Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T21:00:32.647Z Has data issue: false hasContentIssue false
  • English
  • Français

Interfacial electrohydrodynamic solitary waves under horizontal electric fields

Published online by Cambridge University Press:  06 April 2022

Xin Guan  and
Zhan Wang Open the ORCID record for Zhan Wang [Opens in a new window]
Xin Guan
Affiliation:
Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, PR China Department of Mathematics, University College London, London WC1E 6BT, UK
Zhan Wang*
Affiliation:
Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, PR China University of Chinese Academy of Sciences, Beijing 100049, PR China
*
†Email address for correspondence: zwang@imech.ac.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

Interfacial waves between two superimposed dielectric fluid layers under a horizontal electric field are investigated from asymptotic and numerical aspects. The fluid is taken to be inviscid, incompressible and non-conducting in each layer. The competing forces resulting from gravity, surface tension and electric field are all considered. A systematic procedure is proposed to derive model equations of multiple scales in various possible limits from the electrified Euler equations in the framework of the Zakharov–Craig–Sulem formulation. Based on thorough analyses of the Dirichlet–Neumann operators in the long-wave approximation, classic weakly nonlinear models – including the Boussinesq-type system, the Korteweg–de Vries (KdV) equation and its variants, and the Benjamin-type equation – are obtained under different scaling assumptions. In addition, strongly nonlinear models (without the smallness assumption on the wave amplitude) in the Benjamin and Barannyk–Papageorgiou–Petropoulos regimes are derived. In these models, the electric effects are shown to produce dispersive regularisations of long and short waves. A modified boundary integral equation method is developed to compute solitary waves in the original electrified Euler equations. Through comparisons with solitary-wave solutions in the Euler equations, it is found that in various asymptotic regimes, weakly nonlinear models are in overall good agreement when wave amplitudes are small. In contrast, the range of validity of the strongly nonlinear model is much broader. It is shown that the horizontal electric field plays a significant role in the physical system: it expands the range of parameters for the existence of progressive waves, changes the qualitative characteristics of solitary waves and leads to a new type of solitary wave, namely the KdV–wavepacket mixed type.

JFM classification

Waves/Free-surface Flows: Capillary waves Waves/Free-surface Flows: Solitary waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 940 , 10 June 2022 , A15
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Akylas, T.R. 1993 Envelope solitons with stationary crests. Phys. Fluids 5, 789–791.CrossRefGoogle Scholar
Anderson, T.G., Cimpeanu, R., Papageorgiou, D.T. & Petropoulos, P.G. 2017 Electric field stabilization of viscous liquid layers coating the underside of a surface. Phys. Rev. Fluids 2, 054001.CrossRefGoogle Scholar
Baker, G.R., Meiron, D.I. & Orszag, S.A. 1982 Generalized vortex methods for free-surface flow problems. J. Fluid Mech. 123, 477–501.CrossRefGoogle Scholar
Barannyk, L.L., Papageorgiou, D.T. & Petropoulos, P.G. 2012 Suppression of Rayleigh–Taylor instability using electric fields. Maths Comput. Simul. 82, 1008–1016.CrossRefGoogle Scholar
Barannyk, L.L., Papageorgiou, D.T., Petropoulos, P.G. & Vanden-Broeck, J.-M. 2015 Nonlinear dynamics and wall touch-up in unstably stratified multilayer flows in horizontal channels under the action of electric fields. SIAM J. Appl. Maths 75 (1), 92–113.CrossRefGoogle Scholar
Benjamin, T.B. 1992 A new kind of solitary wave. J. Fluid Mech. 245, 401–411.CrossRefGoogle Scholar
Benney, D.J. 1977 A general theory for interactions between short and long waves. Stud. Appl. Maths 56 (1), 81–94.CrossRefGoogle Scholar
Cimpeanu, R., Papageorgiou, D.T. & Petropoulos, P.G. 2014 On the control and suppression of the Rayleigh–Taylor instability using electric fields. Phys. Fluids 26, 022105.CrossRefGoogle Scholar
Coifman, R. & Meyer, Y. 1985 Nonlinear harmonic analysis and analytic dependence. Proc. Symp. Pure Math. 43, 71–78.CrossRefGoogle Scholar
Craig, W. & Sulem, C. 1993 Numerical simulation of gravity waves. J. Comput. Phys. 108, 73–83.CrossRefGoogle Scholar
Easwaran, C.V. 1988 Solitary waves on a conducting fluid layer. Phys. Fluids 31, 3442–3443.CrossRefGoogle Scholar
Eldabe, N.T. 1989 Effect of a tangential electric field on Rayleigh–Taylor instability. J. Phys. Soc. Japan 58, 115–120.CrossRefGoogle Scholar
Fernandez de La Mora, J. & Loscertales, I.G. 1994 The current emitted by highly conducting Taylor cones. J. Fluid Mech. 260, 155–184.CrossRefGoogle Scholar
Gamero-Castaño, M. & Hruby, V. 2001 Electrospray as a source of nanoparticles for efficient colloid thrusters. J. Propul. Power 17 (5), 977–987.CrossRefGoogle Scholar
Gao, T., Milewski, P.A., Papageorgiou, D.T. & Vanden-Broeck, J.-M. 2017 Dynamics of fully nonlinear capillary-gravity solitary waves under normal electric fields. J. Engng Maths 108, 107–122.CrossRefGoogle ScholarPubMed
Gleeson, H., Hammerton, P., Papageorgiou, D.T. & Vanden-Broeck, J.-M. 2007 A new application of the Kortweg–de Vries Benjamin–Ono equation in interfacial electrohydrodynamics. Phys. Fluids 19, 031703.CrossRefGoogle Scholar
Grandison, S., Papageorgiou, D.T. & Vanden-Broeck, J.-M. 2007 Interfacial capillary waves in the presence of electric fields. Euro. J. Mech. (B/Fluids) 26, 404–421.CrossRefGoogle Scholar
Grandison, S., Vanden-Broeck, J.-M., Papageorgiou, D.T., Miloh, T. & Spivak, B. 2007 Axisymmetric waves in electrohydrodynamic flows. J. Engng Maths 62, 133–148.CrossRefGoogle Scholar
Hunt, M.J., Vanden-Broeck, J.-M., Papageorgiou, D.T. & Părău, E.I. 2017 Benjamin–Ono Kadomtsev–Petviashvili's models in interfacial electro-hydrodynamics. Eur. J. Mech. (B/Fluids) 65, 459–463.CrossRefGoogle Scholar
Joshi, A., Radhakrishna, M.C. & Rudraiah, N. 2010 Rayleigh–Taylor instability in dielectric fluids. Phys. Fluids 22, 064102.CrossRefGoogle Scholar
Kochurin, E.A. & Zubarev, N.M. 2018 Gravity-capillary waves on the free surface of a liquid dielectric in a tangential electric field. IEEE Trans. Dielec. Elec. Insul. 25, 1723–1730.CrossRefGoogle Scholar
Laget, O. & Dias, F. 1997 Numerical computation of capillary-gravity interfacial solitary waves. J. Fluid Mech. 349, 221–251.CrossRefGoogle Scholar
Melcher, J.R. 1963 Field-Coupled Surface Waves. MIT Press.Google Scholar
Melcher, J.R. & Schwarz, W.J. 1968 Interfacial relaxation overstability in a tangential electric field. Phys. Fluids 11, 2604.CrossRefGoogle Scholar
Melcher, J.R. & Taylor, G.I. 1969 Electrohydrodynamics: a review of the role of interfacial shear stresses. Annu. Rev. Fluid Mech. 1, 111–146.CrossRefGoogle Scholar
Milewski, P.A., Vanden-Broeck, J.-M. & Wang, Z. 2011 Hydroelastic solitary waves in deep water. J. Fluid Mech. 679, 628–640.CrossRefGoogle Scholar
Papageorgiou, D.T. 2019 Film flows in the presence of electric fields. Annu. Rev. Fluid Mech. 51, 155–187.CrossRefGoogle Scholar
Papageorgiou, D.T., Petropoulos, P.G. & Vanden-Broeck, J.-M. 2005 Gravity capillary waves in fluid layers under normal electric fields. Phys. Rev. E 72 (5), 051601.CrossRefGoogle ScholarPubMed
Papageorgiou, D.T. & Vanden-Broeck, J.-M. 2004 a Large-amplitude capillary waves in electrified fluid sheets. J. Fluid Mech. 508, 71–88.CrossRefGoogle Scholar
Papageorgiou, D.T. & Vanden-Broeck, J.-M. 2004 b Antisymmetric capillary waves in electrified fluid sheets. Eur. J. Appl. Maths 15, 609–623.CrossRefGoogle Scholar
Papageorgiou, D.T. & Vanden-Broeck, J.-M. 2007 Numerical and analytical.studies of non-linear gravity-capillary waves in fluid layers under normal electric fields. IMA J. Appl. Maths 72, 832–653.CrossRefGoogle Scholar
Petviashvili, V.I. 1976 Equation of an extraordinary soliton. Sov. J. Plasma Phys. 2, 257–258.Google Scholar
Schäffer, E., Thurn-Albrecht, T., Russell, T.P. & Steiner, U. 2000 Electrically induced structure formation and pattern transfer. Nature 403, 874–877.CrossRefGoogle ScholarPubMed
Taylor, G.I. 1964 Disintegration of water droplets in an electric field. Proc. R. Soc. Lond. A 280, 383–397.Google Scholar
Taylor, G.I. & McEwan, A.D. 1965 The stability of a horizontal fluid interface in a vertical electric field. J. Fluid Mech. 22, 1–15.CrossRefGoogle Scholar
Tilley, B.S., Petropoulos, P.G. & Papageorgiou, D.T. 2001 Dynamics and rupture of planar electrified liquid sheet. Phys. Fluids 13, 3547–3563.CrossRefGoogle Scholar
Turner, R.E.L. & Vanden-Broeck, J.-M. 1988 Broadening of interfacial solitary waves. Phys. Fluids 31 (9), 2486–2490.CrossRefGoogle Scholar
Wang, Z. 2017 Modelling nonlinear electrohydrodynamic surface waves over three-dimensional conducting fluids. Proc. R. Soc. Lond. A 473, 20160817.Google Scholar
Yang, Q., Li, B.Q., Zhao, Z., Shao, J. & Xu, F. 2016 Numerical analysis of the Rayleigh–Taylor instability in an electric field. J. Fluid Mech. 792, 397–434.CrossRefGoogle Scholar
Zakharov, V.E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 2, 190–194.Google Scholar
Zubarev, N.M. 2004 Nonlinear waves on the surface of a dielectric liquid in a strong tangential electric field. Phys. Lett. A 333, 284–288.CrossRefGoogle Scholar