We theoretically investigate the deformation of a perfect dielectric drop suspended in a second dielectric liquid subject to a uniform electric field. Axisymmetric equilibrium shapes are found by solving simultaneously the Young–Laplace equation at the interface and Laplace equation for the electric field. Analytical solutions are constructed for the governing nonlinear boundary-value problem using domain perturbation method together with a special type of Hermite–Padé approximation. The results show the existence of a critical electric capillary number beyond which no axisymmetric figure is possible.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Yangsoo, S., Kim, C.: Spreading of inkjet droplet of non-Newtonian fluid on solid surface with controlled contact angle at low Weber and Reynolds numbers. J. Non-Newtonian Fluid Mech. 162(1–3), 78–87 (2009)
Bailey, G.: Electrostatic Spraying of Liquids. Wiley, New York (1988)
Bienia, M., Quilliet, C., Vallade, M.: Modification of drop shape controlled by electrowetting. Langmuir 19(22), 9328–9333 (2003)
Kebarle, P.: A brief overview of the present status of the mechanisms involved in electrospray mass spectrometry. J. Mass Spectrom. 35, 804–817 (2000)
Brandenbourger, M., Caps, H., Vitry, Y., Dorbolo, S.: Electrically charged droplets in microgravity. Microgravity Sci. Technol. 29(3), 229–239 (2017)
Song, H., Chen, D.L., Ismagilov, R.F.: Reactions in droplets in microfluidic channels. Angewandte Chemie Iinternational Edition 45, 7336–7356 (2006)
Zaghdoudi, M.C., Lallemand, M.: Electric field effects on pool boiling. J. Enhanced Heat Transf. 9(5–6), 187–208 (2002)
McDonald, J.E.: The shape and aerodynamics of large raindrops. J. Meteorol. 11(6), 478–494 (1954)
Beard, K.V., Feng, J., Chuang, C.C.: A Simple perturbation model for the electrostatic shape of falling drops. J. Atmos. Sci. 46(15), 2404–2418 (1989)
Thiam, A.R., Bremond, N., Bibette, J.: Breaking of an emulsion under an ac electric field. Phys. Rev. Lett. 102, 188304 (2009)
Drelich, J., Bryll, G., Kapczynski, J., Hupka, J., Miller, J.D., Hanson, F.V.: The effect of electric field pulsation frequency on breaking water-in-oil emulsions. Fuel Process. Technol. 31(2), 105–113 (1992)
Bernard Cohen, I.: Benjamin Franklin’s Experiments: A New Edition of Franklin’s Experiments and observations on electricity. Harvard University Press, Cambridge (1941)
Rayleigh, Lord: On the equilibrium of liquid conducting masses charged with electricity. Philos. Mag. 14, 184–186 (1882)
O’Konski, C.T., Thacher, H.C.: The distortion of aerosol droplets by an electric field. J. Phys. Chem. 57, 955–958 (1953)
Allan, R.S., Mason, S.G.: Particle behaviour in shear and electric fields I. Deformation and burst of fluid drops. Proc. R. Soc. Lond. A 267, 45–61 (1962)
Garton, C.G., Krasucki, Z.: Bubbles in insulating liquids: stability in an electric field. Proc. R. Soc. Lond. A 280, 211–226 (1964)
Taylor, G.I.: Studies in electrohydrodynamics: I. The circulation produced in a drop by an electric field. Proc. R. Soc. Lond. A 280, 383–397 (1966)
Taylor, G.I.: Disintegration of water drops in an electric field. Proc. R. Soc. Lond. A 291, 159–166 (1964)
Saville, D.A.: Electrohydrodynamics: the Taylor–Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29, 27–64 (1997)
Rosenkilde, C.E.: A dielectric fluid drop in an electric field. Proc. R. Soc. Lond. A 312, 473–494 (1969)
Ajayi, O.O.: A note on Taylor’s electrohydrodynamic theory. Proc. Roy. Soc. Lond. A 364, 499–507 (1978)
Miksis, M.J.: Shape of a drop in an electric field. Phys. Fluids 24, 1967–1972 (1981)
Dodgson, V., Sozou, C.: The deformation of a liquid drop by an electric field. ZAMP 38, 424–432 (1987)
Sherwood, J.D.: Breakup of fluid droplets in electric and magnetic fields. J. Fluid Mech. 188, 133–146 (1988)
Basaran, O.A., Scriven, L.E.: Axisymmetric shapes and stability of charged drops in an external electric field. Phys. Fluids A 1, 799–809 (1989)
Li, H., Halsey, T.C., Lobkovsky, A.: Singular shape of a fluid drop in an electric or magnetic field. Europhys. Lett. 27, 575–580 (1994)
Feng, J.Q., Scott, T.C.: A computational analysis of electrohydrodynamics of a leaky dielectric drop in an electric field. J. Fluid Mech. 311, 289–326 (1996)
Stone, H.A., Lister, J.R., Brenner, M.P.: Drops with conical ends in electric and magnetic fields. Proc. Roy. Soc. Lond. A 455, 329–347 (1999)
Shaw, S.J., Shaw, S.P.D.M.: Critical strength of an electric field whereby a bubble can adopt a steady shape. Proc. R. Soc. A Math. Phys. Eng. Sci. 465(2110), 3127–3143 (2009)
Bjorklund, E.: The level-set method applied to droplet dynamics in the presence of an electric field. Comput. Fluids 38(2), 358–369 (2009)
Paknemat, H., Pishevar, A.R., Pournaderi, P.: Numerical simulation of drop deformations and breakup modes caused by direct current electric fields. Phys. Fluids 24, 102101 (2012)
Melcher, J.R., Taylor, G.I.: Electrohydrodynamics: a review of the role of interfacial shear stresses. Annu. Rev. Fluid Mech. 1, 111–146 (1969)
Joseph, D.D.: Domain perturbations: the higher order theory of infinitesimal water waves. Arch. Ration. Mech. Anal. 51, 295–303 (1973)
Sattinger D.H.: Topics in stability and bifurcation theory. Springer, Lecture Notes in Mathematics, 309 (1973)
Guttmann, A.J.: Asymptotic analysis of power-series expansions. in Phase Transitions and Critical Phenomena Vol. 13, 1-234, eds C. Domb and J.L Lebowitz (Academic Press, New York) (1989)
Common, A.K.: Applications of Hermite–Padé approximants to water waves and the harmonic oscillator on a lattice. J. Phys. A 15, 3665–3677 (1982)
Makinde, O.: On thermal stability of a reactive third-grade fluid in a channel with convective cooling the walls. Appl. Math. Comput. 213(1), 170–176 (2009)
Er-Riani, M., El Jarroudi, M., Sero-Guillaume, O.: Hermite–Padé approximation approach to shapes of rotating drops. Appl. Math. Mod. 38, 212–220 (2014)
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Filali, Y., Er-Riani, M. & El Jarroudi, M. Deformation of a fluid drop subjected to a uniform electric field. Z. Angew. Math. Phys. 72, 12 (2021). https://doi.org/10.1007/s00033-020-01439-w
- Drop deformation
- Drop breakup
- Electrohydrodynamic effect
- Electric capillary number
Mathematics Subject Classification