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 and λ, respectively, then the toroidal drop with volume 4π/3 and having major radius ρ 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 number Ca_E that defines the ratio of electric stress to surface tension, are functions of R, ρ and λ and are found analytically. Those functions are relatively insensitive to λ, and for ρ ≥ 3, they admit simple approximations, which coincide with those obtained recently for λ = 1. Streamlines inside and outside the toroidal drop for the same R and ρ but different λ are qualitatively similar.