Convective, absolute, and global instabilities of thermocapillary-buoyancy convection in extended layers

We study the linear stability of thermocapillary-buoyancy convection in an extended liquid layer subject to a longitudinal temperature gradient. It is found that by applying the concepts of convective, absolute, and global instabilities, theory agrees well with experiment. Two different effects due to the lateral walls are considered. First, the stationary disturbance due to the end walls induces a steady wave pattern spreading over the whole layer as the zero-frequency mode becomes convectively unstable. Second, virtual reflections of traveling disturbances by the lateral walls provide the feedback necessary for the onset of a global instability. In the simplest case, a global neutrally stable state is formed by a couple of transverse waves propagating at the same frequency in opposite directions, so that spatial amplification of one wave compensates for the attenuation of the other. However, the most dangerous self-sustained disturbance is set up by a couple of mirror symmetric oblique waves propagating purely spanwise. For purely thermocapillary-driven flow the threshold of self-sustained instability is just slightly higher than that of the convective instability. However, for liquids of large Prandtl number a moderate buoyancy effect may cause a significant stabilization of self-sustained oscillatory instability.