Accurate and efficient boundary integral methods for electrified liquid bridge problems

We derive and implement boundary integral methods for axisymmetric liquid bridge problems in the presence of an axial electric field. The liquid bridge is bounded by solid parallel electrodes placed perpendicular to the axis of symmetry and held at a constant potential difference. The fluid is assumed to be nonconducting and has permittivity different from that of the passive surrounding medium. The problem reduces to the solution of two harmonic problems for the fluid and voltage potential inside the bridge and another harmonic problem for the voltage potential outside the bridge. The shape of the moving interface is determined by the imposition of stress, as well as kinematic and electric field boundary conditions, the former condition accounting for discontinuous electric stresses across the interface. We propose fast and highly accurate boundary integral methods based on fast summations of appropriate series representations of axisymmetric Green's functions in bounded geometries. We implement our method to calculate equilibrium shapes for electrified liquid bridges in the absence and presence of gravity. Such calculations appear in the literature using finite element methods, and our boundary integral approach is a fast and accurate alternative.