Interfacial electrohydrodynamic solitary waves under horizontal electric fields. (English) Zbl 1493.76121
Summary: 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.
MSC:
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 76B25 | Solitary waves for incompressible inviscid fluids |
| 76B55 | Internal waves for incompressible inviscid fluids |
| 76B45 | Capillarity (surface tension) for incompressible inviscid fluids |
| 76M45 | Asymptotic methods, singular perturbations applied to problems in fluid mechanics |
| 76M99 | Basic methods in fluid mechanics |
Keywords:
capillary solitary wave; inviscid incompressible dielectric fluid layer; electrified Euler equations; Zakharov-Craig-Sulem model; asymptotic analysis; boundary integral equation methodReferences:
| [1] | Akylas, T.R.1993Envelope solitons with stationary crests. Phys. Fluids5, 789-791. · Zbl 0777.76018 |
| [2] | Anderson, T.G., Cimpeanu, R., Papageorgiou, D.T. & Petropoulos, P.G.2017Electric field stabilization of viscous liquid layers coating the underside of a surface. Phys. Rev. Fluids2, 054001. |
| [3] | Baker, G.R., Meiron, D.I. & Orszag, S.A.1982Generalized vortex methods for free-surface flow problems. J. Fluid Mech.123, 477-501. |
| [4] | Barannyk, L.L., Papageorgiou, D.T. & Petropoulos, P.G.2012Suppression of Rayleigh-Taylor instability using electric fields. Maths Comput. Simul.82, 1008-1016. · Zbl 1243.78010 |
| [5] | Barannyk, L.L., Papageorgiou, D.T., Petropoulos, P.G. & Vanden-Broeck, J.-M.2015Nonlinear dynamics and wall touch-up in unstably stratified multilayer flows in horizontal channels under the action of electric fields. SIAM J. Appl. Maths75 (1), 92-113. · Zbl 1328.76057 |
| [6] | Benjamin, T.B.1992A new kind of solitary wave. J. Fluid Mech.245, 401-411. · Zbl 0779.76013 |
| [7] | Benney, D.J.1977A general theory for interactions between short and long waves. Stud. Appl. Maths56 (1), 81-94. · Zbl 0358.76011 |
| [8] | Cimpeanu, R., Papageorgiou, D.T. & Petropoulos, P.G.2014On the control and suppression of the Rayleigh-Taylor instability using electric fields. Phys. Fluids26, 022105. |
| [9] | Coifman, R. & Meyer, Y.1985Nonlinear harmonic analysis and analytic dependence. Proc. Symp. Pure Math.43, 71-78. |
| [10] | Craig, W. & Sulem, C.1993Numerical simulation of gravity waves. J. Comput. Phys.108, 73-83. · Zbl 0778.76072 |
| [11] | Easwaran, C.V.1988Solitary waves on a conducting fluid layer. Phys. Fluids31, 3442-3443. · Zbl 0654.76105 |
| [12] | Eldabe, N.T.1989Effect of a tangential electric field on Rayleigh-Taylor instability. J. Phys. Soc. Japan58, 115-120. |
| [13] | Fernandez De La Mora, J. & Loscertales, I.G.1994The current emitted by highly conducting Taylor cones. J. Fluid Mech.260, 155-184. |
| [14] | Gamero-Castaño, M. & Hruby, V.2001Electrospray as a source of nanoparticles for efficient colloid thrusters. J. Propul. Power17 (5), 977-987. |
| [15] | Gao, T., Milewski, P.A., Papageorgiou, D.T. & Vanden-Broeck, J.-M.2017Dynamics of fully nonlinear capillary-gravity solitary waves under normal electric fields. J. Engng Maths108, 107-122. · Zbl 1444.76034 |
| [16] | Gleeson, H., Hammerton, P., Papageorgiou, D.T. & Vanden-Broeck, J.-M.2007A new application of the Kortweg-de Vries Benjamin-Ono equation in interfacial electrohydrodynamics. Phys. Fluids19, 031703. |
| [17] | Grandison, S., Papageorgiou, D.T. & Vanden-Broeck, J.-M.2007Interfacial capillary waves in the presence of electric fields. Euro. J. Mech. (B/Fluids)26, 404-421. |
| [18] | Grandison, S., Vanden-Broeck, J.-M., Papageorgiou, D.T., Miloh, T. & Spivak, B.2007Axisymmetric waves in electrohydrodynamic flows. J. Engng Maths62, 133-148. |
| [19] | Hunt, M.J., Vanden-Broeck, J.-M., Papageorgiou, D.T. & Părău, E.I.2017Benjamin-Ono Kadomtsev-Petviashvili’s models in interfacial electro-hydrodynamics. Eur. J. Mech. (B/Fluids)65, 459-463. · Zbl 1408.76579 |
| [20] | Joshi, A., Radhakrishna, M.C. & Rudraiah, N.2010Rayleigh-Taylor instability in dielectric fluids. Phys. Fluids22, 064102. |
| [21] | Kochurin, E.A. & Zubarev, N.M.2018Gravity-capillary waves on the free surface of a liquid dielectric in a tangential electric field. IEEE Trans. Dielec. Elec. Insul.25, 1723-1730. |
| [22] | Laget, O. & Dias, F.1997Numerical computation of capillary-gravity interfacial solitary waves. J. Fluid Mech.349, 221-251. · Zbl 0897.76011 |
| [23] | Melcher, J.R.1963Field-Coupled Surface Waves. MIT Press. |
| [24] | Melcher, J.R. & Schwarz, W.J.1968Interfacial relaxation overstability in a tangential electric field. Phys. Fluids11, 2604. |
| [25] | Melcher, J.R. & Taylor, G.I.1969Electrohydrodynamics: a review of the role of interfacial shear stresses. Annu. Rev. Fluid Mech.1, 111-146. |
| [26] | Milewski, P.A., Vanden-Broeck, J.-M. & Wang, Z.2011Hydroelastic solitary waves in deep water. J. Fluid Mech.679, 628-640. |
| [27] | Papageorgiou, D.T.2019Film flows in the presence of electric fields. Annu. Rev. Fluid Mech.51, 155-187. · Zbl 1412.76013 |
| [28] | Papageorgiou, D.T., Petropoulos, P.G. & Vanden-Broeck, J.-M.2005Gravity capillary waves in fluid layers under normal electric fields. Phys. Rev. E72 (5), 051601. |
| [29] | Papageorgiou, D.T. & Vanden-Broeck, J.-M.2004aLarge-amplitude capillary waves in electrified fluid sheets. J. Fluid Mech.508, 71-88. |
| [30] | Papageorgiou, D.T. & Vanden-Broeck, J.-M.2004bAntisymmetric capillary waves in electrified fluid sheets. Eur. J. Appl. Maths15, 609-623. |
| [31] | Papageorgiou, D.T. & Vanden-Broeck, J.-M.2007Numerical and analytical.studies of non-linear gravity-capillary waves in fluid layers under normal electric fields. IMA J. Appl. Maths72, 832-653. · Zbl 1388.76447 |
| [32] | Petviashvili, V.I.1976Equation of an extraordinary soliton. Sov. J. Plasma Phys.2, 257-258. |
| [33] | Schäffer, E., Thurn-Albrecht, T., Russell, T.P. & Steiner, U.2000Electrically induced structure formation and pattern transfer. Nature403, 874-877. |
| [34] | Taylor, G.I.1964Disintegration of water droplets in an electric field. Proc. R. Soc. Lond. A280, 383-397. · Zbl 0119.21101 |
| [35] | Taylor, G.I. & Mcewan, A.D.1965The stability of a horizontal fluid interface in a vertical electric field. J. Fluid Mech.22, 1-15. |
| [36] | Tilley, B.S., Petropoulos, P.G. & Papageorgiou, D.T.2001Dynamics and rupture of planar electrified liquid sheet. Phys. Fluids13, 3547-3563. |
| [37] | Turner, R.E.L. & Vanden-Broeck, J.-M.1988Broadening of interfacial solitary waves. Phys. Fluids31 (9), 2486-2490. · Zbl 0649.76008 |
| [38] | Wang, Z.2017Modelling nonlinear electrohydrodynamic surface waves over three-dimensional conducting fluids. Proc. R. Soc. Lond. A473, 20160817. · Zbl 1404.76298 |
| [39] | Yang, Q., Li, B.Q., Zhao, Z., Shao, J. & Xu, F.2016Numerical analysis of the Rayleigh-Taylor instability in an electric field. J. Fluid Mech.792, 397-434. · Zbl 1381.76419 |
| [40] | Zakharov, V.E.1968Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys.2, 190-194. |
| [41] | Zubarev, N.M.2004Nonlinear waves on the surface of a dielectric liquid in a strong tangential electric field. Phys. Lett. A333, 284-288. · Zbl 1123.76374 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
