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Pulse evolution for Marangoni-Bénard convection. (English) Zbl 1076.76594
Summary: Marangoni-Bénard convection is the process by which oscillatory waves are generated on an interface due to a change in surface tension. This process, which can be mass or temperature driven, is described by a perturbed Korteweg-de Vries (KdV) equation. For a certain parameter range, this perturbed KdV equation has a solitary wave solution with a unique steady-state amplitude for which the excitation and friction terms in the perturbed KdV equation are in balance. The evolution of an initial \(\text{sech}^ 2\) pulse to the steady-state solitary wave governed by the perturbed KdV equation of Marangoni-Bénard convection is examined. Approximate equations, derived from mass conservation, and momentum evolution for the perturbed KdV equation are used to describe the evolution of the initial pulse into steady-state solitary wave(s) plus dispersive radiation. Initial conditions which result in one or two solitary waves are considered. A phase plane analysis shows that the pulse evolves on two timescales, initially to a solution of the KdV equation, before evolving to the unique steady solitary wave of the perturbed KdV equation. The steady-state solitary wave is shown to be stable. A parameter regime for which the steady-state solitary wave is never reached, with the pulse amplitude increasing without bound, is also examined. The results obtained from the approximate conservation equations are found to be in good agreement with full numerical solutions of the perturbed KdV equation governing Marangoni-Benard convection.

76R05 Forced convection
76E06 Convection in hydrodynamic stability
76D45 Capillarity (surface tension) for incompressible viscous fluids
35Q53 KdV equations (Korteweg-de Vries equations)
Full Text: DOI
[1] Garazo, A.N.; Velarde, M.G., Dissipative Korteweg-de Vries description of Marangoni-Bénard convection, Phys. fluids A, 3, 2295-2300, (1988) · Zbl 0745.76077
[2] Chu, X-.L.; Velarde, M.G., Korteweg-de Vries excitation in Bénard-Marangoni convection, Amer. phys. soc., 43, 1094-1096, (1991)
[3] Weidman, P.D.; Linde, H.; Velarde, M.G., Evidence for solitary wave behaviour in Marangoni-Bénard convection, Phys. fluids A, 4, 921-926, (1992)
[4] Linde, H.; Chu, X.-L.; Velarde, M.G.; Waldhelm, W.W., Wall reflections of solitary waves in Marangoni-Bénard convection, Phys. fluids. A, 5, 3162-3166, (1993)
[5] Kawahara, T., Formation of saturated solitons in a nonlinear dispersive system with instability and dispersion, Phys. rev. lett., 51, 381-383, (1983)
[6] Kawahara, T.; Toh, S., Pulse interactions in an unstable dissipative-dispersive nonlinear system, Phys. fluids, 31, 2103-2111, (1988)
[7] Marchant, T.R., The evolution and interaction of Marangoni-Bénard solitary waves, Wave motion, 23, 307-320, (1996) · Zbl 0936.76534
[8] Kath, W.L.; Smyth, N.F., Soliton evolution and radiation loss for the Korteweg-de Vries equation, Phys. rev. E, 51, 1, 661-670, (1995)
[9] Smyth, N.F.; Worthy, A.L., Solitary wave evolution for mkdv equations, Wave motion, 21, 263-275, (1995) · Zbl 0968.35517
[10] Grimshaw, R., Slowly varying waves. I. Korteweg-de Vries equation, (), 359-375 · Zbl 0414.76017
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