Thermocapillary instabilities in a liquid layer subjected to an oblique temperature gradient: Effect of a prescribed normal temperature gradient at the substrate

We consider thermocapillary instability in a three-dimensional liquid layer with a deformable interface with an ambient gas phase and subjected to an oblique temperature gradient when the temperature gradient at the substrate is prescribed. We demonstrate that this configuration leads to a drastic change in the instability features with respect to those emerging when either a purely vertical temperature gradient (VTG) or a purely horizontal temperature gradient (HTG) is present. In the case of the return flow as the base state, the spanwise long-wave instability mode dominates except for the range of small Bond numbers Bo. Slippage at the substrate has a stabilizing (destabilizing) effect on streamwise (spanwise) long-wave modes in the presence of a HTG. In the case of linear flow as the base state, both streamwise and spanwise long-wave modes play a major role in the instability onset depending on the ratio between the HTG and the VTG η for higher values of the capillary number Ca, e.g., Ca > 0.001. However, for lower values of Ca, e.g., Ca < 0.001, streamwise and spanwise instability modes become finite-waves at large η. In contrast to the return flow, for the linear flow, slippage at the substrate destabilizes both long-wave modes.