Advances in Mathematical Physics

Volume 2017 (2017), Article ID 9312681, 8 pages

https://doi.org/10.1155/2017/9312681

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

School of Architecture Engineering, Neijiang Normal University, Sichuan 641100, China

Correspondence should be addressed to Bo Tao

Received 18 May 2017; Revised 18 September 2017; Accepted 11 October 2017; Published 12 November 2017

Academic Editor: Prabir Daripa

Copyright © 2017 Bo Tao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. 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 [1–10] and the references therein). The electric stresses have great impact on electrified free surfaces or interfaces, which can not only modify wave patterns but also change stability characteristics of the system. These influences are of great practical interest since they can lead to quantity (e.g., heat or mass) transfer enhancement (see, e.g., [5, 11, 12]). 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]. Recent theoretical research in this field has focused on the nonlinear phenomena and corresponding mechanisms, such as nonlinear coherent structures (e.g., [2, 5, 8, 10, 11, 14–20]) and touchdown singularities (e.g., [7, 21]). Considerable effects have been put into the modeling and numerical studies of nonlinear interfacial electrohydrodynamic waves.

For incompressible, inviscid, and irrotational flows, electrified interfacial waves were studied in the nonlinear regime under externally applied electric fields by different groups. The electric field is usually posed parallel or perpendicular to the undisturbed interface and is therefore called tangential or normal electric field, respectively. The orientation of the electric field with respect to the undisturbed interface plays an important role in the system. From the point of view of the linear theory, the dispersion relations are of the forms for two-dimensional gravity-capillary waves in deep water under normal and tangential electric field, respectively, where is the frequency, the wavenumber, the acceleration due to gravity, the surface tension coefficient, the strength of the electric field, the density of the liquid, and the ratio of permittivities. The forms of functions and depend on the specific problem; for example, for normal electric field acting on a perfect conducting fluid ([2, 8]) and for a dielectric fluid under a tangential electric field ([10]). Linearly speaking, a tangential electric field provides a dispersive contribution so as to stabilize the system, while a considerably strong normal electric field can provide energy to a certain range of wavenumbers to induce instability. On the nonlinear side, Tilley et al. [7] proposed a nonlinear long-wave system to study the dynamics of electrified liquid sheets and found that the tangential electric field can delay the formation of the film rupture. More than that, Barannyk et al. showed in [15] that the tangential electric field can even suppress the Rayleigh-Taylor instability in some situations. On the contrary, it can be deduced from the work by Gleeson et al. [16], Papageorgiou et al. [21], Lin et al. [2], and Wang [8] that the normal electric field has a destabilizing effect on the interface.

N. M. Zubarev and O. V. Zubareva [10, 20] and Tao and Guo [22] considered the electrified gas-fluid or vacuum-fluid interface so that they can make the assumption that the permittivity of the fluid was much larger compared to that of gas (the permittivities for pure water and air are 80 and 1, resp.). As a consequence, the actual two-layer problem could be reduced to one layer, and the theoretical and numerical techniques developed for free-surface water wave problems can be generalized to include the electric field.

On the numerical side, most researches focus on periodic waves (see, e.g., [5, 7, 10, 14, 21]); however the studies of electrified solitary waves have begun recently. Easwaran [23] first derived the Korteweg-de Vries (KdV) equation in the context of electrohydrodynamic waves which admits soliton solutions naturally. 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. Barannyk et al. [15] computed interfacial solitary waves based on a system describing long waves between two immiscible dielectric fluids. It is noted that KdV-Benjamin-Ono equation was first derived in the context of interfacial waves between two immiscible fluids when surface tension is sufficiently strong (e.g., [24, 25]).

All the aforementioned papers confine themselves to two-dimensional (2D) problems, but there are relatively fewer studies on the three-dimensional (3D) case. 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. In this paper, we are concerned with a dielectric liquid under a tangential electric field, and restoring forces due to gravity, surface tension, and Maxwell stresses are all taken into account. We will propose weakly nonlinear models (including both unidirectional and bidirectional equations) under Zubarev’s approximation and numerically search for truly 3D solitary waves (which is known as “lumps” in the literature) within these models.

The rest of the paper is structured as follows. In Section 2, the governing equations are described and simplified by making the assumption that the permittivity of the fluid is large. In Section 3, the Dirichlet-Neumann operator is introduced, which plays an important role in deriving weakly nonlinear models. In Section 4, a bidirectional model (the modified Benney-Luke equation) and a unidirectional model (the modified Kadomtsev-Petviashvili equation) are derived under the long-wave approximation. Then 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.

#### 2. Formulation

##### 2.1. Governing Equations

We consider an incompressible, inviscid, and irrotational fluid, moving on a solid wall, under a horizontal electric field with uniform strength in the far field. We introduce the Cartesian coordinates, such that and are horizontal variables, and points upwards with being the undisturbed air-water interface (see Figure 1). We denote the displacement of the air-water interface by ; then the velocity potential of the fluid, which is denoted by , is governed by And the voltage potentials within and above the fluid domain, denoted by and , respectively, are also governed by Laplace’s equation: The kinematic and dynamic boundary conditions at the interface are given by (see, e.g., [10, 22, 26]) where , , and is the ratio of the permittivity of water, , to that of air, . The detailed derivation of the electrostatic pressure at the interface between an ideal dielectric liquid and air (or vacuum) in the presence of free electric charges can be found in [26] or [5]. At the interface , the continuity condition of voltage potentials and the normal stress induced by electricity are, respectively, Following Barannyk et al. [15], the kinematic boundary conditions for and at are of the same form Finally, the far-field conditions for the electric field, complete the whole system.