Skip to main content
SpringerLink
Log in

Exact solutions of the equations of motion of liquid helium with a charged free surface

  • Fluids
  • Published:
Journal of Experimental and Theoretical Physics Aims and scope Submit manuscript

Abstract

The dynamics of the development of instability of the free surface of liquid helium, which is charged by electrons localized above it, is studied. It is shown that, if the charge completely screens the electric field above the surface and its magnitude is much larger than the instability threshold, the asymptotic behavior of the system can be described by the well-known 3D Laplacian growth equations. The integrability of these equations in 2D geometry makes it possible to describe the evolution of the surface up to the formation of singularities, viz., cuspidal point at which the electric field strength, the velocity of the liquid, and the curvature of its surface assume infinitely large values. The exact solutions obtained for the problem of the electrocapillary wave profile at the boundary of liquid helium indicate the tendency to a change in the surface topology as a result of formation of charged bubbles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. W. Cole and M. H. Cohen, Phys. Rev. Lett. 23, 1238 (1969).

    Article  ADS  Google Scholar 

  2. V. B. Shikin, Zh. Éksp. Teor. Fiz. 58, 1748 (1970) [Sov. Phys. JETP 31, 936 (1970)].

    Google Scholar 

  3. L. P. Gor’kov and D. M. Chernikova, Pis’ma Zh. Éksp. Teor. Fiz. 18, 119 (1973) [JETP Lett. 18, 68 (1973)].

    Google Scholar 

  4. D. M. Chernikova, Fiz. Nizk. Temp. 2, 1374 (1976) [Sov. J. Low Temp. Phys. 2, 669 (1976)].

    Google Scholar 

  5. L. Tonks, Phys. Rev. 48, 562 (1935).

    Article  ADS  Google Scholar 

  6. Ya. I. Frenkel’, Zh. Éksp. Teor. Fiz. 6, 347 (1936).

    Google Scholar 

  7. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 8: Electrodynamics of Continuous Media (Nauka, Moscow, 1982; Pergamon, New York, 1984).

    Google Scholar 

  8. L. P. Gor’kov and D. M. Chernikova, Dokl. Akad. Nauk SSSR 228, 829 (1976) [Sov. Phys. Dokl. 21, 328 (1976)].

    Google Scholar 

  9. H. Ikezi, Phys. Rev. Lett. 42, 1688 (1979).

    Article  ADS  Google Scholar 

  10. D. M. Chernikova, Fiz. Nizk. Temp. 6, 1513 (1980) [Sov. J. Low Temp. Phys. 6, 737 (1980)].

    Google Scholar 

  11. V. B. Shikin and Yu. P. Monarkha, Two-Dimensional Charged Systems in Helium (Nauka, Moscow, 1989).

    Google Scholar 

  12. A. P. Volodin, M. S. Khaikin, and V. S. Édel’man, Pis’ma Zh. Éksp. Teor. Fiz. 26, 707 (1977) [JETP Lett. 26, 543 (1977)].

    Google Scholar 

  13. N. M. Zubarev, Pis’ma Zh. Éksp. Teor. Fiz. 71, 534 (2000) [JETP Lett. 71, 367 (2000)].

    Google Scholar 

  14. G. D. Crapper, J. Fluid Mech. 2, 532 (1957).

    ADS  MATH  MathSciNet  Google Scholar 

  15. V. E. Zakharov, Prikl. Mekh. Tekh. Fiz., No.2, 86 (1968).

  16. P. Ya. Polubarinova-Kochina, Dokl. Akad. Nauk SSSR 47, 254 (1945).

    Google Scholar 

  17. D. Bensimon, L. P. Kadanoff, Sh. Liang, et al., Rev. Mod. Phys. 58, 977 (1986).

    Article  ADS  Google Scholar 

  18. A. I. D’yachenko, V. E. Zakharov, and E. A. Kuznetsov, Fiz. Plazmy 22, 916 (1996).

    Google Scholar 

  19. V. E. Zakharov and A. I. Dyachenko, Physica D (Amsterdam) 98, 652 (1996).

    MathSciNet  Google Scholar 

  20. M. B. Mineev-Weinstein and S. P. Dawson, Phys. Rev. E 50, R24 (1994).

    Article  ADS  Google Scholar 

  21. N. M. Zubarev, Phys. Lett. A 243, 128 (1998).

    Article  ADS  Google Scholar 

  22. N. M. Zubarev, Zh. Éksp. Teor. Fiz. 114, 2043 (1998) [JETP 87, 1110 (1998)].

    Google Scholar 

  23. S. D. Howison, SIAM J. Appl. Math. 46, 20 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  24. V. S. Édel’man, Usp. Fiz. Nauk 130, 675 (1980) [Sov. Phys. Usp. 23, 227 (1980)].

    Google Scholar 

  25. G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974; Mir, Moscow, 1977).

    Google Scholar 

  26. N. M. Zubarev, Zh. Éksp. Teor. Fiz. 116, 1990 (1999) [JETP 89, 1078 (1999)].

    Google Scholar 

  27. G. P. Ivantsov, Dokl. Akad. Nauk SSSR 58, 567 (1947).

    Google Scholar 

  28. A. I. Dyachenko, E. A. Kuznetsov, M. D. Spector, and V. E. Zakharov, Phys. Lett. A 221, 73 (1996).

    ADS  Google Scholar 

  29. A. I. D’yachenko, Dokl. Akad. Nauk 376, 27 (2001).

    MathSciNet  Google Scholar 

  30. N. M. Zubarev, Pis’ma Zh. Éksp. Teor. Fiz. 73, 613 (2001) [JETP Lett. 73, 544 (2001)].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Electrophysics, Ural Division, Russian Academy of Sciences, ul. Komsomol’skaya 34, Yekaterinburg, 620016, Russia

    N. M. Zubarev

Authors
  1. N. M. Zubarev
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

__________

Translated from Zhurnal Éksperimental’no\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} \) i Teoretichesko\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} \) Fiziki, Vol. 121, No. 3, 2002, pp. 624–636.

Original Russian Text Copyright © 2002 by Zubarev.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zubarev, N.M. Exact solutions of the equations of motion of liquid helium with a charged free surface. J. Exp. Theor. Phys. 94, 534–544 (2002). https://doi.org/10.1134/1.1469153

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1134/1.1469153

Keywords

Search

Navigation