneously, and we formulate an alternative concerning the role of buoyancy. In fact, it is necessary to consider two fundamentally distinct problems: the classical shallow-convection problem for a non-deformable upper surface with partial account of the Marangoni effect (the RBM problem), and the full BM problem for a deformable free surface without the buoyancy effect. We shall be mostly concerned with the thermocapillary BM instabilities problem on a free-falling vertical film, since most experiments and theories have focused on this (in fact, wave dynamics on an inclined plane is quite analogous). For a thin film we consider three main situations in relation to the magnitude of the characteristic Reynolds number (Re) and we derive various model equations. These model equations are analyzed from various points of view but the central intent of this paper is to elucidate the role of the Marangoni number on the evolution of the free surface in space and time. Finally, some recent numerical results are presented.

Fulltext pdf (542 KB)
Fulltext is also available at DOI: 10.1070/PU1998v041n03ABEH000374
PACS: 44.25.+f, 44.30.+v, 47.10.+g, 47.27.+i (all)
DOI: 10.1070/PU1998v041n03ABEH000374
URL: https://ufn.ru/en/articles/1998/3/b/
000073306100002
Citation: Zeytounian R Kh "The Benard-Marangoni thermocapillary-instability problem" Phys. Usp. 41 241–267 (1998)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Зейтунян Р Х «Проблема термокапиллярной неустойчивости Бенара-Марангони» УФН 168 259–286 (1998); DOI: 10.3367/UFNr.0168.199803b.0259

References (85) ↓ Cited by (47) Similar articles (20)

  1. Levich V G, Krylov V S Ann. Rev. Fluid Mech. 1 293 (1969)
  2. Sarpkaya T Ann. Rev. Fluid Mech. 28 83 (1996)
  3. Bénard H Rev. Gen. Sci. Pures Appl. 11 1261 (1900)
  4. Davis S H, in Annual Review of Fluid Mechanics (Eds J L Lumley, M Van Dyke, H L Reed) Vol. 19 (Palo Alto, CA: Annual Reviews, 1987) p. 403
  5. Sterning C V, Scriven L E AIChE J 5 514 (1959)
  6. Levich V G Physicochemical Hydrodynamics (Englewood Cliffs, NJ: Prentice-Hall, 1962)
  7. Kenning D B R Appl. Mech. Rev. 21 1101 (1968)
  8. Norman C, Pomeau Y, Velarde M G Rev. Mod. Phys. 49 581 (1977)
  9. Velarde M G, Chu X-L, in Flow, Diffusion, and Rate Processes (Advances in Thermodynamics, Vol. 6, Eds S Sieniutycz, P Salamon, New York: Taylor and Francis, 1992) p. 110-145
  10. Probstein R F Physicochemical Hydrodynamics: An Introduction (Second Edition, New York: Wiley, 1994)
  11. Waves on Fluid Interfaces (Proc. of a Symposium, University of Wisconsin-Madison, 1982, Ed. Meyer R E, New York: Academic Press, 1983)
  12. Lin S P, Wanh C Y "Modelling Wavy Film Flows", in Encyclopedia of Fluid Mechanics (Ed. N P Cheremisinoff) Vol. 1 (Houston: Gulf Pub. Co., Book Division, 1986 ) p. 931-951
  13. Trifonov Yu Ya, Tsvelodub O Yu J. Fluid Mech. 229 531 (1991)
  14. Prokopiou Th, Cheng M, Chang H-Ch J. Fluid Mech. 222 665 (1991)
  15. Chang H-Ch Ann. Rev. Fluid. Mech. 26 103 (1994)
  16. Chang H-Ch, Demekhin E A Advances in Applied Mechanics 32 1 (1996)
  17. Tan M J, Bankoff S G, Davis S H Phys. Fluids A 2 313 (1990)
  18. Shtilman L, Sivashinsky G Physica D 52 477 (1991)
  19. Garazo A N, Velarde M G Phys. Fluids A 3 (10) 2295 (1991); 4 (5) 1089 (1992)
  20. Joo S W, Davis S H, Bankoff S G J. Fluid Mech. 230 117 (1991)
  21. Oron A, Rosenau Ph J. Phys. II (Paris) 2 131 (1992)
  22. Ryabitskiî E A Izv. Ros. Akad. Nauk, Mekh. Zhidk. Gaza 27 (3) 19 (1992)
  23. Villers D, Platten J K J. Fluid Mech. 234 487 (1992)
  24. O’Brien S B G M J. Fluid Mech. 254 649 (1993)
  25. Prakash A, Fujita D, Koster J N Eur. J. Mech. B/Fluids 12 (1) 15 (1993)
  26. Ezersky A B et al. Phys. Rev. A 42 1126 (1993)
  27. Ezersky A B et al. Phys. Rev. E 47 1126 (1993)
  28. Ezersky A B et al. Phys. Rev. E 48 4414 (1993)
  29. Parmentier P M, Regnier V C, Lebon G Int. J. Heat Mass Transfer 36 2417 (1993)
  30. Mundrane M, Zebib A Phys. Fluids 5 (4) 810 (1993)
  31. Oron A, Rosenau Ph J. Fluid Mech. 273 361 (1994)
  32. Golovin A A, Nepomnyashchy A A, Pismen L M Phys. Fluids 6 (1) 34 (1994)
  33. Nepomnyashchy A A, Velarde M G Phys. Fluids 6 (1) 187 (1994)
  34. Wilson S K J. of Engineering Math. 28 427 (1994)
  35. Thess. A, Orszag S A J. Fluid Mech. 283 201 (1995)
  36. Stebe K J, Barthes-Biesel D J. Fluid Mech. 286 25 (1995)
  37. Oron A, Deissler R J, Duh J C Eur. J. Mech. B/Fluids 14 (6) 737 (1995)
  38. Joo S W J. Fluid Mech. 293 127 (1995)
  39. Tanny J, Chen C C, Chen C F J. Fluid Mech. 303 1 (1995)
  40. Batitchev V A PMM59 (6) 1003 (1995)
  41. Zeytounian R Kh, in Asymptotic Modelling in Fluid Mechanics Vol. 442 (Heidelberg: Springer-Verlag, 1995) p. 153
  42. Ryabitskiî E A Izv. Ros. Akad. Nauk, Mekh. Zhidk. Gaza 31 (1) 1 (1996)
  43. Mercier J F, Normand C Phys. Fluids 8 (6) 1433 (1996)
  44. Savino R, Monti R Phys. Fluids 8 (11) 2906 (1996)
  45. Or A C, Kelly R E Bull. Am. Phys. Soc. 40 (12) 1949 (1996)
  46. Oron A, Bankoff S G, Davis S H Phys. Fluids 8 (12) 3433 (1996)
  47. Zeytounian R Kh Int. J. of Engineering Sci. 27 (11) 1361 (1989)
  48. Zeytounian R Kh Int. J. of Engineering Sci. (in press, 1997)
  49. Dauby P C, Lebon G J. Fluid. Mech. 329 25 (1996)
  50. Vince J-M On des propagatives dans des systemes convectives soumis a des effets de tension superficielle These de Doctorat de l’Universite de Paris (1994)
  51. Shkadov V Ya Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza (2) 20 (1968)
  52. Demekhin Ye A, Tokarev G Yu, Shkadov V Ya Physica D 52 338 (1991)
  53. Joseph D D, Renardy Y Y Fundamental of Two Fluids Dynamics Part I Mathematical Theory and Applications (New York: Springer-Verlag, 1993)
  54. Pavithran S, Redeekopp L G Studies in Appl. Math. 93 209 (1994)
  55. Drazin P G, Reid W H Hydrodynamic Stability (Cambridge; New York: Cambridge University Press, 1981)
  56. Zeytounian R Kh The Bénard-Marangoni Thermocapillary Instability Problem: Vertical Falling Film for High and Low Reynolds Numbers (unpublished manuscript)
  57. Alekseenko S K et al. AIChE J. 31 1446 (1985)
  58. Landau L D Dokl. Acad. Sci. USSR 44 311 (1944); also Collected Papers of L D Landau (Oxford: Perganom, 1965) p. 387-391
  59. Stuart J T J. Fluid Mech. 9 353 (1960)
  60. Newell A C Lectures in Appl. Math. 15 157 (1974)
  61. Stewartson K, Stuart J T J. Fluid Mech. 48 529 (1971)
  62. Lorenz E N Atmospheric Sci. 20 130 (1963)
  63. Zghal M Waves on the Surface of a Falling Liquid Film: Marangoni Effect Doctoral Thesis, L.M.L University of Lille I (unpublished manuscript)
  64. Cheng M, Chang H-Ch Phys. Fluids 7 (1) 34 (1995)
  65. Liu J, Gollub J P Phys. Rev. Lett. 70 2289 (1993)
  66. Benney D J J. Math. Phys. 45 150 (1996)
  67. Whitham G B Linear and Nonlinear Waves (New York: Wiley, 1974)
  68. Hopf E Comm. Pure and Appl. Math. 3 201 (1950)
  69. Hooper A P, Grimshaw R Phys. Fluids 28 37 (1985)
  70. Chang H-Ch, Demekhin E A, Kopelevich D I J. Fluid Mech. 250 433 (1993)
  71. Chang H-Ch, Demekhin E A, Kopelevich D I Physica D 63 299 (1993)
  72. Hyman J M, Nicolaenko B, Zaleski S Physica D 23 265 (1986)
  73. Zeytounian R Kh Arch. Mech. 41 (2/3) 383 (Warszawa, 1989)
  74. Tatsumi T, in Turbulence and Chaotic Phenomena in Fluids (Proc. of the International Symposium, Kyoto, Japan, 1983, Ed. T Tatsumi, Amsterdam: North Holland, 1984)
  75. Kawahara T, Toh S Phys. Fluids 31 2103 (1988)
  76. Christov C I, Velarde M G Physica D 86 323 (1995)
  77. Zeytounian R Kh Phys. Usp. 38 1333 (1995)
  78. Kudryashev N A Prikl. Math. Mekh. 54 (3) 450 (1990)
  79. Rednikov A Ye et al. Acta Applicandae Mathematicae 39 457 (1995)
  80. Liu J, Gollub J P Phys. Fluids 6 (5) 1702 (1994)
  81. Balmforth N J, Ierley G R, Spiegel E A SIAM J. Appl. Math. 54 (5) 1291 (1994)
  82. Balmforth N J Ann. Rev. Fluid. Mech. 27 335 (1995)
  83. Barthelet P, Charry F, Fabre J J. Fluid Mech. 303 23 (1995)
  84. Cheng M, Chang H-Ch Phys. Fluids A 2 (8) 1364 (1990)
  85. Kazhdan D et al. Phys. Fluids 7 (11) 2679 (1995)
© 1918–2024 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions