Marangoni effects on a thin liquid film coating a sphere with axial or radial thermal gradients

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 σ on temperature T. 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 either with 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. This suggests a potential method for the indirect measurement of d σ / d T by monitoring the thickness profile of the thin film. In the case of a radial temperature gradient, a stability analysis reveals the most unstable non-axisymmetric modes on an initially uniform coating film.