Abstract
Overstability for simultaneous surface-tension- and buoyancy-driven instability in a horizontal infinite liquid layer is theoretically investigated by means of a small disturbance analysis. Formulation and results are given in dimensionless forms. Critical wavenumbers, time constants, and Marangoni numbers are computed. Besides the influence of Prandtl, Bond, and crispation numbers, the modifications induced by interfacial viscosities, heat transfer at the free surface, buoyancy with respect to a pure Marangoni mechanism, and different thermal conditions at the rigid wall, are included in the analysis. The case of exchange of stability is considered as a special case of overstability. This work provides a generalization of Takashima’s work [J. Phys. Soc. Jpn. 50, 2745, 2751 (1981)] concerning a pure Marangoni mechanism (with less general conditions).
Highlights
The present work follows a long line of studies concern ing RBM (Rayleigh-Benard-Marangoni) computations, anchored through a direct filiation in a line of works that goes back to the original theoretical master analysis of Ray leigh.[1]
The primary interest was in the coupling between Marangoni and Rayleigh numbers for overstability, the opportunity has been taken to include other ingredients in the analysis (Prandtl, Bond, and crispa tion numbers, modifications induced by interfacial viscos ities and heat transfer at the free surface, and different ther mal conditions at the rigid wall)
We suggest that the mechanism of inhibition of overstability by buoyancy might be physically understood by remembering that, for tension-driven convection, the free surface is depressed above a hot stream while it is elevated for buoyancy-driven convection.[51 4] this last statement concerns the case of exchange of stability, it indicates a conflict in the direction of deformation of the free surface between the buoyancy case and the surface tension case which might be the cause of the overstability inhibition by buoyancy
Summary
The present work follows a long line of studies concern ing RBM (Rayleigh-Benard-Marangoni) computations, anchored through a direct filiation in a line of works that goes back to the original theoretical master analysis of Ray leigh.[1]. The primary interest was in the coupling between Marangoni and Rayleigh numbers for overstability, the opportunity has been taken to include other ingredients in the analysis (Prandtl, Bond, and crispa tion numbers, modifications induced by interfacial viscos ities and heat transfer at the free surface, and different ther mal conditions at the rigid wall). This results in a large dimension of the parameter space, but only a selected and exemplifying set of results will be displayed. Crystal growth typically involves nonmotionless basic states
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