TY - JOUR
T1 - A leaky dielectric drop with conical ends in an electric field
AU - Zabarankin, Michael
N1 - Publisher Copyright:
© 2019 Society for Industrial and Applied Mathematics
PY - 2019
Y1 - 2019
N2 - There has long been interest in the existence of stationary shapes with conical ends for drops freely suspended in an ambient fluid and subjected to a uniform electric field. If the two phases are leaky dielectric (slightly conducting) viscous incompressible fluids with ratios of the phases’ conductivities, dielectric constants, and viscosities being R, Q, and λ, respectively, then a stationary shape with a conical end determined by angle ϑ0 may exist only when R and ϑ0 are related and when electric capillary number CaE, defining the ratio of electric stress to interfacial tension, is a certain function of R, Q, and ϑ0. Of particular interest is a spindle shape that can be defined by a single parameter, e.g., ϑ0. For each λ, one can express R and CaE in terms of ϑ0 and Q from those two conditions and then can find ϑ0 and Q that make a spindle shape as close to being stationary as possible. For corresponding R, Q, and CaE, a spindle shape with ϑ0 = 0.9275π (the angle of the tangent cone at the spindle’s vertex is 2(π − ϑ0) = 26.1◦) is on par with known stationary smooth shapes in terms of the value of the normal velocity on the interface and in terms of streamline behavior near the shape.
AB - There has long been interest in the existence of stationary shapes with conical ends for drops freely suspended in an ambient fluid and subjected to a uniform electric field. If the two phases are leaky dielectric (slightly conducting) viscous incompressible fluids with ratios of the phases’ conductivities, dielectric constants, and viscosities being R, Q, and λ, respectively, then a stationary shape with a conical end determined by angle ϑ0 may exist only when R and ϑ0 are related and when electric capillary number CaE, defining the ratio of electric stress to interfacial tension, is a certain function of R, Q, and ϑ0. Of particular interest is a spindle shape that can be defined by a single parameter, e.g., ϑ0. For each λ, one can express R and CaE in terms of ϑ0 and Q from those two conditions and then can find ϑ0 and Q that make a spindle shape as close to being stationary as possible. For corresponding R, Q, and CaE, a spindle shape with ϑ0 = 0.9275π (the angle of the tangent cone at the spindle’s vertex is 2(π − ϑ0) = 26.1◦) is on par with known stationary smooth shapes in terms of the value of the normal velocity on the interface and in terms of streamline behavior near the shape.
KW - Boundary-integral equation
KW - Conical ends
KW - Electric field
KW - Leaky dielectric drop
KW - Stationary shape
KW - Stokes flow
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U2 - 10.1137/18M1211088
DO - 10.1137/18M1211088
M3 - Article
AN - SCOPUS:85073617492
SN - 0036-1399
VL - 79
SP - 1768
EP - 1796
JO - SIAM Journal on Applied Mathematics
JF - SIAM Journal on Applied Mathematics
IS - 5
ER -