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Nonlinear three-dimensional electrohydrodynamic interactions of viscous leaky dielectric drops

Michael A. McDougall1,*, Stephen K. Wilson1,2,†, and Debasish Das1,‡

  • 1Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, United Kingdom
  • 2Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom

  • *Contact author: michael.mcdougall@strath.ac.uk
  • †Contact author: sw3197@bath.ac.uk
  • ‡Contact author: debasish.das@strath.ac.uk
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Phys. Rev. Fluids 11, 103702 – Published 19 October, 2026

DOI: https://doi.org/10.1103/lrkw-739x

Abstract

When a drop of a leaky dielectric fluid is suspended in another fluid and subjected to a uniform dc electric field, it becomes polarized, leading to tangential electric stresses that drive fluid motion both inside and outside the drop. In the presence of a second drop, the dynamics of the first drop are altered due to electrohydrodynamic interactions with the second, causing the drops to translate due to dielectrophoretic forces and hydrodynamic interactions. We present a semianalytical nonlinear three-dimensional small deformation theory for a pair of identical, widely separated leaky dielectric drops suspended in a weakly conducting fluid. This theory is valid under conditions of large drop separation, high drop viscosity, and high surface tension, ensuring that the drops remain nearly spherical. The resulting model is semiempirical, with selected higher-order terms retained rather than obtained through a systematic asymptotic expansion. For the first time, we develop a model within the Taylor-Melcher leaky dielectric framework that incorporates both transient charge relaxation and convection. This allows the model to capture the transition to Quincke rotation, a symmetry-breaking phenomenon in which drops begin to spontaneously rotate in sufficiently strong fields. We derive and numerically integrate coupled nonlinear ordinary differential equations for the dipole moments, shapes, and positions of the drops. Our results show good quantitative agreement with previous numerical and experimental work in the limit of zero charge relaxation and convection, and the model reduces to existing models for isolated drops and interacting solid spheres in the relevant limits. We also discuss the hysteresis in the onset of Quincke rotation of isolated drops observed in experiments. Various trajectories for pairs of drops undergoing Quincke rotation are presented, along with results for fixed drops, including their shape, the nature of electrohydrodynamic interactions (attractive or repulsive), and the conditions leading to Quincke rotation. In particular, it is shown that the onset of Quincke rotation for a pair of drops is qualitatively different from that for an isolated drop due to electrohydrodynamic interactions and from that for a pair of solid spheres due to straining flows present only in drops.

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References (67)

  1. J. R. Melcher and G. I. Taylor, Electrohydrodynamics: A review of the role of interfacial shear stresses, Annu. Rev. Fluid Mech. 1, 111 (1969).
  2. 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).
  3. O. A. Basaran, H. Gao, and P. P. Bhat, Nonstandard inkjets, Annu. Rev. Fluid Mech. 45, 85 (2013).
  4. J. Seyed-Yagoobi, Electrohydrodynamic pumping of dielectric liquids, J. Electrost. 63, 861 (2005).
  5. J. S. Eow and M. Ghadiri, Electrostatic enhancement of coalescence of water droplets in oil: A review of the technology, Chem. Eng. J. 85, 357 (2002).
  6. K. J. Ptasinski and P. J. A. M. Kerkhof, Electric field driven separations: Phenomena and applications, Sep. Sci. Technol. 27, 995 (1992).
  7. C. T. O'Konski and H. C. Thacher, The distortion of aerosol droplets by an electric field, J. Phys. Chem. 57, 955 (1953).
  8. R. S. Allan and S. G. Mason, Particle behaviour in shear and electric fields I. Deformation and burst of fluid drops, Proc. R. Soc. A 267, 45 (1962).
  9. C. G. Garton and Z. Krasucki, Bubbles in insulating liquids: Stability in an electric field, Proc. R. Soc. A 280, 211 (1964).
  10. G. I. Taylor, Studies in electrohydrodynamics. I. The circulation produced in a drop by an electric field, Proc. R. Soc. A 291, 159 (1966).
  11. D. A. Saville, Electrohydrodynamics: The Taylor–Melcher leaky dielectric model, Annu. Rev. Fluid Mech. 29, 27 (1997).
  12. D. T. Papageorgiou, Film flows in the presence of electric fields, Annu. Rev. Fluid Mech. 51, 155 (2019).
  13. P. M. Vlahovska, Electrohydrodynamics of drops and vesicles, Annu. Rev. Fluid Mech. 51, 305 (2019).
  14. S. Torza, R. G. Cox, and S. G. Mason, Electrohydrodynamic deformation and bursts of liquid drops, Philos. Trans. R. Soc. A 269, 295 (1971).
  15. O. O. Ajayi, A note on Taylor's electrohydrodynamic theory, Proc. R. Soc. A 364, 499 (1978).
  16. J. A. Lanauze, L. M. Walker, and A. S. Khair, Nonlinear electrohydrodynamics of slightly deformed oblate drops, J. Fluid Mech. 774, 245 (2015).
  17. D. Das and D. Saintillan, Electrohydrodynamics of viscous drops in strong electric fields: Numerical simulations, J. Fluid Mech. 829, 127 (2017).
  18. G. G. Peng, R. Brandão, E. Yariv, and O. Schnitzer, Equatorial blowup and polar caps in drop electrohydrodynamics, Phys. Rev. Fluids 9, 083701 (2024).
  19. G. G. Peng and O. Schnitzer, Bistability and charge-density blowup in the onset of drop Quincke rotation, Phys. Rev. Fluids 10, L081701 (2025).
  20. Q. Brosseau and P. M. Vlahovska, Streaming from the equator of a drop in an external electric field, Phys. Rev. Lett. 119, 034501 (2017).
  21. B. W. Wagoner, P. M. Vlahovska, M. T. Harris, and O. A. Basaran, Electrohydrodynamics of lenticular drops and equatorial streaming, J. Fluid Mech. 925, A36 (2021).
  22. W. Weiler, Zur Darstellung elektrischer Kraftlinien, Z. Phys. Chem. Unterricht 6, 194 (1893).
  23. G. Quincke, Ueber Rotationen im constanten electrischen Felde, Ann. Phys. 295, 417 (1896).
  24. I. Turcu, Electric field induced rotation of spheres, J. Phys. A: Math. Gen. 20, 3301 (1987).
  25. T. B. Jones, Quincke rotation of spheres, IEEE Trans. Ind. Appl. IA-20, 845 (1984).
  26. D. Das and D. Saintillan, Electrohydrodynamic interaction of spherical particles under Quincke rotation, Phys. Rev. E 87, 043014 (2013).
  27. Y. Dolinsky and T. Elperin, Dipole interaction of the Quincke rotating particles, Phys. Rev. E 85, 026608 (2012).
  28. H. He, P. F. Salipante, and P. M. Vlahovska, Electrorotation of a viscous droplet in a uniform direct current electric field, Phys. Fluids 25, 032106 (2013).
  29. D. Das and D. Saintillan, A three-dimensional small-deformation theory for electrohydrodynamics of dielectric drops, J. Fluid Mech. 914, A22 (2021).
  30. P. F. Salipante and P. M. Vlahovska, Electrohydrodynamic rotations of a viscous droplet, Phys. Rev. E 88, 043003 (2013).
  31. P. F. Salipante and P. M. Vlahovska, Electrohydrodynamics of drops in strong uniform dc electric fields, Phys. Fluids 22, 112110 (2010).
  32. M. Firouznia, S. H. Bryngelson, and D. Saintillan, A spectral boundary integral method for simulating electrohydrodynamic flows in viscous drops, J. Comput. Phys. 489, 112248 (2023).
  33. D. Das and D. Saintillan, A nonlinear small-deformation theory for transient droplet electrohydrodynamics, J. Fluid Mech. 810, 225 (2017).
  34. A. Varshney, S. Ghosh, S. Bhattacharya, and A. Yethiraj, Self organization of exotic oil-in-oil phases driven by tunable electrohydrodynamics, Sci. Rep. 2, 738 (2012).
  35. A. Varshney, S. Gohil, M. Sathe, S. Rao R V, J. B. Joshi, S. Bhattacharya, A. Yethiraj, and S. Ghosh, Multiscale flow in an electro-hydrodynamically driven oil-in-oil emulsion, Soft Matter 12, 1759 (2016).
  36. S. K. Tadavani, J. R. Munroe, and A. Yethiraj, The effect of confinement on the electrohydrodynamic behavior of droplets in a microfluidic oil-in-oil emulsion, Soft Matter 12, 9246 (2016).
  37. C. Sozou, Electrohydrodynamics of a pair of liquid drops, J. Fluid Mech. 67, 339 (1975).
  38. J. C. Baygents, N. J. Rivette, and H. A. Stone, Electrohydrodynamic deformation and interaction of drop pairs, J. Fluid Mech. 368, 359 (1998).
  39. M. Zabarankin, Small deformation theory for two leaky dielectric drops in a uniform electric field, Proc. R. Soc. A 476, 20190517 (2020).
  40. C. Sorgentone, J. I. Kach, A. S. Khair, L. M. Walker, and P. M. Vlahovska, Numerical and asymptotic analysis of the three-dimensional electrohydrodynamic interactions of drop pairs, J. Fluid Mech. 914, A24 (2021).
  41. J. I. Kach, L. M. Walker, and A. S. Khair, Prediction and measurement of leaky dielectric drop interactions, Phys. Rev. Fluids 7, 013701 (2022).
  42. C. Sorgentone and P. M. Vlahovska, Tandem droplet locomotion in a uniform electric field, J. Fluid Mech. 951, R2 (2022).
  43. J. I. Kach, L. M. Walker, and A. S. Khair, Nonequilibrium structure formation in electrohydrodynamic emulsions, Soft Matter 19, 9179 (2023).
  44. Q. Dong, Z. Xie, X. Zhou, J. Lu, and Z. Wang, Collective propulsion of viscous drop pairs based on Quincke rotation in a uniform electric field, Phys. Fluids 36, 017134 (2024).
  45. A. Bricard, J.-B. Caussin, N. Desreumaux, O. Dauchot, and D. Bartolo, Emergence of macroscopic directed motion in populations of motile colloids, Nature (London) 503, 95 (2013).
  46. G. Hetsroni and S. Haber, The flow in and around a droplet or bubble submerged in an unbound arbitrary velocity field, Rheol. Acta 9, 488 (1970).
  47. J. M. Rallison, The deformation of small viscous drops and bubbles in shear flows, Annu. Rev. Fluid Mech. 16, 45 (1984).
  48. G. Hetsroni and S. Haber, Low Reynolds number motion of two drops submerged in an unbounded arbitrary velocity field, Int. J. Multiphase Flow 4, 1 (1978).
  49. H. Lamb, Hydrodynamics (Cambridge University Press, Cambridge, UK, 1924).
  50. S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications (Courier Corporation, North Chelmsford, MA, 2013).
  51. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media (Prentice-Hall, Hoboken, NJ, 1965).
  52. T. B. Jones, Dielectrophoretic force calculation, J. Electrost. 6, 69 (1979).
  53. T. C. Halsey, Electrorheological fluids, Science 258, 761 (1992).
  54. See Supplemental Material at http://link.aps.org/supplemental/10.1103/lrkw-739x for experimental data extracted from references cited in the main text, MATLAB codes used to reproduce the results, and movies corresponding to Figs. 5 and 7.
  55. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, 2nd ed. (CRC Press, Boca Raton, FL, 2015).
  56. The MathWorks, Inc., Optimization toolbox version: 24.2 (r2024b) (2024), url: https://www.mathworks.com/.
  57. G. Bossis and J. F. Brady, Dynamic simulation of sheared suspensions. I. General method, J. Chem. Phys. 80, 5141 (1984).
  58. K. Takamura, H. L. Goldsmith, and S. G. Mason, The microrheology of colloidal dispersions: XII. Trajectories of orthokinetic pair-collisions of latex spheres in a simple electrolyte, J. Colloid Interface Sci. 82, 175 (1981).
  59. A. Bricard, J.-B. Caussin, D. Das, C. Savoie, V. Chikkadi, K. Shitara, O. Chepizhko, F. Peruani, D. Saintillan, and D. Bartolo, Emergent vortices in populations of colloidal rollers, Nat. Commun. 6, 7470 (2015).
  60. D. Saintillan, Nonlinear interactions in electrophoresis of ideally polarizable particles, Phys. Fluids 20, 067104 (2008).
  61. W.-F. Hu, M.-C. Lai, and Y.-N. Young, A hybrid immersed boundary and immersed interface method for electrohydrodynamic simulations, J. Comput. Phys. 282, 47 (2015).
  62. J. M. López-Herrera, S. Popinet, and M. A. Herrada, A charge-conservative approach for simulating electrohydrodynamic two-phase flows using volume-of-fluid, J. Comput. Phys. 230, 1939 (2011).
  63. D. Das and D. Saintillan, On the absence of collective motion in a bulk suspension of spontaneously rotating dielectric particles, Soft Matter 19, 6825 (2023).
  64. L. Lobry and E. Lemaire, Viscosity decrease induced by a DC electric field in a suspension, J. Electrost. 47, 61 (1999).
  65. H. Nganguia, Y.-N. Young, P. M. Vlahovska, J. Blawzdziewicz, J. Zhang, and H. Lin, Equilibrium electro-deformation of a surfactant-laden viscous drop, Phys. Fluids 25, 092106 (2013).
  66. H. Nganguia, O. S. Pak, and Y.-N. Young, Effects of surfactant transport on electrodeformation of a viscous drop, Phys. Rev. E 99, 063104 (2019).
  67. H. Nganguia, D. Das, O. S. Pak, and Y.-N. Young, Influence of surface viscosities on the electrodeformation of a prolate viscous drop, Soft Matter 19, 776 (2023).

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Phys. Rev. Fluids 11, 103702– Published 19 October, 2026
  • Received 16 May 2025
  • Accepted 17 August 2026
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