Theoretical and Mathematical Physics

, Volume 196, Issue 3, pp 1380–1391 | Cite as

Construction of Exact Solutions for Equilibrium Configurations of the Boundary of a Conducting Liquid Deformed By an External Electric Field

  • N. M. ZubarevEmail author
  • O. V. Zubareva


In a two-dimensional plane-symmetric formulation, we consider the problem of the equilibrium configurations of the free surface of a conducting capillary liquid placed in an external electric field. We find a one-parameter family of exact solutions of the problem according to which the fluid takes the shape of a blade. Such a configuration provides formally unlimited local field amplification: the field strength is maximum at the edge of the blade and drops to zero at the periphery. For a given potential difference between the liquid and the flat electrode located above it, we find threshold values of the electric field strength at the edge of the liquid blade, the radius of curvature of the edge, and the distance from the edge to the electrode limiting the region of existence of the solutions.


equilibrium configuration exact solution free surface conductive fluid surface tension electrostatic force conformal map method 


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  1. 1.
    L. Tonks, “A theory of liquid surface rupture by a uniform electric field,” Phys. Rev., 48, 562–568 (1935).ADSCrossRefGoogle Scholar
  2. 2.
    Ya. Frenkel, “On the Tonks’ theory of the rupture of the surface of liquid by a uniform electric field in vacuum,” Soviet JETP, 6, 348–350 (1936).zbMATHGoogle Scholar
  3. 3.
    J. R. Melcher, Field-Coupled Surface Waves, MIT Press, Cambridge, Mass. (1963).Google Scholar
  4. 4.
    A. A. Kuznetsov and M. D. Spektor, “Existence of a hexagonal relief on the surface of a dielectric fluid in an external electrical field,” Soviet JETP, 44, 136–141 (1976).ADSGoogle Scholar
  5. 5.
    N. M. Zubarev and O. V. Zubareva, “Dynamics of the free surface of a conducting liquid in a near-critical electric field,” Tech. Phys., 46, 806–814 (2001).CrossRefGoogle Scholar
  6. 6.
    M. D. Gabovich, “Liquid-metal ion emitters,” Sov. Phys. Usp., 26, 447–455 (1983).ADSCrossRefGoogle Scholar
  7. 7.
    A. I. Zhakin, “Electrohydrodynamics of charged surfaces,” Phys. Usp., 56, 141–163 (2013).ADSCrossRefGoogle Scholar
  8. 8.
    N. M. Zubarev, “Formation of root singularities on the free surface of a conducting fluid in an electric field,” Phys. Lett. A, 243, 128–131 (1998)ADSCrossRefGoogle Scholar
  9. 9.
    L. M. Baskin, A. V. Batrakov, S. A. Popov, and D. I. Proskurovsky, “Electrohydrodynamic phenomena on the explosive-emission liquid-metal cathode,” IEEE Trans. Dielectr. Electr. Insul., 2, 231–236 (1995).CrossRefGoogle Scholar
  10. 10.
    W. Driesel, C. Dietzsch, and R. Mühle, “In situ observation of the tip shape of AuGe liquid alloy ion sources using a high voltage transmission electron microscope,” J. Vac. Sci. Technol. B, 14, 3367–380 (1996).CrossRefGoogle Scholar
  11. 11.
    N. M. Zubarev, “Formation of conic cusps at the surface of liquid metal in electric field,” JETP Lett., 73, 544–548 (2001).ADSCrossRefGoogle Scholar
  12. 12.
    V. B. Shikin, “Instability and reconstruction of a charged liquid surface,” Phys. Usp., 54, 1203–1225 (2011).ADSCrossRefGoogle Scholar
  13. 13.
    A. A. Levchenko, G. V. Kolmakov, L. P. Mezhov-Deglin, M. G. Mikhailov, and A. B. Trusov, “Static phenomena at the charged surface of liquid hydrogen,” Low Temperature Phys., 25, 242 (1999).ADSCrossRefGoogle Scholar
  14. 14.
    J. W. S. Rayleigh, “X X. On the equilibrium of liquid conducting masses charged with electricity,” Philos. Mag., 14, 184–186 (1882).CrossRefGoogle Scholar
  15. 15.
    M. J. Miksis, “Shape of a drop in an electric field,” Phys. Fluids, 24, 1967–1972 (1981).ADSCrossRefzbMATHGoogle Scholar
  16. 16.
    N. A. Pelekasis, J. A. Tsamopoulos, and G. D. Manolis, “Equilibrium shapes and stability of charged and conducting drops,” Phys. Fluids A, 2, 1328–1340 (1990).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    N. M. Zubarev and O. V. Zubareva, “Equilibrium configurations of uncharged conducting liquid jets in a transverse electric field,” Phys. A, 385, 35–45 (2007).CrossRefGoogle Scholar
  18. 18.
    G. I. Taylor, “Disintegration of water drops in an electric field,” Proc. Roy. Soc. London Ser. A, 280, 383–397 (1964).ADSCrossRefzbMATHGoogle Scholar
  19. 19.
    N. M. Zubarev, “Exact solution of the problem of the equilibrium configuration of the charged surface of a liquid metal,” JETP, 89, 1078–1085 (1999)ADSCrossRefGoogle Scholar
  20. 19a.
    “Exact solution for the steady-state surface profile of a liquid metal in an external electric field,” Tech. Phys. Lett., 25, 920–921 (1999).Google Scholar
  21. 20.
    N. M. Zubarev, “On the problem of the existence of a singular stationary profile for the charged surface of a conducting liquid,” Tech. Phys. Lett., 27, 217–219 (2001).ADSCrossRefGoogle Scholar
  22. 21.
    M. I. Gurevich, The Theory of Jets in an Ideal Fluid [in Russian], Nauka, Moscow (1979); English transl. prev. ed. (Intl. Series Monogr. Pure Appl. Math., Vol. 93), Pergamon, Oxford (1966).Google Scholar
  23. 22.
    G. D. Crapper, “An exact solution for progressive capillary waves of arbitrary amplitude,” J. Fluid Mech., 2, 532–540 (1957).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 23.
    N. M. Zubarev, O. V. Zubareva, and P. K. Ivanov, “Exact solutions for equilibrium surface configurations of a conducting liquid in the electric field of a charged straight filament,” Tech. Phys. Lett., 35, 967–969 (2009).ADSCrossRefGoogle Scholar
  25. 24.
    N. B. Volkov, M. Zubarev, and V. Zubareva, “Exact solutions to the problem on the shape of an uncharged conducting liquid jet in a transverse electric field,” JETP, 122, 950–955 (2016).ADSCrossRefGoogle Scholar
  26. 25.
    J. A. Shercliff, “Magnetic shaping of molten metal columns,” Proc. Roy. Soc. London Ser. A, 375, 455–473 (1981).ADSCrossRefGoogle Scholar
  27. 26.
    W. Kinnersley, “Exact large amplitude capillary waves on sheets of fluid,” J. Fluid Mech., 77, 229–241 (1976).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 27.
    M. G. Blyth and J.-M. Vanden-Broeck, “Magnetic shaping of a liquid metal column and deformation of a bubble in a vortex flow,” SIAM J. Appl. Math., 66, 174–186 (2005).MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Institute for Electrophysics, Urals BranchRASEkaterinburgRussia
  2. 2.Lebedev Physical InstituteRASMoscowRussia

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