A new application of the Korteweg–de Vries Benjamin-Ono equation in interfacial electrohydrodynamics

Abstract

We consider waves on a layer of finite depth governed by the Euler equations in the presence of gravity, surface tension, and vertical electric fields. We use perturbation theory to identify canonical scalings and to derive a Korteweg-de Vries Benjamin-Ono equation arising in interfacial electrohydrodynamics. When the Bond number is equal to 1 / 3, dispersion disappears and the equation reduces to the Benjamin-Ono equation. In the additional limit of vanishing electric fields, we show how to obtain a new evolution equation that contains third-and fifth-order dispersion as well as a nonlocal electric field term.Classical nonlinear water wave theories leading to evolution equations such as that of Korteweg-de Vries ͑KdV͒ 1 are important in uncovering nonlinear mechanisms and establishing an analytical foundation for further studies, e.g., direct computations. Our main interest here is the novel derivation of such nonlinear models including the influence of a vertical electric field. The importance of interfacial electrohydrodynamics phenomena has been reviewed by Melcher and Taylor, 2 and recent applications can be found in processes such as the electrostatic liquid film radiator ͑Griffing et al. 3 and references therein͒ and lithographically induced self-assembly using liquid films on microfluidic scales ͑Wu and Russel 4 and references therein͒. In some of our recent studies, 5-8 we considered the effect of horizontal and vertical electric fields on nonlinear features ͑such as traveling waves or film rupture͒ of inviscid liquid sheets and layers, by direct computations of the Euler equations and reduced long wave models; comparison between the two indicates that long wave models do quite well in capturing flow quantities such as wave speeds, for example.We consider an inviscid, incompressible, and irrotational flow in a liquid layer ͑region 1͒, bounded below by a wall electrode at y = 0 and above by a free surface at y = h 0 + ͑x , t͒, where h 0 is the mean layer thickness. The fluid motion is described by a velocity potential ͑x , y , t͒ satisfying Laplace's equation in region 1. Surface tension with coefficient and gravity, g, are included. We denote the voltage potential by V and choose V =0 at y = 0. A vertical electric field is imposed by requiring V ϳ E 0 y as y → ϱ, with E 0 a constant. We assume that the fluid in region 1 is a perfect conductor so that V = 0 everywhere in it. The region y Ͼ h 0 + , denoted by region 2, is occupied by a hydrodynamically passive dielectric having permittivity . The voltage potential V͑x , y , t͒ satisfies Laplace's equation there.Given a typical velocity c 0 = ͱ gh 0 , typical free surface amplitude a, and a typical horizontal length scale l, we define dimensionless variables as follows:Here, y ͑1,2͒ denote the vertical coordinates in regions 1 and 2, respectively. We also definewhich represent an amplitude and depth parameters, respectively ͑see Keller 9 and Whitham 10 ͒. In terms of the variables ͑1͒ and ͑2͒, the governing equations and boundary conditions ͑dropping the pr...

Existence of solitary travelling waves in interfacial electrohydrodynamics

“…A normal electric field has been considered [3,4,5,6] when the field is due to parallel electrodes with separation distance comparable to the depth of the fluid. When the normal electric field tends to a constant far from the surface, corresponding to parallel electrodes with very large separation distance, a Kortweg-deVries Benjamin-Ono type equation is obtained for particular wavelength and amplitude scalings [7,8]. In the present paper a more general analysis for the normal electric field case derivation is provided, which allows for a wider range of wavelength and amplitude scalings and includes the effect of electrode separation distance.…”

“…We consider this case to allow direct comparison with earlier works [7,8]. However this case should correspond to the limit d → ∞, the exact nature of the limit emerges as part of the analysis presented in §3.…”

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

“…More than that, Barannyk et al showed in [15] that the tangential electric field can even suppress the RayleighTaylor 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 vacuumfluid 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.).…”

“…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][15][16][17][18][19][20]) and touchdown singularities (e.g., [7,21]). Considerable effects have been put into the modeling and numerical studies of nonlinear interfacial electrohydrodynamic waves.…”

A study of the effects of electric field on two-dimensional inviscid nonlinear free surface flows generated by moving disturbances

“…For E b = 0, it can be viewed as a forced Benjamin-Ono Korteweg de Vries equation. It generalises the unforced equation derived in [5].…”