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Deformation of leaky-dielectric fluid globules under strong electric fields: boundary layers and jets at large Reynolds numbers

Published online by Cambridge University Press:  14 October 2013

Ory Schnitzer ,
Itzchak Frankel  and
Ehud Yariv
Ory Schnitzer
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel
Itzchak Frankel
Affiliation:
Department of Aerospace Engineering, 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: [email protected]
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Abstract

In Taylor’s analysis of electrohydrodynamic drop deformation by a uniform electric field (Proc. R. Soc. Lond. A, vol. 291, 1966, pp. 159–166) inertia is neglected at the outset, resulting in fluid velocities that scale as the square of the applied-field magnitude. For large (i.e. millimetric) drops, with increasing field strength the Reynolds number predicted by this scaling may actually become large, suggesting the need for a complementary large-Reynolds-number investigation. Balancing viscous stresses and electrical shear forces in this limit reveals a different velocity scaling, with the $4/ 3$-power of the applied-field magnitude. For simplicity, we focus upon the flow about a spherical gas bubble. It is essentially confined to two boundary layers propagating from the poles to the equator, where they collide to form a radial jet. The transition occurs over a small deflection region about the equator where the flow is effectively inviscid. The deviation of the bubble shape from the original sphericity is quantified by the capillary number given by the ratio of a characteristic Maxwell stress to Laplace’s pressure. At leading order the bubble deforms owing to: (i) the surface distribution of the Maxwell stress, associated with the familiar electric-field profile; (ii) the hydrodynamic boundary-layer pressure, engendered here by centrifugal forces; and (iii) the intense pressure distribution acting over the narrow equatorial deflection zone, appearing on the bubble scale as a concentrated load. Remarkably, the unique flow topology and associated scalings allow the obtaining of a closed-form expression for the deformation through the mere application of integral mass and momentum balances. On the bubble scale, the concentrated pressure load is manifested in the appearance of a non-smooth equatorial dimple.

JFM classification

Boundary Layers: Boundary Layers Drops and Bubbles: Drops and Bubbles Drops and Bubbles: Electrohydrodynamic effects
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 734 , 10 November 2013 , R3
Copyright
©2013 Cambridge University Press 

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References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Baygents, J. C. & Saville, D. A. 1989 The circulation produced in a drop by an electric field: a high field strength electrokinetic model. In Drops & Bubbles, Third International Colloquium, Monterey 1988 (ed. Wang, T.), vol. 7, pp. 7–17. AIP Conference Proceedings.Google Scholar
Feng, J. Q. 1999 Electrohydrodynamic behaviour of a drop subjected to a steady uniform electric field at finite electric Reynolds number. Proc. R. Soc. Lond. A 455 (1986), 2245–2269.CrossRefGoogle Scholar
Feng, J. Q. & Scott, T. C. 1996 A computational analysis of electrohydrodynamics of a leaky dielectric drop in an electric field. J. Fluid Mech. 311, 289–326.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.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics: Course of Theoretical Physics. Pergamon.Google Scholar
Melcher, J. R. & Taylor, G. I. 1969 Electrohydrodynamics: a review of the role of interfacial shear stresses. Annu. Rev. Fluid Mech. 1, 111–146.Google Scholar
Moore, D. W. 1963 The boundary layer on a spherical gas bubble. J. Fluid Mech. 16 (2), 161–176.Google Scholar
Salipante, P. F. & Vlahovska, P. M. 2010 Electrohydrodynamics of drops in strong uniform dc electric fields. Phys. Fluids 22, 112110.CrossRefGoogle Scholar
Saville, D. A. 1997 Electrohydrodynamics: the Taylor–Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29, 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
Stewart, M. B. & Morrison, F. A. 1979 Small Reynolds number electrohydrodynamic flow around drops and the resulting deformation. J. Appl. Mech. 46 (3), 510–512.Google Scholar
Stewartson, K. 1958 On rotating laminar boundary layers. In Boundary Layer Research, Freiburg, Germany, 1957 (ed. H. Görtler), pp. 59–71. IUTAM Symposium, Springer.CrossRefGoogle Scholar
Taylor, G. 1966 Studies in electrohydrodynamics. I. The circulation produced in a drop by electrical field. Proc. R. Soc. Lond. A 291 (1425), 159–166.Google Scholar
Torza, S., Cox, R. G. & Mason, S. G. 1971 Electrohydrodynamic deformation and burst of liquid drop. Phil. Trans. R. Soc. A 269 (1198), 295–319.Google Scholar
Yariv, E. & Rhodes, D. 2013 Electrohydrodynamic drop deformation by strong electric fields: slender-body analysis. SIAM J. Appl. Math. (in press).CrossRefGoogle Scholar
Zholkovskij, E. K., Masliyah, J. H. & Czarnecki, J. 2002 An electrokinetic model of drop deformation in an electric field. J. Fluid Mech. 472, 1–27.Google Scholar