Liquid toroidal drop under uniform electric field

The problem of a stationary liquid toroidal drop freely suspended in another fluid and subjected to an electric field uniform at infinity is addressed analytically. Taylor’s discriminating function implies that, when the phases have equal viscosities and are assumed to be slightly conducting (leaky dielectrics), a spherical drop is stationary whenQ =(2R 2 +3R +2)/(7R 2 ), whereR andQ are ratios of the phases’ electric conductivities and dielectric constants, respectively. This condition holds for any electric capillary number, CaE , that defines the ratio of electric stress to surface tension. Pairam and Fernández-Nieves showed experimentally that, in the absence of external forces (CaE =0), a toroidal drop shrinks towards its centre, and, consequently, the drop can be stationary only for some CaE >0. This work findsQ and CaE such that, under the presence of an electric field and with equal viscosities of the phases, a toroidal drop having major radiusρ and volume 4π /3 is qualitatively stationary—the normal velocity of the drop’s interface is minute and the interface coincides visually with a streamline. The foundQ and CaE depend onR andρ , and for largeρ , e.g.ρ ≥3, they have simple approximations:Q ∼(R 2 +R +1)/(3R 2 ) andCa E ∼ 3 3 π ρ / 2   ( 6  ln  ⁡ ρ + 2  ln ⁡ [ 96 π ] − 9 ) / ( 12  ln  ⁡ ρ + 4  ln ⁡ [ 96 π ] − 17 )   ( R + 1 ) 2 / ( R − 1 ) 2 .