TY - JOUR
T1 - Liquid toroidal drop under uniform electric field
AU - Zabarankin, Michael
N1 - Publisher Copyright:
©2017 The Author(s) Published by the Royal Society. All rights reserved.
PY - 2017
Y1 - 2017
N2 - 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=(2R2 + 3R + 2)/(7R2), where R and Q 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 finds Q and CaE 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 4π/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 CaE depend on R and ρ, and for large ρ, e.g. ρ ≥ 3, they have simple approximations: Q∼(R2+ R + 1)/(3R2) and CaE ∼3 √ 3πρ/2 (6 lnρ + 2 ln[96π] - 9)/(12 ln ρ + 4 ln[96π] - 17) (R + 1)2/(R - 1)2.
AB - 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=(2R2 + 3R + 2)/(7R2), where R and Q 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 finds Q and CaE 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 4π/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 CaE depend on R and ρ, and for large ρ, e.g. ρ ≥ 3, they have simple approximations: Q∼(R2+ R + 1)/(3R2) and CaE ∼3 √ 3πρ/2 (6 lnρ + 2 ln[96π] - 9)/(12 ln ρ + 4 ln[96π] - 17) (R + 1)2/(R - 1)2.
KW - Analytical solution
KW - Electric field
KW - Stationary shape
KW - Stokes flow
KW - Toroidal drop
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U2 - 10.1098/rspa.2016.0633
DO - 10.1098/rspa.2016.0633
M3 - Article
AN - SCOPUS:85029811045
SN - 1364-5021
VL - 473
JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
IS - 2202
M1 - 20160633
ER -