A leaky dielectric drop with conical ends in an electric field

There has long been interest in the existence of stationary shapes with conical ends for drops freely suspended in an ambient fluid and subjected to a uniform electric field. If the two phases are leaky dielectric (slightly conducting) viscous incompressible fluids with ratios of the phases’ conductivities, dielectric constants, and viscosities being R, Q, and λ, respectively, then a stationary shape with a conical end determined by angle ϑ₀ may exist only when R and ϑ₀ are related and when electric capillary number Ca_E, defining the ratio of electric stress to interfacial tension, is a certain function of R, Q, and ϑ₀. Of particular interest is a spindle shape that can be defined by a single parameter, e.g., ϑ₀. For each λ, one can express R and Ca_E in terms of ϑ₀ and Q from those two conditions and then can find ϑ₀ and Q that make a spindle shape as close to being stationary as possible. For corresponding R, Q, and Ca_E, a spindle shape with ϑ₀ = 0.9275π (the angle of the tangent cone at the spindle’s vertex is 2(π − ϑ₀) = 26.1^◦) is on par with known stationary smooth shapes in terms of the value of the normal velocity on the interface and in terms of streamline behavior near the shape.