The Free Transmission Problem for a Water Droplet

Water droplets on insulating material influence strongly the aging process of the material and the material looses its hydrophobic and insulating properties. The shape of the droplets signifies the state of the aging material. The present paper discusses a numerical procedure to calculate the droplet shape in an electric field generated by constant voltage between two electrodes. This leads to a transmission problem on the free surface of the droplet. It contains two sub-problems, first finding the droplet shape in a given electric field via an evolution problem, and second calculating the electric field for a given droplet shape via finite elements. The force density acting on the droplet shape depends on the electric field, and the electric field depends on the droplet shape. Both sub-problems have to be solved simultaneously. The typical shapes of the droplets are shown for several voltages. Finally a comparison between the behaviours of a dielectric droplet of pure rainwater and a conductive droplet of water with environmental additives is presented.