Flow stability in shallow droplets subject to localized heating of the bottom plate

We investigate theoretically the stability of thermo-capillary convection within a droplet when subject to localized heating from below. To model the droplet, we use a mathematical model based on lubrication theory. We formulate a base-state droplet profile, and we examine its stability with respect to small-amplitude perturbations in the azimuthal direction. Such linear stability analysis reveals that the base state is stable across a wide parameter space. We carry out transient simulations in three spatial dimensions: the simulations reveal that when the heating is slightly off-centered with respect to the droplet center, vortices develop within the droplet. The vortices persist when the contact line is pinned. These findings are consistent with experimental studies of locally heated sessile droplets.