Liquid toroidal drop under uniform electric field

Abstract

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 when Q =(2 R 2 +3 R +2)/(7 R 2 ), where R and Q are ratios of the phases’ electric conductivities and dielectric constants, respectively. This condition holds for any electric capillary number, Ca E , that defines the ratio of electric stress to surface tension. Pairam and Fernández-Nieves showed experimentally that, in the absence of external forces (Ca E =0), a toroidal drop shrinks towards its centre, and, consequently, the drop can be stationary only for some Ca E >0. This work finds Q and Ca E such that, under the presence of an electric field and with equal viscosities of the phases, a toroidal drop having major radius ρ and volume 4 π /3 is qualitatively stationary—the normal velocity of the drop’s interface is minute and the interface coincides visually with a streamline. The found Q and Ca E depend on R and ρ , and for large ρ , e.g. ρ ≥3, they have simple approximations: Q ∼( R 2 + R +1)/(3 R 2 ) and Ca 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 .