  We investigate theoretically the stability of thermo-capillary convection
within a droplet when heated by a point source 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 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 point-heated sessile droplets.
