Nonlinear wave interactions on the surface of a conducting fluid under vertical electric fields. (English) Zbl 1510.76209
Summary: In this paper, we are concerned with capillary-gravity waves propagating on a two-dimensional conducting fluid under the effect of an electric field imposed in the direction perpendicular to the undisturbed free surface. This work aims to investigate the impact of the electric fields on the nonlinear wave interactions in this context. The full system is mathematically difficult to solve since it is a nonlinear, two-layered, free boundary problem, and the interface dynamics results from the strong coupling between the Euler equations for the lower fluid layer and an electric contribution from the upper gas layer. To investigate electrohydrodynamic wave interactions, we propose a novel numerical scheme based on a time-dependent conformal map and an interpolation technique to conduct unsteady simulations of the fully nonlinear electrified Euler equations of the two-layered problem. To gain analytical insights, we first derive weakly nonlinear envelope equations based on the method of multiple scales for the resonant triad, long-short wave interaction, and modulation of a single-mode wavetrain (a special case of resonant quartet interaction). Summative remarks are made to illustrate the transition from 3-wave interactions to 4-wave interactions and vice versa, occurring when the coefficients of the associated nonlinear Schrödinger equation become singular. The fully nonlinear results obtained by the prescribed numerical method are compared with the predictions by the weakly nonlinear theories, and good agreement is achieved.
MSC:
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 76B45 | Capillarity (surface tension) for incompressible inviscid fluids |
| 76M40 | Complex variables methods applied to problems in fluid mechanics |
Keywords:
gravity-capillary wave; Euler equations; free boundary problem; time-dependent conformal map; interpolation methodReferences:
| [1] | Melcher, J. R.; Taylor, G. I., Electrohydrodynamics: A review of the role of interfacial shear stresses, Annu. Rev. Fluid Mech., 1, 111-146 (1969) |
| [2] | Papageorgiou, D. T., Film flows in the presence of electric fields, Ann. Rev. Fluid Mech., 51, 155-187 (2019) · Zbl 1412.76013 |
| [3] | Taylor, G.; McEwan, A., The stability of a horizontal fluid interface in a vertical electric field, J. Fluid Mech., 22, 1, 1-15 (1965) · Zbl 0129.20604 |
| [4] | Melcher, J. R.; Schwarz, W. J., Interfacial relaxation over stability in a tangential electric field, Phys. Fluids, 11, 12, 2604-2616 (1968) |
| [5] | Barannyk, L. L.; Papageorgiou, D. T.; Petropoulos, P. G., Suppression of Rayleigh-Taylor instability using electric fields, Math. Comput. Simulation, 82, 1008-1016 (2012) · Zbl 1243.78010 |
| [6] | Cimpeanu, R.; Papageorgiou, D. T.; Petropoulos, P. G., On the control and suppression of the Rayleigh-Taylor instability using electric fields, Phys. Fluids, 26, Article 022105 pp. (2014) · Zbl 1321.76029 |
| [7] | Guan, X.; Wang, Z., Interfacial electrohydrodynamic solitary waves under horizontal electric fields, J. Fluid Mech., 940, A15 (2022) · Zbl 1493.76121 |
| [8] | Zubarev, N. M.; Kochurin, E., Nonlinear dynamics of the interface between fluids at the suppression of Kelvin-Helmholtz instability by a tangential electric field, JETP Lett., 104, 275-280 (2016) |
| [9] | Gao, T.; Milewski, P. A.; Papageorgiou, D. T.; Vanden-Broeck, J.-M., Dynamics of fully nonlinear capillary-gravity solitary waves under normal electric fields, J. Engrg. Math., 108, 107-122 (2018) · Zbl 1444.76034 |
| [10] | Gao, T.; Doak, A.; Vanden-Broeck, J.-M.; Wang, Z., Capillary-gravity waves on a dielectric fluid of finite depth under normal electric field, Eur. J. Mech. B Fluids, 77, 98-107 (2019) · Zbl 1472.76015 |
| [11] | Easwaran, C., Solitary waves on a conducting fluid layer, Phys. Fluids, 31, 3442-3443 (1988) · Zbl 0654.76105 |
| [12] | 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, Article 031703 pp. (2007) · Zbl 1146.76394 |
| [13] | Wang, Z., Modelling nonlinear electrohydrodynamic surface waves over three-dimensional conducting fluids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 473, Article 20160817 pp. (2017) · Zbl 1404.76298 |
| [14] | Lin, Z.; Zhu, Y.; Wang, Z., Local bifurcation of electrohydrodynamic waves on a conducting fluid, Phys. Fluids, 29, Article 032107 pp. (2017) |
| [15] | 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, Quart. J. Mech. Appl. Math., 73, 3, 231-250 (2020) · Zbl 1472.76025 |
| [16] | Phillips, O. M., On the dynamics of unsteady gravity waves of finite amplitude, I: The elementary interactions, J. Fluid Mech., 9, 2, 193-217 (1960) · Zbl 0094.41101 |
| [17] | Zakharov, V. E., Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9, 190-194 (1968) |
| [18] | Benney, D. J.; Newell, A. C., The propagation of nonlinear wave envelopes, J. Math. Phys., 46, 1-4, 133-139 (1967) · Zbl 0153.30301 |
| [19] | McGoldrick, L. F., Resonant interactions among capillary-gravity waves, J. Fluid Mech., 21, 305-331 (1965) · Zbl 0125.44204 |
| [20] | Simmons, W. F., A variational method for weak resonant wave interactions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 309, 1499, 551-577 (1969) |
| [21] | Case, K. M.; Chiu, S. C., Three-wave resonant interactions of gravity-capillary waves, Phys. Fluids, 20, 5, 742-745 (1977) · Zbl 0397.76013 |
| [22] | McGoldrick, L. F., On Wilton’s ripples: a special case of resonant interactions, J. Fluid Mech., 42, 1, 193-200 (1970) · Zbl 0208.56403 |
| [23] | Benney, D. J., A general theory for interactions between short and long waves, Stud. Appl. Math., 56, 1, 81-94 (1977) · Zbl 0358.76011 |
| [24] | Djordjevic, V. D.; Redekopp, L. G., On two-dimensional packets of capillary-gravity waves, J. Fluid Mech., 79, 4, 703-714 (1977) · Zbl 0351.76016 |
| [25] | Craig, W.; Guyenne, P.; Sulem, C., The surface signature of internal waves, J. Fluid Mech., 710, 277-303 (2012) · Zbl 1275.76059 |
| [26] | Dyachenko, A. I.; Zakharov, V. E.; Kuznetsov, E. A., Nonlinear dynamics of the free surface of an ideal fluid, Plasma Phys. Rep., 22, 10, 829-840 (1996) |
| [27] | Craik, A. D.D., Wave Interactions and Fluid Flows (1985), Cambridge University Press · Zbl 0581.76002 |
| [28] | Kaup, D. J., The solution of the general initial value problem for the full three-dimensional three-wave resonant interaction, Physica D, 3, 1-2, 374-395 (1981) · Zbl 1194.35449 |
| [29] | Jones, M. C.W., Nonlinear stability of resonant capillary-gravity waves, Wave Motion, 15, 3, 267-283 (1992) · Zbl 0745.76022 |
| [30] | Gao, T.; Milewski, P. A.; Wang, Z., Capillary-gravity solitary waves on water of finite depth interacting with a linear shear current, Stud. Appl. Math., 147, 3, 1036-1057 (2021) · Zbl 1481.76054 |
| [31] | Milewski, P. A.; Vanden-Broeck, J.-M.; Wang, Z., Dynamics of steep two-dimensional gravity-capillary solitary waves, J. Fluid Mech., 664, 466-477 (2010) · Zbl 1221.76052 |
| [32] | Vanden-Broeck, J.-M., Gravity-Capillary Free-Surface Flows (2010), Cambridge University Press · Zbl 1202.76002 |
| [33] | Tanaka, M., Maximum amplitude of modulated wavetrain, Wave Motion, 2, 6, 559-568 (1990) |
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