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The elongated shape of a dielectric drop deformed by a strong electric field

Published online by Cambridge University Press:  29 November 2010

DOV RHODES  and
EHUD YARIV
DOV RHODES
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel
EHUD YARIV*
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel
*
†Email address for correspondence: udi@technion.ac.il
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Abstract

A dielectric drop is suspended within a dielectric liquid and is exposed to a uniform electric field. Due to polarization forces, the drop deforms from its initial spherical shape, becoming prolate in the field direction. At strong electric fields, the drop elongates significantly, becoming long and slender. For moderate ratios of the permittivities of the drop and surrounding liquid, the drop ends remain rounded. The slender limit was originally analysed by Sherwood (J. Phys. A, vol. 24, 1991, p. 4047) using a singularity representation of the electric field. Here, we revisit it using matched asymptotic expansions. The electric field within the drop is continued into a comparable solution in the ‘inner’ region, at the drop cross-sectional scale, which is itself matched into the singularity representation in the ‘outer’ region, at the drop longitudinal scale. The expansion parameter of the problem is the elongated drop slenderness. In contrast to familiar slender-body analysis, this parameter is not provided by the problem formulation, and must be found throughout the course of the solution. The drop aspect-ratio scaling, with the 6/7-power of the electric field, is identical to that found by Sherwood (J. Phys. A, vol. 24, 1991, p. 4047). The predicted drop shape is compared with the boundary-integral solutions of Sherwood (J. Fluid Mech., vol. 188, 1988, p. 133). While the agreement is better than that found by Sherwood (J. Phys. A, vol. 24, 1991, p. 4047), the weak logarithmic decay of the error terms still hinders an accurate calculation. We obtain the leading-order correction to the drop shape, improving the asymptotic approximation.

JFM classification

Drops and Bubbles: Electrohydrodynamic effects
Type
Papers
Information
Journal of Fluid Mechanics , Volume 664 , 10 December 2010 , pp. 286 - 296
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Acrivos, A. & Lo, T. S. 1978 Deformation and breakup of a single slender drop in an extensional flow. J. Fluid Mech. 86, 641–672.CrossRefGoogle Scholar
Allan, R. S. & Mason, S. G. 1962 Particle behaviour in shear and electric fields. I. Deformation and burst of fluid drops. Proc. Roy. Soc. Lond. A 267 (1328), 45–61.Google Scholar
Batchelor, G. K. 1970 Slender-body theory for particles of arbitrary cross-section in Stokes flow. J. Fluid Mech. 44, 419–441.Google Scholar
Cox, R. G. 1970 The motion of long slender bodies in a viscous fluid. Part 1. General theory. J. Fluid Mech. 44, 791–810.Google Scholar
Dubash, N. & Mestel, A. J. 2007 a Behaviour of a conducting drop in a highly viscous fluid subject to an electric field. J. Fluid Mech. 581, 469–493.Google Scholar
Dubash, N. & Mestel, A. J. 2007 b Breakup behavior of a conducting drop suspended in a viscous fluid subject to an electric field. Phys. Fluids 19, 072101.CrossRefGoogle Scholar
Ha, J. W. & Yang, S. M. 2000 Deformation and breakup of Newtonian and non-Newtonian conducting drops in an electric field. J. Fluid Mech. 405, 131–156.CrossRefGoogle Scholar
Higuera, F. J. 2008 Breakup of a supported drop of a viscous conducting liquid in a uniform electric field. Phys. Rev. E 78 (1), 16314.Google Scholar
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.Google Scholar
Hinch, E. J. & Acrivos, A. 1980 Long slender drops in a simple shear flow. J. Fluid Mech. 98 (2), 305–328.Google Scholar
Lac, E. & Homsy, G. M. 2007 Axisymmetric deformation and stability of a viscous drop in a steady electric field. J. Fluid Mech. 590, 239–264.CrossRefGoogle Scholar
Li, H., Halsey, T. C. & Lobkovsky, A. 1994 Singular shape of a fluid drop in an electric or magnetic field. Europhys. Lett. 27, 575–580.Google Scholar
Macky, W. A. 1931 Some investigations on the deformation and breaking of water drops in strong electric fields. Proc. R. Soc. Lon. A 133 (822), 565–587.Google Scholar
Melcher, J. R. & Taylor, G. I. 1969 Electrohydrodynamics: a review of the role of interfacial shear stresses. Annu. Rev. Fluid Mech. 1 (1), 111–146.Google Scholar
Miksis, M. J. 1981 Shape of a drop in an electric field. Phys. Fluids 24, 1967.Google Scholar
Saville, D. A. 1997 Electrohydrodynamics: the Taylor–Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29 (1), 27–64.Google Scholar
Sherwood, J. D. 1988 Breakup of fluid droplets in electric and magnetic fields. J. Fluid Mech. 188, 133–146.Google Scholar
Sherwood, J. D. 1991 The deformation of a fluid drop in an electric field: a slender-body analysis. J. Phys. A 24, 4047–4053.Google Scholar
Stone, H. A., Lister, J. R. & Brenner, M. P. 1999 Drops with conical ends in electric and magnetic fields. Proc. R. Soc. Lond. A 455 (1981), 329.Google Scholar