On the Onset of Marangoni Convection in a Rotating Fluid Layer

Abstract

Pearson’s pioneering theoretical analysis of the onset of Marangoni (surface-tension-driven) convection in a fluid layer was extended by Vidal and Acrivos to include the effect of rotation. Their analysis was further extended to include other aspects of the problem by McConaghy and Finlayson, Namikawa et al., Takashima and Namikawa and Sarma. Kaddame and Lebon obtained the critical points for the onset of steady and oscillatory Marangoni convection with rotation for the case when the Marangoni number, M > 0. While Char et al. and Chang and Chiang studied the onset of oscillatory convection in the M 0. Consider a horizontal layer of quiescent fluid of thickness d which is unbounded in the horizontal xand y-directions. The layer is kept rotating uniformly around the vertical z axis with a constant angular velocity and subject to a uniform vertical temperature gradient. The surface tension takes the form 1⁄4 0 ðT T0Þ, where 0 and T0 are reference values, and is the rate of change of surface tension with the temperature. The deformation of the interface is represented by the relation z 1⁄4 d þ ðx; y; tÞ, wherein ðx; y; tÞ is an a-priori unknown deformation with respect to the mean thickness d. In the reference state, the fluid is at rest with respect to the rotating axes and heat is conducted between the lower boundary (z 1⁄4 0) and the interface. When motion sets in, the velocity v 1⁄4 ðu; v;wÞ, pressure p and temperature T fields obey the usual balance equations of mass, momentum and energy, r v 1⁄4 0; ð1Þ @v @t þ ðv rÞvþ 2 v 1⁄4 1 rpþ rv; ð2Þ