Thermocapillary instabilities in a liquid layer subjected to an oblique temperature gradient

Stability analysis of a liquid layer subjected to an oblique temperature gradient (OTG) is carried out. The general linear stability analysis reveals a stabilization effect of the imposed horizontal component (horizontal temperature gradient, HTG) of the OTG on the long-wave instabilities introduced by the vertical component (vertical temperature gradient, VTG) of the OTG. This stabilization is due to the VTG induced by the prescribed HTG, which counteracts the imposed VTG. The induced VTG arises due to the presence of advection of the energy. As a result of the stabilization, the long-wave mode forms an island of instability in the $\eta$–$Ma_c$ plane, where $\eta$ and $Ma_c$ are the ratio of the strength of the imposed HTG to imposed VTG components of the OTG, and the critical Marangoni number, respectively. However, for sufficiently high $\eta$, a new class of modes emerge with the critical Marangoni number scaling as $Ma_c \sim 1/\eta$. These modes originate as a result of the interaction between the thermocapillary flow caused by the imposed HTG on the one hand, and the VTG on the other, and remain the dominant modes of instability at higher $\eta$. The long-wave analysis is carried out and, in its framework, the nonlinear evolution equation is derived, and, based on it, linear and weakly nonlinear analyses are performed. An increase in $\eta$ changes the type of bifurcation from subcritical to supercritical. The numerical solution of the evolution equation around the critical parameter values validates the predictions of the weakly nonlinear analysis. The present study illustrates a possible use of imposing the HTG to prevent dry-spot formation and rupture of the film caused by the imposed VTG.