FINITE-AMPLITUDE REGIMES OF THE SHORT-WAVE MARANGONI-BENARD CONVECTIVE INSTABILITY

A model of the infinite Prandtl number thermocapillary instability in layers of infinite depth is developed in the framework of the amplitud e equations formalism. Making use of eigenfunctions at a given Marango ni number Ma as a basis for the nonlinear problem, rather than the neu tral stability functions, it is shown that third-order equations may v isibly be extrapolated rather far above the threshold. In particular, results are obtained about the wavelength selection problem between fa stest growing modes (wave numbers around k(max) similar to Ma(1/2) for a zero free surface Blot number) and critical modes (k(c)-->0 and Ma( c)-->0). Transient numerical integration of the equations reveals an u nbounded growth of the mean wavelength, thus indicating the absence of an intrinsic wavelength for this physical system. This is explained i n terms of the mean (horizontally averaged) temperature profile distor tion by convection. The final steady state of this evolution (imposed wavelength) is then approximated analytically. Earlier results about t he competition between rolls and hexagonal patterns are qualitatively recovered. These solutions are then investigated in the limit Ma-->inf inity, where power law relationships are derived for main convective q uantities. In particular, a saturation behavior is obtained for a quan tity (the bulk temperature decrease), which can be considered as a mea sure of the heat transport increase due to convection.