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Electrohydrodynamics of lenticular drops and equatorial streaming

Published online by Cambridge University Press:  31 August 2021

Brayden W. Wagoner ,
Petia M. Vlahovska Open the ORCID record for Petia M. Vlahovska [Opens in a new window] ,
Michael T. Harris  and
Osman A. Basaran Open the ORCID record for Osman A. Basaran [Opens in a new window]
Brayden W. Wagoner
Affiliation:
Davidson School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA
Petia M. Vlahovska
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208, USA
Michael T. Harris
Affiliation:
Davidson School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA
Osman A. Basaran*
Affiliation:
Davidson School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA
*
†Email address for correspondence: obasaran@purdue.edu
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Abstract

Drops subjected to electric fields can deform into singular shapes exhibiting apparent sharp tips. At high field strengths, a perfectly conducting drop surrounded by a perfectly insulating exterior fluid deforms into a prolate-shaped drop with conical ends and can exist in hydrostatic equilibrium. On the conical ends, capillary stress, which is due to the out-of-plane curvature and is singular, balances electric normal stress which is also singular. If the two phases are not perfect conductors/insulators but are both leaky dielectrics and the drop is much more conducting and viscous than the exterior, electric tangential stress disrupts the hydrostatic force balance and leads to jet emission from the cone's apex. If, however, the physical situation is inverted so that a weakly conducting, slightly viscous drop is immersed in a highly conducting, more viscous exterior, the drop deforms into an oblate lens-like profile before eventually becoming unstable. In experiments, the equator of a lenticular drop superficially resembles a wedge prior to instability. Such a drop disintegrates by equatorial streaming by ejecting a thin liquid sheet from its equator. We show theoretically by performing a local analysis that a lenticular drop's equatorial profile can be a wedge only if an approximate form of the surface charge transport equation – continuity of normal current condition – is used. Moreover, we demonstrate via numerical simulation that such wedge-shaped drops do not become unstable and therefore cannot emit equatorial sheets. We then show by transient simulations how equatorial streaming can occur when charge transport along the interface is analysed without approximation.

JFM classification

Drops and Bubbles: Electrohydrodynamic effects Drops and Bubbles: Drops

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JFM Papers
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Journal of Fluid Mechanics , Volume 925 , 25 October 2021 , A36
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© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Abbott, J.P. 1978 An efficient algorithm for the determination of certain bifurcation points. J. Comput. Appl. Maths 4 (1), 19–27.CrossRefGoogle Scholar
Anna, S.L., Bontoux, N. & Stone, H.A. 2003 Formation of dispersions using ‘flow focusing’ in microchannels. Appl. Phys. Lett. 82 (3), 364–366.CrossRefGoogle Scholar
Barrero, A. & Loscertales, I.G. 2007 Micro-and nanoparticles via capillary flows. Annu. Rev. Fluid Mech. 39, 89–106.CrossRefGoogle Scholar
Basaran, O.A. & Scriven, L.E. 1982 Profiles of electrified drops and bubbles. In Proceedings of the Second International Colloquium on Drops and Bubbles (ed. D. H. Le Croissette), pp. 322–329. National Aeronautics and Space Administration.Google Scholar
Basaran, O.A. & Scriven, L.E. 1988 The Taylor pump: viscous-free surface flow driven by electric shear stress. Chem. Engng Commun. 67 (1), 259–273.CrossRefGoogle Scholar
Basaran, O.A. & Scriven, L.E. 1990 Axisymmetric shapes and stability of pendant and sessile drops in an electric field. J. Colloid Interface Sci. 140 (1), 10–30.CrossRefGoogle Scholar
Basaran, O.A. & Wohlhuter, F.K. 1992 Effect of nonlinear polarization on shapes and stability of pendant and sessile drops in an electric (magnetic) field. J. Fluid Mech. 244, 1–16.CrossRefGoogle Scholar
Bentenitis, N. & Krause, S. 2005 Droplet deformation in DC electric fields: the extended leaky dielectric model. Langmuir 21 (14), 6194–6209.CrossRefGoogle Scholar
Berkenbusch, M.K., Cohen, I. & Zhang, W.W. 2008 Liquid interfaces in viscous straining flows: numerical studies of the selective withdrawal transition. J. Fluid Mech. 613, 171–203.CrossRefGoogle Scholar
Betelú, S.I., Fontelos, M.A., Kindelán, U. & Vantzos, O. 2006 Singularities on charged viscous droplets. Phys. Fluids 18 (5), 051706.CrossRefGoogle Scholar
Blake, T.D. & Ruschak, K.J. 1979 A maximum speed of wetting. Nature 282 (5738), 489–491.CrossRefGoogle Scholar
Brosseau, Q. & Vlahovska, P.M. 2017 Streaming from the equator of a drop in an external electric field. Phys. Rev. Lett. 119 (3), 034501.CrossRefGoogle Scholar
Brown, R.A. & Scriven, L.E. 1980 The shapes and stability of captive rotating drops. Phil. Trans. R. Soc. Lond. A 297 (1429), 51–79.Google Scholar
Burton, J.C. & Taborek, P. 2011 Simulations of Coulombic fission of charged inviscid drops. Phys. Rev. Lett. 106 (14), 144501.CrossRefGoogle ScholarPubMed
Castro-Hernández, E., Campo-Cortés, F. & Gordillo, J.M. 2012 Slender-body theory for the generation of micrometre-sized emulsions through tip streaming. J. Fluid Mech. 698, 423–445.CrossRefGoogle Scholar
Christodoulou, K.N. & Scriven, L.E. 1992 Discretization of free surface flows and other moving boundary problems. J. Comput. Phys. 99 (1), 39–55.CrossRefGoogle Scholar
Cohen, I. & Nagel, S.R. 2002 Scaling at the selective withdrawal transition through a tube suspended above the fluid surface. Phys. Rev. Lett. 88 (7), 074501.CrossRefGoogle Scholar
Collins, R.T. 2008 Electrohydrodynamics of free surface flows. PhD thesis, Purdue University, IN.Google Scholar
Collins, R.T., Jones, J.J., Harris, M.T. & Basaran, O.A. 2008 Electrohydrodynamic tip streaming and emission of charged drops from liquid cones. Nat. Phys. 4 (2), 149–154.CrossRefGoogle Scholar
Collins, R.T., Sambath, K., Harris, M.T. & Basaran, O.A. 2013 Universal scaling laws for the disintegration of electrified drops. Proc. Natl Acad. Sci. USA 110 (13), 4905–4910.CrossRefGoogle ScholarPubMed
Das, D. & Saintillan, D. 2017 a Electrohydrodynamics of viscous drops in strong electric fields: numerical simulations. J. Fluid Mech. 829, 127–152.CrossRefGoogle Scholar
Das, D. & Saintillan, D. 2017 b A nonlinear small-deformation theory for transient droplet electrohydrodynamics. J. Fluid Mech. 810, 225–253.CrossRefGoogle Scholar
Deen, W.M. 1998 Analysis of Transport Phenomena. Oxford University Press.Google Scholar
Deshmukh, S.D. & Thaokar, R.M. 2013 Deformation and breakup of a leaky dielectric drop in a quadrupole electric field. J. Fluid Mech. 731, 713–733.CrossRefGoogle Scholar
Esmaeeli, A. & Sharifi, P. 2011 Transient electrohydrodynamics of a liquid drop. Phys. Rev. E 84 (3), 036308.CrossRefGoogle ScholarPubMed
Evangelio, A., Campo-Cortés, F. & Gordillo, J.M. 2016 Simple and double microemulsions via the capillary breakup of highly stretched liquid jets. J. Fluid Mech. 804, 550–577.CrossRefGoogle 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. & Basaran, O.A. 1994 Shear flow over a translationally symmetric cylindrical bubble pinned on a slot in a plane wall. J. Fluid Mech. 275, 351–378.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.CrossRefGoogle Scholar
Fenn, J.B., Mann, M., Meng, C.K., Wong, S.F. & Whitehouse, C.M. 1989 Electrospray ionization for mass spectrometry of large biomolecules. Science 246 (4926), 64–71.CrossRefGoogle ScholarPubMed
Fernández de La Mora, J. 2007 The fluid dynamics of Taylor cones. Annu. Rev. Fluid Mech. 39, 217–243.CrossRefGoogle Scholar
Fontelos, M.A., Kindelán, U. & Vantzos, O. 2008 Evolution of neutral and charged droplets in an electric field. Phys. Fluids 20 (9), 092110.CrossRefGoogle Scholar
Ganán-Calvo, A.M., López-Herrera, J.M., Herrada, M.A., Ramos, A. & Montanero, J.M. 2018 Review on the physics of electrospray: from electrokinetics to the operating conditions of single and coaxial Taylor cone-jets, and AC electrospray. J. Aero. Sci. 125, 32–56.CrossRefGoogle Scholar
Gilbert, W. 1958 De magnete (first published in Latin in 1600 and translated by P.F. Mottelay in 1893). Dover.Google Scholar
Gordillo, J.M., Sevilla, A. & Campo-Cortés, F. 2014 Global stability of stretched jets: conditions for the generation of monodisperse micro-emulsions using coflows. J. Fluid Mech. 738, 335–357.CrossRefGoogle Scholar
Harris, M.T., Scott, T.C. & Byers, C.H. 1992 Method and apparatus for the production of metal oxide powder. US Patent 5,122,360.Google Scholar
Huh, C. & Scriven, L.E. 1971 Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35 (1), 85–101.CrossRefGoogle Scholar
Joffre, G., Prunet-Foch, B., Berthomme, S. & Cloupeau, M. 1982 Deformation of liquid menisci under the action of an electric field. J. Electrost. 13 (2), 151–165.CrossRefGoogle Scholar
Kamal, C., Sprittles, J.E., Snoeijer, J.H. & Eggers, J. 2019 Dynamic drying transition via free-surface cusps. J. Fluid Mech. 858, 760–786.CrossRefGoogle Scholar
Kistler, S.F. & Scriven, L.E. 1983 Coating flows. In Computational Analysis of Polymer Processing (ed. J.R.A. Pearson & S.M. Richardson), vol. 1, pp. 243–299. Springer.CrossRefGoogle 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
Lanauze, J.A., Walker, L.M. & Khair, A.S. 2013 The influence of inertia and charge relaxation on electrohydrodynamic drop deformation. Phys. Fluids 25 (11), 112101.CrossRefGoogle Scholar
Lanauze, J.A., Walker, L.M. & Khair, A.S. 2015 Nonlinear electrohydrodynamics of slightly deformed oblate drops. J. Fluid Mech. 774, 245–266.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 (8), 575–580.CrossRefGoogle Scholar
Macky, W.A. 1930 The deformation of soap bubbles in electric fields. Math. Proc. Camb. Phil. Soc. 26 (3), 421–428.CrossRefGoogle Scholar
Macky, W.A. 1931 Some investigations on the deformation and breaking of water drops in strong electric fields. Proc. R. Soc. Lond. A 133 (822), 565–587.Google Scholar
Marín, A.G. 2021 The Saturnian droplet. J. Fluid Mech. 908, F1.CrossRefGoogle Scholar
Marín, A.G., Loscertales, I.G. & Barrero, A. 2008 Conical tips inside cone-jet electrosprays. Phys. Fluids 20 (4), 042102.CrossRefGoogle Scholar
Marín, A.G., Loscertales, I.G., Marquez, M. & Barrero, A. 2007 Simple and double emulsions via coaxial jet electrosprays. Phys. Rev. Lett. 98 (1), 014502.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Michell, J.H. 1899 On the direct determination of stress in an elastic solid, with application to the theory of plates. Proc. Lond. Math. Soc. 1 (1), 100–124.CrossRefGoogle Scholar
Miksis, M.J. 1981 Shape of a drop in an electric field. Phys. Fluids 24 (11), 1967–1972.CrossRefGoogle Scholar
Nolan, J.J. 1924 The breaking of water-drops by electric fields. Proc. R. Irish Acad. A 37, 28–39.Google Scholar
Notz, P.K. & Basaran, O.A. 2004 Dynamics and breakup of a contracting liquid filament. J. Fluid Mech. 512, 223–256.CrossRefGoogle Scholar
Ptasinski, K.J. & Kerkhof, P.J.A.M. 1992 Electric field driven separations: phenomena and applications. Sep. Sci. Technol. 27 (8–9), 995–1021.CrossRefGoogle Scholar
Ramos, A. & Castellanos, A. 1994 a Conical points in liquid-liquid interfaces subjected to electric fields. Phys. Lett. A 184 (3), 268–272.CrossRefGoogle Scholar
Ramos, A. & Castellanos, A. 1994 b Equilibrium shapes and bifurcation of captive dielectric drops subjected to electric fields. J. Electrost. 33 (1), 61–86.CrossRefGoogle Scholar
Rayleigh, Lord 1882 On the equilibrium of liquid conducting masses charged with electricity. Phil. Mag. 14 (87), 184–186.CrossRefGoogle Scholar
Sambath, K. & Basaran, O.A. 2014 Electrohydrostatics of capillary switches. AIChE J. 60 (4), 1451–1459.CrossRefGoogle Scholar
Saville, D.A. 1997 Electrohydrodynamics: the Taylor–Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29 (1), 27–64.CrossRefGoogle Scholar
Scott, T.C., DePaoli, D.W. & Sisson, W.G. 1994 Further development of the electrically driven emulsion-phase contactor. Ind. Engng Chem. Res. 33 (5), 1237–1244.CrossRefGoogle Scholar
Scott, T.C. & Wham, R.M. 1988 Surface area generation and droplet size control in solvent extraction systems utilizing high intensity electric fields. US Patent 4,767,515.Google Scholar
Scott, T.C. & Wham, R.M. 1989 An electrically driven multistage countercurrent solvent extraction device: the emulsion-phase contactor. Ind. Engng Chem. Res. 28 (1), 94–97.CrossRefGoogle Scholar
Scriven, L.E. & Suszynski, W.J. 1990 Take a closer look at coating problems. Chem. Engng Prog. 86 (9), 24–29.Google Scholar
Simpkins, P.G. & Kuck, V.J. 2000 Air entrapment in coatings by way of a tip-streaming meniscus. Nature 403 (6770), 641–643.CrossRefGoogle ScholarPubMed
Smith, C.V. & Melcher, J.R. 1967 Electrohydrodynamically induced spatially periodic cellular stokes-flow. Phys. Fluids 10 (11), 2315–2322.CrossRefGoogle Scholar
Suryo, R. & Basaran, O.A. 2006 Tip streaming from a liquid drop forming from a tube in a co-flowing outer fluid. Phys. Fluids 18 (8), 082102.CrossRefGoogle Scholar
Taylor, G.I. 1934 The formation of emulsions in definable fields of flow. Proc. R. Soc. Lond. A 146 (858), 501–523.Google Scholar
Taylor, G.I. 1964 Disintegration of water drops in an electric field. Proc. R. Soc. Lond. A 280 (1382), 383–397.Google Scholar
Taylor, G.I. 1966 Studies in electrohydrodynamics. I. The circulation produced in a drop by an electric field. Proc. R. Soc. Lond. A 291 (1425), 159–166.Google Scholar
Tseng, Y.-H. & Prosperetti, A. 2015 Local interfacial stability near a zero vorticity point. J. Fluid Mech. 776, 5–36.CrossRefGoogle Scholar
Ungar, L.H. & Brown, R.A. 1982 The dependence of the shape and stability of captive rotating drops on multiple parameters. Phil. Trans. R. Soc. Lond. A 306 (1493), 347–370.Google Scholar
Vlahovska, P.M. 2019 Electrohydrodynamics of drops and vesicles. Annu. Rev. Fluid Mech. 51, 305–330.CrossRefGoogle Scholar
Wagoner, B.W., Vlahovska, P.M., Harris, M.T. & Basaran, O.A. 2020 Electric-field-induced transitions from spherical to discocyte and lens-shaped drops. J. Fluid Mech. 904, R4.CrossRefGoogle Scholar
Waterman, L.C. 1965 Electrical coalescers: theory and practice. Chem. Engng Prog. 61 (18), 51–57.Google Scholar
Wilson, C.T.R. & Taylor, G.I. 1925 The bursting of soap-bubbles in a uniform electric field. Math. Proc. Camb. Phil. Soc. 22 (5), 728–730.CrossRefGoogle Scholar
Wohlhuter, F.K. & Basaran, O.A. 1992 Shapes and stability of pendant and sessile dielectric drops in an electric field. J. Fluid Mech. 235, 481–510.CrossRefGoogle Scholar
Yamaguchi, Y., Chang, C.J. & Brown, R.A. 1984 Multiple buoyancy-driven flows in a vertical cylinder heated from below. Phil. Trans. R. Soc. Lond. A 312 (1523), 519–552.Google Scholar
Zabarankin, M. 2013 A liquid spheroidal drop in a viscous incompressible fluid under a steady electric field. SIAM J. Appl. Maths 73 (2), 677–699.CrossRefGoogle Scholar
Zahn, M. & Shumovich, R. 1985 Labyrinthine instability in dielectric fluids. IEEE Trans. Ind. Applics. IA-21 (1), 53–61.CrossRefGoogle Scholar
Zeleny, J. 1914 The electrical discharge from liquid points, and a hydrostatic method of measuring the electric intensity at their surfaces. Phys. Rev. 3 (2), 69–91.CrossRefGoogle Scholar
Zeleny, J. 1917 Instability of electrified liquid surfaces. Phys. Rev. 10 (1), 1–6.CrossRefGoogle Scholar