Skip to main content
SpringerLink
Log in

Well-posedness of electrohydrodynamic waves under vertical electric field

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

This paper is devoted to the problem of wave propagation on a perfect conducting fluid under a vertical electric field, a kind of electrohydrodynamic waves. Three competing restoring forces resulting from gravity, surface tension, and normal electric field are taken into account. There are many numerical and experimental results on electrohydrodynamic waves, but the system’s well-posedness, which is a fundamental problem, has not been investigated in the past. We shall show the system considered is locally well posed based on energy estimates in proper Sobolev spaces and careful examinations of the Dirichlet–Neumann operators.

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. Abou El Magd, A.M., El Sayed, F.: Nonlinear electrohydrodynamic Rayleigh–Taylor instability. part 1. A perpendicular field in the absence of surface charges. J. Fluid Mech. 129, 473–494 (1983)

    Article  MathSciNet  Google Scholar 

  2. Ambrose, D.M., Masmoudi, N.: The zero surface tension limit of two-dimensional water waves. Commun. Pure Appl. Math. 58(10), 1287–1315 (2005)

    Article  MathSciNet  Google Scholar 

  3. Ambrose, D.M., Masmoudi, N.: The zero surface tension limit of three-dimensional water waves. Indiana Univ. Math. J. 58(2), 479–521 (2009)

    Article  MathSciNet  Google Scholar 

  4. Barannyk, L.L., Papageorgiou, D.T., Petropoulos, P.G.: Suppression of Rayleigh–Taylor instability using electric fields. Math. Comput. Simul. 82, 1008–1016 (2012)

    Article  MathSciNet  Google Scholar 

  5. Beyer, K., Günther, M.: On the Cauchy problem for a capillary drop. I. Irrotational motion. Math. Methods Appl. Sci. 21, 1149–1183 (1998)

    Article  MathSciNet  Google Scholar 

  6. Christodoulou, D., Lindblad, H.: On the motion of the free surface of a liquid. Comm. Pure Appl. Math. 53(12), 1536–1602 (2000)

    Article  MathSciNet  Google Scholar 

  7. Coutand, D., Shkoller, S.: Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Am. Math. Soc. 20(3), 829–930 (2007)

    Article  MathSciNet  Google Scholar 

  8. Craig, W., Sulem, C.: Numerical simulation of gravity waves. J. Comput. Phys. 108, 73–83 (1993)

    Article  MathSciNet  Google Scholar 

  9. Elhefnawy, A.R.F.: Nonlinear electrohydrodynamic Kelvin–Helmholtz instability under the influence of an oblique electric field. Phys. A 182, 419–435 (1992)

    Article  MathSciNet  Google Scholar 

  10. Lannes, D.: Well-posedness of the water-waves equations. J. Am. Math. Soc. 18(3), 605–654 (2005)

    Article  MathSciNet  Google Scholar 

  11. Hunt, M.J., Vanden-Broeck, J.-M., Papageorgiou, D.T., Părău, E.I.: Benjamin-Ono Kadomtsev-Petviashvili’s models in interfacial electro-hydrodynamics. Eur. J. Mech. B. Fluids 65, 459–463 (2017)

    Article  MathSciNet  Google Scholar 

  12. Lannes, D.: The Water Waves Problem: Mathematical Analysis and Asymptotics, vol. 188. American Mathematical Society, Providence (2013)

    MATH  Google Scholar 

  13. Lannes, D.: A stability criterion for two-fluid interfaces and applications. Arch. Ration. Mech. Anal. 208(2), 481–567 (2013)

    Article  MathSciNet  Google Scholar 

  14. Lindblad, H.: Well-posedness for the motion of an incompressible liquid with free surface boundary. Ann. Math. 162, 109–194 (2005)

    Article  MathSciNet  Google Scholar 

  15. Melcher, J.R., Schwarz, W.J.: Interfacial relaxation overstability in a tangential electric field. Phys. Fluids 11, 2604 (1968)

    Article  Google Scholar 

  16. Papageorgiou, D.T., Petropoulos, P.G., Vanden-Broeck, J.M.: Gravity capillary waves in fluid layer under normal electric field. Phys. Rev. E. 72, 051601 (2005)

    Article  MathSciNet  Google Scholar 

  17. Papageorgiou, D.T., Vanden-Broeck, J.M.: Large-amplitude capillary waves in electrified fluid sheets. J. Fluid Mech. 508, 71–88 (2004)

    Article  MathSciNet  Google Scholar 

  18. Shatah, J., Zeng, C.: A priori estimates for fluid interface problems. Commun. Pure Appl. Math. 61(6), 848–876 (2008)

    Article  MathSciNet  Google Scholar 

  19. Shatah, J., Zeng, C.: Local well-posedness for fluid interface problems. Arch. Ration. Mech. Anal. 199(2), 653–705 (2011)

    Article  MathSciNet  Google Scholar 

  20. Taylor, G.I., McEwan, A.D.: The stability of a horizontal fluid interface in a vertical electric field. J. Fluid Mech. 22, 1–15 (1965)

    Article  Google Scholar 

  21. Tilley, B.S., Petropoulos, P.G., Papageorgiou, D.T.: Dynamics and rupture of planar electrified liquid sheets. Phys. Fluids 13, 3547–3563 (2001)

    Article  Google Scholar 

  22. Wang, Z.: Modelling nonlinear electrohydrodynamic surface waves over three-dimensional conducting fluids. Proc. R. Soc. A 473, 20160817 (2017)

    Article  MathSciNet  Google Scholar 

  23. Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130(1), 39–72 (1997)

    Article  MathSciNet  Google Scholar 

  24. Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Am. Math. Soc. 12(2), 445–495 (1999)

    Article  MathSciNet  Google Scholar 

  25. Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 2, 190–194 (1968)

    Google Scholar 

Download references

Acknowledgements

Jiaqi Yang is supported by the Fundamental Research Funds for the Central Universities, G2020KY05205.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710072, Shaanxi, People’s Republic of China

    Jiaqi Yang

Authors
  1. Jiaqi Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiaqi Yang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, J. Well-posedness of electrohydrodynamic waves under vertical electric field. Z. Angew. Math. Phys. 71, 171 (2020). https://doi.org/10.1007/s00033-020-01402-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-020-01402-9

Keywords

Mathematics Subject Classification

Search

Navigation