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Finite-amplitude regimes of the short-wave Marangoni-Benard convective instability

par Colinet, Pierre ;Legros, Jean Claude ;Kamotani, Yasuhiro;Dauby, Pierre C.;Lebon, Georgy
Référence Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 52, 3, page (2603-2616)
Publication Publié, 1995

Article révisé par les pairs

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Résumé :

A model of the infinite Prandtl number thermocapillary instability in layers of infinite depth is developed in the framework of the amplitude equations formalism. Making use of eigenfunctions at a given Marangoni number Ma as a basis for the nonlinear problem, rather than the neutral stability functions, it is shown that third-order equations may visibly be extrapolated rather far above the threshold. In particular, results are obtained about the wavelength selection problem between fastest growing modes (wave numbers around kmaxMa1/2 for a zero free surface Biot number) and critical modes (kc0 and Mac0). Transient numerical integration of the equations reveals an unbounded growth of the mean wavelength, thus indicating the absence of an intrinsic wavelength for this physical system. This is explained in terms of the mean (horizontally averaged) temperature profile distortion by convection. The final steady state of this evolution (imposed wavelength) is then approximated analytically. Earlier results about the competition between rolls and hexagonal patterns are qualitatively recovered. These solutions are then investigated in the limit Ma, where power law relationships are derived for main convective quantities. In particular, a saturation behavior is obtained for a quantity (the bulk temperature decrease), which can be considered as a measure of the heat transport increase due to convection. © 1995 The American Physical Society.