Multiple-scale analysis of oscillatory thermocapillary convection of high Prandtl number fluids in a rectangular cavity

Abstract A lot of research work has been directed toward the study of oscillatory thermocapillary convection for the past about 20 years. However, the real mechanism for the onset of oscillatory thermocapillary convection is still not fully understood. The reasons are as follows. If, to most people, the Marangoni number, Ma , is the only parameter to describe the onset of oscillatory thermocapillary convection, why it cannot correlate the ground-based and microgravity experimental data properly? If the “surface-deformation number” or the so-called “ S -parameter” is the parameter which can correlate the experimental data properly, why it cannot be derived without conjectures? To resolve this deficiency, multiple-scale analysis is applied to determine, among others, the characteristic length, time, and velocity scales for an unsteady, two-dimensional thermocapillary convection with a deformable free surface in a rectangular cavity. For flow situations with A 2 M a ,0 ≫O(1), A 2 R σ ,0 ≫O(1), and Pr >O(1), the ratio of the free surface-variation time scale to the convective time scale of the main surface flow turns out to be the so-called “ S -parameter”. The S -parameter thus implies physically a delayed cooling effect of the return flow on the surface flow. Therefore, the most important contribution of the present study is to provide a theoretical basis for the derivation of the S -parameter and, hopefully, guidance for the study of oscillatory thermocapillary convection.