Droplet coalescence on a sloped fibre
Transporting droplets along cylindrical surfaces is an emerging area of research with a wide range of practical applications such as fog collection and filtration. Recent experimental studies by Feng et al. (2024) have revealed a wealth of new droplet coalescence dynamics on a pre-wetted cylindrical fibre, illustrating the need for more advanced theory. In this paper, we study both the early-stage and late-stage dynamics of droplet coalescence on a sloped cylindrical fibre. In the early stage, when the minimum thickness of the liquid bridge connecting two droplets is small, we propose a lubrication-based model to analyze the capillary-driven self-similar dynamics of the liquid bridge. Our theory predicts the bridge growth rate and demonstrates that asymmetry in the adjoining contact angles contributes to the migration of the merged droplet, showing good agreement with experimental observations. In the late stage, as droplet merging progresses, inertia effects become significant and lead to rapid mass transfer. To capture this regime, we develop a weighted residual model for liquid films flowing on a sloped pre-wetted cylinder, incorporating key physical effects such as gravity, low-to-moderate inertia, surface tension, and intermolecular forces. This model explains the experimentally observed directional self-propelled transport during droplet coalescence, where the larger droplet migrates toward the smaller one. Numerical simulations based on this model further support these findings, showing good agreement between the predicted coalescence-induced droplet migration and experimental observations.