Numerical Study of Falling Droplets in a Vertical Electric Field

It is of fundamental importance to study and understand the behavior of falling droplets. External fields, such as electric and magnetic fields, are considered promising methods for controlling falling droplets, especially at small scales. Although some experimental and theoretical work on falling droplets in an electric field has been conducted, a fully numerical model for electrohydrodynamic falling droplets is still lacking. In this article, we proposed a phase field numerical model and studied the falling droplets in a vertical electric field. In particular, the influence of the electric field on the velocity and interfacial morphology of the falling droplets was investigated. It was found that the vertical electric field can elongate the droplet in the vertical direction and increase its falling velocity. With a weak or no electric field (the electrical capillary number Ca_E < 1.0), an indentation appears on the top of the falling droplet, and the droplet evolves into a bowl-like structure. In contrast, a strong electric field (Ca_E ≥ 1.0) causes a protrusion on the droplet, transforming it into a thumbtack-like shape. The numerical model and the obtained results can improve our understanding of falling droplets and highlight potential ways to regulate their behaviors.