Electrohydrodynamics of lenticular drops and equatorial streaming

Abstract

Drops subjected to electric fields can deform into singular shapes exhibiting apparent sharp tips. At high field strengths, a perfectly conducting drop surrounded by a perfectly insulating exterior fluid deforms into a prolate-shaped drop with conical ends and can exist in hydrostatic equilibrium. On the conical ends, capillary stress, which is due to the out-of-plane curvature and is singular, balances electric normal stress which is also singular. If the two phases are not perfect conductors/insulators but are both leaky dielectrics and the drop is much more conducting and viscous than the exterior, electric tangential stress disrupts the hydrostatic force balance and leads to jet emission from the cone's apex. If, however, the physical situation is inverted so that a weakly conducting, slightly viscous drop is immersed in a highly conducting, more viscous exterior, the drop deforms into an oblate lens-like profile before eventually becoming unstable. In experiments, the equator of a lenticular drop superficially resembles a wedge prior to instability. Such a drop disintegrates by equatorial streaming by ejecting a thin liquid sheet from its equator. We show theoretically by performing a local analysis that a lenticular drop's equatorial profile can be a wedge only if an approximate form of the surface charge transport equation – continuity of normal current condition – is used. Moreover, we demonstrate via numerical simulation that such wedge-shaped drops do not become unstable and therefore cannot emit equatorial sheets. We then show by transient simulations how equatorial streaming can occur when charge transport along the interface is analysed without approximation.