Skip to main content
Springer Nature Link
Log in

The Dynamics of Periodic Traveling Interfacial Electrohydrodynamic Waves: Bifurcation and Secondary Bifurcation

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Abstract

In this paper, we consider two-dimensional periodic capillary-gravity waves traveling under the influence of a vertical electric field. The full system is a nonlinear, two-layered, free boundary problem. The interface dynamics are derived by coupling Euler equations for the velocity field of the fluid with voltage potential equations governing the electric field. We first introduce the naive flattening technique to transform the free boundary problem into a fixed boundary problem. We then prove the existence of small-amplitude electrohydrodynamic waves with constant vorticity using local bifurcation theory. Moreover, we show that these electrohydrodynamic waves are formally stable in the linearized sense. Furthermore, we obtain a secondary bifurcation curve that emerges from the primary branch, consisting of ripple solutions on the interface. As far as we know, such solutions in electrohydrodynamics are established for the first time. It is worth noting that the electric field \(E_0\) plays a key role in controlling the shapes and types of waves on the interface.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

  • Ambrose, D.M., Strauss, W.A., Wright, J.D.: Global bifurcation theory for periodic traveling interfacial gravity-capillary waves. Ann. Inst. H. Poincaré C Anal. NonLinéaire 33, 1081–1101 (2016)

    Article  MathSciNet  Google Scholar 

  • Amick, C.J., Turner, R.E.L.: Small internal waves in two-fluid systems. Arch. Ration. Mech. Anal. 108, 111–139 (1989)

    Article  MathSciNet  Google Scholar 

  • Chu, J.F., Ding, Q.X., Escher, J.: Variational formulation of rotational steady water waves in two-layer flows. J. Math. Fluid Mech. 23, 91 (2021)

    Article  MathSciNet  Google Scholar 

  • Constantin, A.: Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis. CBMS-NSF Conference Series in Applied Mathematics 81. SIAM, Philadelphia (2011)

    Book  Google Scholar 

  • Constantin, A., Strauss, W.: Exact steady periodic water waves with vorticity. Commun. Pure Appl. Math. 57, 481–527 (2004)

    Article  MathSciNet  Google Scholar 

  • Constantin, A., Strauss, W.: Stability properties of steady water waves with vorticity. Commun. Pure Appl. Math. 60, 911–950 (2007)

    Article  MathSciNet  Google Scholar 

  • Crandall, M.G., Rabinowitz, P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)

    Article  MathSciNet  Google Scholar 

  • Crandall, M.G., Rabinowitz, P.H.: Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Ration. Mech. Anal. 52, 161–180 (1973)

    Article  MathSciNet  Google Scholar 

  • Dai, G., Li, F., Zhang, Y.: Bifurcation structure and stability of steady gravity water waves with constant vorticity. J. Differ. Equ. 332, 306–332 (2022)

    Article  MathSciNet  Google Scholar 

  • Dai, G., Gao, S., Ma, R., Zhang, Y.: Stability properties of nontrivial periodic water waves for fixed-depth rotational equatorial flows. Anal. Math. Phys. 14, 6 (2024)

    Article  MathSciNet  Google Scholar 

  • Doak, A., Gao, T., Vanden-Broeck, J.M., Kandola, J.J.S.: Capillary-gravity waves on the interface of two dielectric fluid layers under normal electric fields. Q. J. Mech. Appl. Math. 73, 231–250 (2020)

    Article  MathSciNet  Google Scholar 

  • Doak, A., Gao, T., Vanden-Broeck, J.M.: Global bifurcation of capillary-gravity dark solitary waves on the surface of a conducting fluid under normal electric fields. Q. J. Mech. Appl. Math. 75, 215–234 (2022)

    Article  MathSciNet  Google Scholar 

  • Dubreil-Jacotin, M.L.: Sur la détermination rigoureuse des ondes permanentes périodiques d’ampleur finie. J. Math. Pures Appl. 13, 217–291 (1934)

    Google Scholar 

  • Ehrnström, M., Escher, J., Wahlén, E.: Steady water waves with multiple critical layers. SIAM J. Math. Anal. 43, 1436–1456 (2011)

    Article  MathSciNet  Google Scholar 

  • Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)

    Book  Google Scholar 

  • Gao, T., Wang, Z., Vanden-Broeck, J.M.: Nonlinear wave interactions on the surface of a conducting fluid under vertical electric fields. Phys. D 446, 133651 (2023)

    Article  MathSciNet  Google Scholar 

  • Gleeson, H., Hammerton, P., Papageorgiou, D.T., Vanden-Broeck, J.M.: A new application of the Korteweg–de Vries Benjamin–Ono equation in interfacial electrohydrodynamics. Phys. Fluids 19, 031703 (2007)

    Article  Google Scholar 

  • Henry, D., Matioc, B.V.: On the existence of steady periodic capillary-gravity stratified water waves. Ann. Sc. Norm. Super. Pisa CI. Sci. 12, 955–974 (2013)

    MathSciNet  Google Scholar 

  • Hunt, J.M., Dutykh, D.: Free surface flows in electrohydrodynamics with a constant vorticity distribution. Water Waves 3, 297–317 (2021)

    Article  MathSciNet  Google Scholar 

  • Jones, M., Toland, J.F.: The bifurcation and secondary bifurcation of capillary-gravity waves. Proc. R. Soc. Lond. Ser. A 399, 391–417 (1985)

    Article  MathSciNet  Google Scholar 

  • Kielhöfer, H.: Bifurcation theory. An introduction with applications to partial differential equations, 2nd edn. Applied Mathematical Sciences. Springer, New York, p. 156 (2012)

  • Lin, Z., Zhu, Y., Wang, Z.: Local bifurcation of electrohydrodynamic waves on a conducting fluid. Phys. Fluids 29, 032107 (2017)

    Article  Google Scholar 

  • Martin, C.I., Matioc, B.V.: Existence of Wilton ripples for water waves with constant vorticity and capillary effects. SIAM J. Appl. Math. 73, 1582–1595 (2013)

    Article  MathSciNet  Google Scholar 

  • Melcher, J.R., Taylor, G.I.: Electrohydrodynamics: a review of the role of interfacial shear stresses. Annu. Rev. Fluid Mech. 1, 111–146 (1969)

    Article  Google Scholar 

  • Papageorgiou, D.T.: Film flows in the presence of electric fields. Ann. Rev. Fluid Mech. 51, 155–187 (2019)

    Article  MathSciNet  Google Scholar 

  • Reeder, J., Shinbrot, M.: On Wilton ripples. II. Rigorous results. Arch. Ration. Mech. Anal. 77, 321–347 (1981)

    Article  MathSciNet  Google Scholar 

  • Shearer, M.: Secondary bifurcation near a double eigenvalue. SIAM J. Math. Anal. 11, 365–389 (1980)

    Article  MathSciNet  Google Scholar 

  • Smit Vega Garcia, M., Vărvǎrucǎ, E., Weiss, G.S.: Singularities in axisymmetric free boundaries for electrohydrodynamic equations. Arch. Ration. Mech. Anal. 222, 573–601 (2016)

    Article  MathSciNet  Google Scholar 

  • Sinambela, D.: Large-amplitude solitary waves in two-layer density stratified water. SIAM J. Math. Anal. 53, 4812–4864 (2021)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Stokes, G.G.: On the theory of oscillatory waves. Trans. Camb. Philos. Soc. 8, 441–455 (1847)

    Google Scholar 

  • Wahlén, E.: Steady water waves with a critical layer. J. Differ. Equ. 246, 2468–2483 (2009)

    Article  MathSciNet  Google Scholar 

  • Walsh, S., Bühler, O., Shatah, J.: Steady water waves in the presence of wind. SIAM J. Math. Anal. 45, 2182–2227 (2013)

    Article  MathSciNet  Google Scholar 

  • Wang, Z.: Modelling nonlinear electrohydrodynamic surface waves over three-dimensional conducting fluids. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 473, 20160817 (2017)

    MathSciNet  Google Scholar 

  • Wheeler, M.H.: On stratified water waves with critical layers and Coriolis forces. Discrete Contin. Dyn. Syst. 39, 4747–4770 (2019)

    Article  MathSciNet  Google Scholar 

  • Wilton, J.R.: On ripples. Philos. Mag. 29, 688–700 (1915)

    Article  Google Scholar 

  • Vanden-Broeck, J.M.: Gravity-Capillary Free-Surface Flows. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

Download references

Acknowledgements

Dai was supported by National Natural Science Foundation of China (No. 12371110). Xu was supported by the Postdoctoral Science Foundation of China (2023M731381). Zhang was supported by National Natural Science Foundation of China (No. 12301133), the Postdoctoral Science Foundation of China (No. 2023M741441, No. 2024T170353) and Jiangsu Education Department (No. 23KJB110007). The authors would like to express their gratitude to Professor Ian Tobasco for his serious and responsible handling of the manuscript. The authors would also like to thank the referees for their valuable comments and suggestions.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Guowei Dai

  2. School of Mathematical Sciences, Jiangsu University, Zhenjiang, 212013, People’s Republic of China

    Fei Xu & Yong Zhang

Authors
  1. Guowei Dai
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Fei Xu
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Yong Zhang
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Yong Zhang.

Ethics declarations

Conflict of interest

All authors have the same contribution to the article and enjoy equal status, and all authors declare that they have no conflict of interest.

Additional information

Communicated by Ian Tobasco.

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dai, G., Xu, F. & Zhang, Y. The Dynamics of Periodic Traveling Interfacial Electrohydrodynamic Waves: Bifurcation and Secondary Bifurcation. J Nonlinear Sci 34, 99 (2024). https://doi.org/10.1007/s00332-024-10085-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00332-024-10085-9

Keywords

Mathematics Subject Classification