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

## TL;DR

This paper models the dynamics of a thin liquid film on a sphere under gravity, surface tension, and thermal gradients, analyzing stability and flow patterns influenced by Marangoni effects through numerical simulations.

## Contribution

It derives a nonlinear PDE incorporating Marangoni effects for a spherical thin film and analyzes stability under axial and radial thermal gradients with numerical simulations.

## Key findings

- Steady states depend on the balance of gravity and Marangoni effects.
- Axial gradients lead to uniform or localized film accumulation.
- Radial gradients induce unstable non-axisymmetric modes.

## Abstract

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.