Compound Drop Shape Analysis with the Neumann Number

A compound droplet is composed of a traditional pendant drop (PD) or sessile drop (SD) sharing the interface with an immiscible phase of comparable sizes, which could be a solid particle, a gas bubble, or most often another droplet of an immiscible liquid. Over the past decade, the study of compound droplets has attracted increasing attention because of extensive applications in many research fields, such as complex fluids, microfluidics, foam and emulsion, and biomedical applications. Among all technical difficulties in characterizing compound droplets, a central problem in surface science is the prediction of its equilibrium shape, which requires knowledge of complicated boundary conditions. Existing dimensionless groups, such as the Bond number traditionally used to evaluate the shape of PDs and SDs, largely fail in predicting the shape of compound droplets. Here, we propose an alternative Bond number, termed the Neumann number, to characterize the shape of compound droplets. Using three dimensionless groups, that is, the Neumann number, the Bond number, and the Worthington number, we have quantitatively predicted and analyzed the shape of traditional PDs/SDs and various compound droplets, including a PD with a spherical particle suspending at the drop apex, a SD with its apex disturbed by a vertical cylinder, and a spherical SD (no gravity) with its apex disturbed by a fluid lens. It is found that the Neumann number can be readily adapted to quantitatively predict and analyze the shape of PDs/SDs and compound droplets.