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Long-wave Marangoni convection in a thin film heated from below
S Shklyaev1, A A Alabuzhev, M Khenner
1Department of Chemical Engineering, California Institute of Technology, 1200 California Boulevard, Pasadena, California 91125, USA.
Abstract:
We consider long-wave Marangoni convection in a liquid layer atop a substrate of low thermal conductivity, heated from below. We demonstrate that the critical perturbations are materialized at the wave number K ∼ √Bi, where Bi is the Biot number which characterizes the weak heat flux from the free surface. In addition to the conventional monotonic mode, a novel oscillatory mode is found. Applying the K ∼ √Bi scaling, we derive a new set of amplitude equations. Pattern selection on square and hexagonal lattices shows that supercritical branching is possible. A large variety of stable patterns is found for both modes of instability. Finite-amplitude one-dimensional solutions of the set, corresponding to either steady or traveling rolls, are studied numerically; a complicated sequence of bifurcations is found in the former case. The emergence of an oscillatory mode in the case of heating from below and stable patterns with finite-amplitude surface deformation are shown in this system for the first time.
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