Elastica of a pendant droplet: Analytical solution in two dimension

The profile analysis of a pendant droplet is of great value for both fundamental science and engineering applications. In this study, we analytically investigated the configuration of the pendant drop, and found that it is similar to the elastica of a slender beam with large displacement. First, the energy formulation of the droplet-substrate system was presented, then the Young-Laplace equation and Young's equation were derived based on the variation with movable boundary conditions. Next the phase plane analysis was performed to provide a clear map on the existence and multiple forms of solutions. In succession, the morphology of the pendant droplet was explicitly solved in terms of elliptic integrals. The necking point, the adhesion and detachment conditions of the droplet, and the parameter analogies between the pendant drop and the elastica were discussed. Finally, we extended the obtained solutions to analyze the pendant droplet hanging on a substrate with special curvatures. These analyses can be beneficial to the design of new superhydrophobic materials, micro-fluidics, and some analogy experiments.