  We study the time evolution of a thin liquid film coating the outer surface
of a sphere in the presence of gravity, surface tension and thermal gradients.
We derive the fourth-order nonlinear partial differential equation that models
the thin film dynamics, including Marangoni terms arising from the dependence
of surface tension on temperature. We consider two different imposed
temperature distributions with axial or radial thermal gradients. We analyze
the stability of a uniform coating under small perturbations and carry out
numerical simulations in COMSOL for a range of parameter values. In the case of
an axial temperature gradient, we find steady states with either uniform film
thickness, or with the fluid accumulating at the bottom or near the top of the
sphere, depending on the total volume of liquid in the film, dictating whether
gravity or Marangoni effects dominate. In the case of a radial temperature
gradient, a stability analysis reveals the most unstable non-axisymmetric modes
on an initially uniform coating film.
