P. Colinet et al., FINITE-AMPLITUDE REGIMES OF THE SHORT-WAVE MARANGONI-BENARD CONVECTIVE INSTABILITY, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 52(3), 1995, pp. 2603-2616
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.