Model Equations for Three-Dimensional Nonlinear Water Waves under Tangential Electric Field

Bo Tao

doi:10.1155/2017/9312681

Abstract

We are concerned with gravity-capillary waves propagating on the surface of a three-dimensional electrified liquid sheet under a uniform electric field parallel to the undisturbed free surface. For simplicity, we make an assumption that the permittivity of the fluid is much larger than that of the upper-layer gas; hence, this two-layer problem is reduced to be a one-layer problem. In this paper, we propose model equations in the shallow-water regime based on the analysis of the Dirichlet-Neumann operator. The modified Benney-Luke equation and Kadomtsev-Petviashvili equation will be derived, and the truly three-dimensional fully localized traveling waves, which are known as “lumps” in the literature, are numerically computed in the Benney-Luke equation.

Highlights

Interfacial electrohydrodynamic waves have many important applications in mechanical, chemical, and electrical industries, such as electrospray ionization, cooling systems, coating process, and electrowetting
The study of the stability of interfacial electrohydrodynamic waves was initiated by Melcher [3] and Taylor and McEwan [13], and the role of interfacial stresses resulting from electrodes was reviewed by Melcher and Taylor [4]
Considerable effects have been put into the modeling and numerical studies of nonlinear interfacial electrohydrodynamic waves

Summary

Introduction

Interfacial electrohydrodynamic waves have many important applications in mechanical, chemical, and electrical industries, such as electrospray ionization, cooling systems, coating process, and electrowetting (see [1,2,3,4,5,6,7,8,9,10] and the references therein). Hammerton [17] studied solitary waves in the KdV-Benjamin-Ono equation, a model arising from a conducting liquid sheet under a normal electric field generated by two electrodes with a sufficiently large separation distance. Recent study on 3D electrohydrodynamic modeling has focused on conducting fluids under normal electric fields: Hunt et al [18] proposed a one-way model with weak transverse variations which is called the Benjamin-Ono-Kadomtsev-Petviashvili equation, Aliev and Yurchenko [14] established a reduced system for the same setup, and Wang [8] developed fully nonlinear numerical models and weakly nonlinear theories for electrohydrodynamic surface waves in the Hamiltonian framework based on analyses of the Dirichlet-Neumann operators. The main numerical results are followed, including the typical profiles of solitary waves, the bifurcation diagrams, and the formation of lumps resulting from the transverse instability of plane solitary waves

Formulation

Dirichlet-Neumann Operators

Long-Wave Models

Conclusion