Electrohydrodynamic drop deformation by strong electric fields: Slender-body analysis

Slender-body approximations are utilized to analyze drop elongation by a uniformly applied electric field. The Taylor-Melcher model of leaky-dielectric liquids is employed, with electrohydrodynamic flow animation by electrical shear stresses at the free surface. Using the drop slenderness as the small asymptotic parameter, separate asymptotic expansions of the pertinent fields are presented in "inner" and "outer" regions, respectively, corresponding to the drop cross-sectional and longitudinal scales, as well as an additional expansion in the drop phase. For a given shape, both the electric potential and flow field are calculated. Asymptotic matching is possible only for low drop viscosity. The normal-stress condition on the free surface provides a scaling relation between the slenderness parameter and the dimensionless electric field, expressed as a capillary number. The predicted slenderness scaling, inversely with the 6/7-power of the electric field, is the same as that appropriate for dielectric liquids. Within that scaling, the normal-stress condition provides a secondorder ordinary differential equation governing the drop shape. The existence of a solution to this equation necessitates the satisfaction of a new inequality expressed in terms of the conductivity and permittivity ratios. The slender-body formulation does not provide a sufficient number of boundary conditions for this equation. We propose that a unique solution can be found via matching to a separate local solution, valid near the drop tips, where the slender-body approximation breaks down. We have not been able to calculate that separate solution.