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Electrohydrodynamics of viscous drops in strong electric fields: numerical simulations

Published online by Cambridge University Press:  14 September 2017

Debasish Das  and
David Saintillan Open the ORCID record for David Saintillan [Opens in a new window]
Debasish Das
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA
David Saintillan*
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA
*
†Email address for correspondence: dstn@ucsd.edu
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Abstract

Weakly conducting dielectric liquid drops suspended in another dielectric liquid and subject to an applied uniform electric field exhibit a wide range of dynamical behaviours contingent on field strength and material properties. These phenomena are best described by the Melcher–Taylor leaky dielectric model, which hypothesizes charge accumulation on the drop–fluid interface and prescribes a balance between charge relaxation, the jump in ohmic currents from the bulk and charge convection by the interfacial fluid flow. Most previous numerical simulations based on this model have either neglected interfacial charge convection or restricted themselves to axisymmetric drops. In this work, we develop a three-dimensional boundary element method for the complete leaky dielectric model to systematically study the deformation and dynamics of liquid drops in electric fields. The inclusion of charge convection in our simulations permits us to investigate drops in the Quincke regime, in which experiments have demonstrated a symmetry-breaking bifurcation leading to steady electrorotation. Our simulation results show excellent agreement with existing experimental data and small-deformation theories.

JFM classification

Low-Reynolds-number flows: Boundary integral methods Drops and Bubbles: Drops Drops and Bubbles: Electrohydrodynamic effects
Type
Papers
Information
Journal of Fluid Mechanics , Volume 829 , 25 October 2017 , pp. 127 - 152
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© 2017 Cambridge University Press 

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Footnotes

‡

Present address: Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK.

References

Ajayi, O. O. 1978 A note on Taylor’s electrohydrodynamic theory. Proc. R. Soc. Lond. A 364, 499–507.Google Scholar
Allan, R. S. & Mason, S. G. 1962 Particle behaviour in shear and electric fields. I. Deformation and burst of fluid drops. Proc. R. Soc. Lond. A 267, 45–61.Google Scholar
Bandopadhyay, A., Mandal, S., Kishore, N. K. & Chakraborty, S. 2016 Uniform electric-field-induced lateral migration of a sedimenting drop. J. Fluid Mech. 792, 553–589.Google Scholar
Basaran, O. A., Gao, H. & Bhat, P. P. 2013 Nonstandard inkjets. Annu. Rev. Fluid Mech. 45, 85–113.Google Scholar
Baygents, J. C., Rivette, N. J. & Stone, H. A. 1998 Electrohydrodynamic deformation and interaction of drop pairs. J. Fluid Mech. 368, 359–375.CrossRefGoogle Scholar
Bjorklund, E. 2009 The level-set method applied to droplet dynamics in the presence of an electric field. Comput. Fluids 38, 358–369.CrossRefGoogle Scholar
Blanchard, D. C. 1963 The electrification of the atmosphere by particles from bubbles in the sea. Prog. Oceanogr. 1, 73–202.Google Scholar
Brazier-Smith, P. R. 1971 Stability and shape of isolated and pairs of water drops in an electric field. Phys. Fluids 14, 1–6.CrossRefGoogle Scholar
Brazier-Smith, P. R., Jennings, S. G. & Latham, J 1971 An investigation of the behaviour of drops and drop-pairs subjected to strong electrical forces. Proc. R. Soc. Lond. A 325, 363–376.Google Scholar
Castellanos, A. 2014 Electrohydrodynamics. Springer.Google Scholar
Das, D. & Saintillan, D. 2013 Electrohydrodynamic interaction of spherical particles under Quincke rotation. Phys. Rev. E 87, 043014.CrossRefGoogle ScholarPubMed
Das, D. & Saintillan, D. 2017 A nonlinear small-deformation theory for transient droplet electrohydrodynamics. J. Fluid Mech. 810, 225–253.Google Scholar
Dommersnes, P., Mikkelsen, A. & Fossum, J. 2016 Electro-hydrodynamic propulsion of counter-rotating pickering drops. J. Eur. Phys. J. Spec. Top. 225, 699–706.CrossRefGoogle Scholar
Dubash, N. & Mestel, A. J. 2007a Behaviour of a conducting drop in a highly viscous fluid subject to an electric field. J. Fluid Mech. 581, 469–493.CrossRefGoogle Scholar
Dubash, N. & Mestel, A. J. 2007b Breakup behavior of a conducting drop suspended in a viscous fluid subject to an electric field. Phys. Fluids 19, 072101.Google Scholar
Eow, J. S. & Ghadiri, M. 2002 Electrostatic enhancement of coalescence of water droplets in oil: a review of the technology. Chem. Engng J. 85, 357–368.CrossRefGoogle Scholar
Esmaeeli, A. & Sharifi, P. 2011 Transient electrohydrodynamics of a liquid drop. Phys. Rev. E 84, 036308.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, 2245–2269.CrossRefGoogle Scholar
Feng, J. Q. 2002 A 2D electrohydrodynamic model for electrorotation of fluid drops. J. Colloid Interface Sci. 246, 112–121.CrossRefGoogle ScholarPubMed
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
Ha, J.-W. & Yang, S.-M. 2000a Deformation and breakup of Newtonian and non-Newtonian conducting drops in an electric field. J. Fluid Mech. 405, 131–156.Google Scholar
Ha, J.-W. & Yang, S.-M. 2000b Electrohydrodynamics and electrorotation of a drop with fluid less conductive than that of the ambient fluid. Phys. Fluids 12, 764–772.CrossRefGoogle Scholar
Ha, J.-W. & Yang, S.-M. 2000c Rheological responses of oil-in-oil emulsions in an electric field. J. Rheol. 44, 235–256.CrossRefGoogle Scholar
Harris, F. E. & O’Konski, C. T. 1957 Dielectric properties of aqueous ionic solutions at microwave frequencies. J. Phys. Chem. 61, 310–319.Google Scholar
Haywood, R. J., Renksizbulut, M. & Raithby, G. D. 1991 Transient deformation of freely-suspended liquid droplets in electrostatic fields. AIChE J. 37, 1305–1317.Google Scholar
He, H., Salipante, P. F. & Vlahovska, P. M. 2013 Electrorotation of a viscous droplet in a uniform direct current electric field. Phys. Fluids 25, 032106.Google Scholar
Hirata, T., Kikuchi, T., Tsukada, T. & Hozawa, M. 2000 Finite element analysis of electrohydrodynamic time-dependent deformation of dielectric drop under uniform DC electric field. J. Chem. Engng Japan 33, 160–167.CrossRefGoogle Scholar
Hu, W.-F., Lai, M.-C. & Young, Y.-N. 2015 A hybrid immersed boundary and immersed interface method for electrohydrodynamic simulations. J. Comput. Phys. 282, 47–61.CrossRefGoogle Scholar
Hu, C. & Shu, C.-W. 1999 Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150, 97–127.Google Scholar
Huang, Z.-M., Zhang, Y.-Z., Kotaki, M. & Ramakrishna, S. 2003 A review on polymer nanofibers by electrospinning and their applications in nanocomposites. Compos. Sci. Technol. 63, 2223–2253.CrossRefGoogle Scholar
Jaswon, M. A. 1963 Integral equation methods in potential theory. I. Proc. R. Soc. Lond. A 275, 23–32.Google Scholar
Jones, T. B. 1984 Quincke rotation of spheres. IEEE Trans. Ind. Applics IA‐20, 845–849.Google Scholar
Kennedy, M. R., Pozrikidis, C. & Skalak, R. 1994 Motion and deformation of liquid drops, and the rheology of dilute emulsions in simple shear flow. Comput. Fluids 23, 251–278.Google Scholar
Kim, S. & Karrila, S. J. 2013 Microhydrodynamics: Principles and Selected Applications. Dover.Google Scholar
Krause, S. & Chandratreya, P. 1998 Electrorotation of deformable fluid droplets. J. Colloid Interface Sci. 206, 10–18.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
Lanauze, J. A., Walker, L. M. & Khair, A. S. 2013 The influence of inertia and charge relaxation on electrohydrodynamic drop deformation. Phys. Fluids 25, 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
Landau, L. D., Lifshitz, E. M. & Pitaevskiì, L. P. 1984 Electrodynamics of Continuous Media. Elsevier.Google Scholar
Laser, D. J. & Santiago, J. G. 2004 A review of micropumps. J. Micromech. Microengng 14, R35.Google Scholar
Loewenberg, M. & Hinch, E. J. 1996 Numerical simulation of a concentrated emulsion in shear flow. J. Fluid Mech. 321, 395–419.Google Scholar
López-Herrera, J. M., Popinet, S. & Herrada, M. A. 2011 A charge-conservative approach for simulating electrohydrodynamic two-phase flows using volume-of-fluid. J. Comput. Phys. 230, 1939–1955.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
Miksis, M. J. 1981 Shape of a drop in an electric field. Phys. Fluids 24, 1967–1972.Google Scholar
O’Konski, C. T. & Thacher, H. C. 1953 The distortion of aerosol droplets by an electric field. J. Phys. Chem. 57, 955–958.Google Scholar
Pannacci, N., Lemaire, E. & Lobry, L. 2007 Rheology and structure of a suspension of particles subjected to quincke rotation. Rheol. Acta 46, 899–904.CrossRefGoogle Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.Google Scholar
Pozrikidis, C. 2002 A Practical Guide to Boundary Element Methods with the Software Library BEMLIB. CRC Press.Google Scholar
Pozrikidis, C. 2011 Introduction to Theoretical and Computational Fluid Dynamics. Oxford University Press.Google Scholar
Quincke, G. 1896 Über rotationen im constanten electrischen Felde. Ann. Phys. Chem. 59, 417–486.Google Scholar
Rallison, J. M. & Acrivos, A. 1978 A numerical study of the deformation and burst of a viscous drop in an extensional flow. J. Fluid Mech. 89, 191–200.Google Scholar
Saad, Y. & Schultz, M. H. 1986 GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869.CrossRefGoogle Scholar
Salipante, P. F. & Vlahovska, P. M. 2010 Electrohydrodynamics of drops in strong uniform DC electric fields. Phys. Fluids 22, 112110.CrossRefGoogle Scholar
Salipante, P. F. & Vlahovska, P. M. 2013 Electrohydrodynamic rotations of a viscous droplet. Phys. Rev. E 88, 043003.Google Scholar
Sato, H., Kaji, N., Mochizuki, T. & Mori, Y. H. 2006 Behavior of oblately deformed droplets in an immiscible dielectric liquid under a steady and uniform electric field. Phys. Fluids 18, 127101.Google Scholar
Schnitzer, O. & Yariv, E. 2015 The Taylor–Melcher leaky dielectric model as a macroscale electrokinetic description. J. Fluid Mech. 773, 1–33.Google Scholar
Schramm, L. L. 1992 Emulsions: Fundamentals and Applications in the Petroleum Industry. American Chemical Society.Google Scholar
Sellier, A. 2006 On the computation of the derivatives of potentials on a boundary by using boundary-integral equations. Comput. Meth. Appl. Mech. Engng 196, 489–501.CrossRefGoogle Scholar
Sherwood, J. D. 1988 Breakup of fluid droplets in electric and magnetic fields. J. Fluid Mech. 188, 133–146.Google Scholar
Shkadov, V. Y. & Shutov, A. A. 2002 Drop and bubble deformation in an electric field. Fluid Dyn. 37, 713–724.Google Scholar
Simpson, G. C. 1909 On the electricity of rain and its origin in thunderstorms. Phil. Trans. R. Soc. Lond. A 209, 379–413.Google Scholar
Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering flows in small devices: microfluidics toward a lab-on-a-chip. Annu. Rev. Fluid Mech. 36, 381–411.Google Scholar
Supeene, G., Koch, C. R. & Bhattacharjee, S. 2008 Deformation of a droplet in an electric field: nonlinear transient response in perfect and leaky dielectric media. J. Colloid Interface Sci. 318, 463–476.Google Scholar
Symm, G. T. 1963 Integral equation methods in potential theory. II. Proc. R. Soc. Lond. A 275, 33–46.Google Scholar
Taylor, G. I. 1964 Disintegration of water drops in an electric field. Proc. R. Soc. Lond. A 280, 383–397.Google Scholar
Taylor, G. I. 1966 Studies in electrohydrodynamics. I. The circulation produced in a drop by electrical field. Proc. R. Soc. Lond. A 291, 159–166.Google Scholar
Taylor, G. I. 1969 Electrically driven jets. Proc. R. Soc. Lond. A 313, 453–475.Google Scholar
Torza, S., Cox, R. G. & Mason, S. G. 1971 Electrohydrodynamic deformation and burst of liquid drops. Phil. Trans. R. Soc. Lond. A 269, 295–319.Google Scholar
Tyatyushkin, A. N.2017 Unsteady electrorotation of a drop in a constant electric field. arXiv:1703.00434 [physics.flu-dyn].Google Scholar
Varshney, A., Ghosh, S., Bhattacharya, S. & Yethiraj, A. 2012 Self organization of exotic oil-in-oil phases driven by tunable electrohydrodynamics. Sci. Rep. 2, 738.Google Scholar
Varshney, A., Gohil, S., Sathe, M., Rv, S. R., Joshi, J. B., Bhattacharya, S., Yethiraj, A. & Ghosh, S. 2016 Multiscale flow in an electro-hydrodynamically driven oil-in-oil emulsion. Soft Matt. 12, 1759–1764.Google Scholar
Veerapaneni, S. 2016 Integral equation methods for vesicle electrohydrodynamics in three dimensions. J. Comput. Phys. 326, 278–289.CrossRefGoogle Scholar
Vlahovska, P. M. 2011 On the rheology of a dilute emulsion in a uniform electric field. J. Fluid Mech. 670, 481–503.Google Scholar
Xu, X. & Homsy, G. M. 2006 The settling velocity and shape distortion of drops in a uniform electric field. J. Fluid Mech. 564, 395–414.Google Scholar
Yariv, E. & Almog, Y. 2016 The effect of surface-charge convection on the settling velocity of spherical drops in a uniform electric field. J. Fluid Mech. 797, 536–548.Google Scholar
Yariv, E. & Frankel, I. 2016 Electrohydrodynamic rotation of drops at large electric Reynolds numbers. J. Fluid Mech. 788, R2.Google Scholar
Yon, S. & Pozrikidis, C. 1998 A finite-volume/boundary-element method for flow past interfaces in the presence of surfactants, with application to shear flow past a viscous drop. Comput. Fluids 27, 879–902.Google Scholar
Zabarankin, M. 2013 A liquid spheroidal drop in a viscous incompressible fluid under steady electric field. SIAM J. Appl. Math. 73, 677–699.Google Scholar
Zhang, J., Zahn, J. D. & Lin, H. 2013 Transient solution for droplet deformation under electric fields. Phys. Rev. E 87, 043008.Google Scholar
Zinchenko, A. Z. & Davis, R. H. 2000 An efficient algorithm for hydrodynamical interaction of many deformable drops. J. Comput. Phys. 157, 539–587.Google Scholar
Zinchenko, A. Z., Rother, M. A. & Davis, R. H. 1997 A novel boundary-integral algorithm for viscous interaction of deformable drops. Phys. Fluids 9, 1493–1511.Google Scholar

Das and Saintillan supplementary material movie 1

Movie showing the deformation and flow field in the simulations of figure 4

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