Nonlinear Methods of Analyzing the Marangoni Instability

Investigations have been made [26,27] into the stabihty of a liquid layer with limited thickness bounded by a cylinder with a circular or rectangular lateral cross section, this layer having an upper interface common with the bulk of an unmovable gas and the side wall of the cylindrical container being heat-proof, provided the bottom wall is maintained at a higher temperature then the upper one. Within the framework of the Boussinesq approximation for a temperature-dependent density it was found that the limit of the capillary number is close to zero = is the dynamic 

The critical Marangoni number is calculated as a function of the Rayleigh, Prandtl, and Biot numbers together with the aspect ratio (i.e., the ratio of the container radius to the mean depth of a liquid. 

The stability criteria for multiple nonlinear steady states near the critical Marangoni number are determined. 

The configurations, the characteristic dimensions and the amphtudes of the velocities for axisymmetrical circulational cells, as well as the influence of the Prandtl number upon the stability of the periodic movement in those cells are estabhshed. 

The transitions between two steady convective states of equal probability corresponding to the binary double proper values for the critical Marangoni number are predicted and analyzed. 

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