Influence of the prandtl number on the location of recirculation eddies in thermocapillary flows

Two-dimensional thermocapillary-driven flow in a horizontal cavity of large aspect ratio is considered. Of particular interest is the asymptotic matching between the known parallel core flow solution and the non-parallel flow near the ends of the cavity. The influence of the end regions on the core flow is described by the superposition of spatial disturbances to the core flow solution. The stability analysis of this perturbed state leads to an eigenvalue problem for the complex wave number of the disturbances. The end flow structures are related to the nature of the lowest eigenvalue: when purely real it describes a boundary layer regime whereas a non-vanishing imaginary part reveals the existence of periodic structures known as recirculation rolls or eddies. It is found that, depending on the Prandtl number value, Pr, recirculation eddies can exist either near the hot wall (Pr ≫ 1) or the cold wall (Pr ≪ 1). Our results provide a direct interpretation to the different behaviors observed in previous experimental and numerical studies.