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A nonlinear stability analysis of the Bénard-Marangoni problem. (English) Zbl 0571.76036
The authors give a nonlinear analysis of Bénard-Marangoni convection in a horizontal fluid layer of infinite extent. The resulting equations are solved by using the Gorkov-Malkus-Veronis technique, which consists of developing the steady solution in terms of a small parameter measuring the deviation from the marginal state. The present work constitutes a generalization of an earlier work of A. Schlüter, D. Lortz and F. Busse [ibid. 23, 129-144 (1965; Zbl 0134.218)] where only buoyancy-driven instabilities were treated. Here, however, both buoyancy and temperature-dependent surface-tension effects are considered. The authors determine the band of allowed steady states of convection near the onset of convection as a function of the Marangoni number and the wave number, and they display supercritical as well as subcritical zones of instability. As a result, it is found that hexagons are allowable flow patterns.
Reviewer: J.Burbea

MSC:
76E15 Absolute and convective instability and stability in hydrodynamic stability
76E30 Nonlinear effects in hydrodynamic stability
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[1] DOI: 10.1146/annurev.fl.07.010175.000351
[2] DOI: 10.1017/S0022112064000763 · Zbl 0123.42204
[3] DOI: 10.1017/S0022112058000410 · Zbl 0082.39603
[4] E.S.A., European Space Agency Rep. 98 pp 527– (1981)
[5] DOI: 10.1017/S0022112080000274__S0022112080000274
[6] DOI: 10.1017/S0022112069002217 · Zbl 0214.25404
[7] DOI: 10.1088/0034-4885/41/12/003
[8] Busse, J. Maths and Phys. 46 pp 140– (1967) · Zbl 0204.28401
[9] Block, Nature 178 pp 650– (1956)
[10] Bénard, Rev. Gen. Sci. Pure Appl. 11 pp 1261– (1900)
[11] DOI: 10.1063/1.1761716
[12] DOI: 10.1017/S002211206000116X · Zbl 0096.21102
[13] Lebon, Bull. Acad. R. Belg. 66 pp 520– (1980)
[14] DOI: 10.1007/BF01176278 · Zbl 0517.76042
[15] DOI: 10.1016/0017-9310(79)90057-7 · Zbl 0394.76072
[16] DOI: 10.1007/BF00266474 · Zbl 0141.43803
[17] Gorkov, Sov. Phys. JETP 6 pp 311– (1957)
[18] DOI: 10.1017/S0022112066000727
[19] Serrin, J. Fluid Mech. 3 pp 1– (1959)
[20] DOI: 10.1017/S0022112064000751 · Zbl 0123.42203
[21] DOI: 10.1017/S0022112065001271 · Zbl 0134.21801
[22] DOI: 10.1017/S002211206700134X · Zbl 0234.76066
[23] Rosenblat, J. Fluid Mech. 120 pp 91– (1982)
[24] DOI: 10.1017/S0022112058000616 · Zbl 0082.18804
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