Heat transfer, vapor diffusion, and Stefan flow around levitating droplets near a heated liquid surface

We consider a slowly condensing droplet levitating near a surface of evaporating liquid and develop a mathematical model to describe diffusion, heat transfer, and fluid flow in the system. The method of separation of variables in bipolar coordinates is used to obtain series expansions for all physical quantities. This framework allows us to determine temperature profiles and condensation rates at the surface of the droplet, as well as calculate the Stokes force for the conditions of levitation. We find that the dependence of the equilibrium concentration on temperature is necessary to accurately model the phase change at the surface of the liquid. The condensation of vapor leads to the temperature in the droplet being, on average, higher than the surrounding air. Due to the temperature gradient in the droplet, condensation is higher at the top of the droplet and lower at the bottom.