Thermocapillarity Motion of a Droplet within a Permeable Cavity

The thermocapillarity motion of a non-deformable spherical droplet embedded in a concentric permeable spherical cavity, filled with a Newtonian viscous fluid, and subjected to a uniformly prescribed temperature gradient, is investigated analytically. The energy and momentum field equations are resolved within the quasi-steady limit, considering small Péclet and Reynolds numbers. Additionally, in this investigation, it is assumed that the capillary number at the droplet interface is small, ensuring the perpetuation of the droplet's spherical shape throughout its motion.We have derived normalized thermocapillarity velocity results across a broad spectrum of relative thermal conductivity values, cavity permeabilities, and viscosity ratios. The obtained normalized thermocapillarity velocity is emphasized using graphs and tables, allowing for a comparison with existing literature data. Additionally, specific cases available in the literature have been examined to further validate our findings. This study is inspired by diverse flow situations, including particle deposition in processes like reverse osmosis, dialysis, and within various biological organs where fluid traverses' membranes or cell cavity walls.