Buoyant-thermocapillary instabilities of differentially heated liquid layers

The stability of buoyant-thermocapillary-driven flows in a fluid layer subjected to a horizontal temperature gradient is analyzed. Our purpose is the modelization of recent experimental results obtained for a fluid of Prandtl number Pr=7, by Daviaud and Vince [Phys. Rev. E 48, 4432 (1993)], who observed a transition between traveling waves and stationary rolls when the height of fluid is increased. Our model takes into account several effects that were examined separately in previous studies. The relative importance of buoyancy and thermocapillarity is controlled by the ratio W of Marangoni number to Rayleigh number. The fluid layer is bounded below by a rigid plane whose temperature varies linearly along the direction of the thermal gradient. A Biot number is introduced to describe heat transfer at the top free surface. Our stability analysis establishes the existence of a transition between stationary and oscillatory modes when W exceeds a value W0 which is a function of the Biot number. Moreover, two types of oscillatory modes have been identified which differ by the range of variation of their critical parameters (wave number, frequency, angle of propagation) versus W.