Exact solution for the electric field associated with charge caps on a leaky dielectric droplet at high electric Reynolds number

In a recent paper [Peng , ], the phenomenon of electric charge cap formation on the surface of a leaky dielectric droplet in an electric field was identified. The surface charge caps, which form at the droplet poles when charge relaxation is faster inside the droplet phase than in the ambient, are stagnant and perfectly conducting. Using asymptotic methods, a nonstandard, inhomogeneous, mixed boundary value problem satisfied by the droplet field potential in this high electric Reynolds number limit was derived by Peng and then solved numerically. The present article gives an exact solution to that mixed boundary value problem. This solution resolves fully the square-root singularities in the potential at the edges of the electric caps that can cause convergence difficulties in numerical schemes. Knowledge of the solution in analytical form is valuable because it feeds into a nonlinear problem for the coupled hydrodynamics. Moreover, the theoretical approach here is potentially extendible to finding the fields associated with electric charge cap formation in other settings.