A leaky dielectric drop with conical ends in an electric field

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Abstract

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.

Original languageEnglish
Pages (from-to)1768-1796
Number of pages29
JournalSIAM Journal on Applied Mathematics
Volume79
Issue number5
DOIs
StatePublished - 2019

Keywords

  • Boundary-integral equation
  • Conical ends
  • Electric field
  • Leaky dielectric drop
  • Stationary shape
  • Stokes flow

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