Toroidal drop under electric field: arbitrary drop-to-ambient fluid viscosity ratio

In the absence of external forces, a liquid toroidal drop freely suspended in another fluid shrinks towards its centre. It is shown that if the two phases are slightly conducting viscous incompressible fluids with the drop-to-ambient fluid ratios of electric conductivities, dielectric constants and viscosities to be 1/R ,Q andλ , respectively, then the toroidal drop with volume 4π /3 and having major radiusρ can become almost stationary when subjected to a uniform electric field aligned with the drop’s axis of symmetry. In this case,Q and electric capillary numberCa E that defines the ratio of electric stress to surface tension, are functions ofR ,ρ andλ and are found analytically. Those functions are relatively insensitive toλ , and forρ ≥3, they admit simple approximations, which coincide with those obtained recently forλ =1. Streamlines inside and outside the toroidal drop for the sameR andρ but differentλ are qualitatively similar.