The role of surface charge convection in the electrohydrodynamics and breakup of prolate drops

The deformation of a weakly conducting, ‘leaky dielectric’, drop in a density matched, immiscible weakly conducting medium under a uniform direct current (DC) electric field is quantified computationally. We exclusively consider prolate drops, for which the drop elongates in the direction of the applied field. Furthermore, for the majority of this study, we assume the drop and medium to have equal viscosities. Using axisymmetric boundary integral computations, we delineate drop deformation and breakup regimes in the $Ca_{E}-Re_{E}$ parameter space, where $Ca_{E}$ is the electric capillary number (ratio of the electric to capillary stresses); and $Re_{E}$ is the electric Reynolds number (ratio of charge relaxation to flow time scales), which characterizes the strength of surface charge convection along the interface. For so-called ‘prolate A’ drops, where the surface charge is convected towards the ‘poles’ of the drop, we demonstrate that increasing $Re_{E}$ reduces the critical capillary number for breakup. Moreover, surface charge convection is the cause of an abrupt transition in the breakup mode of a drop from end pinching, where the drop elongates and develops bulbs at its ends that eventually detach, to a breakup mode characterized by the formation of conical ends. On the contrary, the deformation of ‘prolate B’ drops, where the surface charge is convected away from the poles, is essentially unaffected by the magnitude of $Re_{E}$.