Nonlinear analysis and solitary waves for two superposed streaming electrified fluids of uniform depths with rigid boundaries

Abstract

The nonlinear modulation of the interfacial waves of two superposed dielectric fluids with uniform depths and rigid horizontal boundaries, under the influence of constant normal electric fields and uniform horizontal velocities, is investigated using the multiple-time scales method. It is found that the behavior of small perturbations superimposed on traveling wave trains can be described by a nonlinear Schrödinger equation in a frame of reference moving with the group velocity. Wave-like solutions to this equation are examined, and different types of localized excitations (envelope solitary waves) are shown to exist. It is shown that when these perturbations are neutrally stable and sufficiently long, solutions to the nonlinear Schrödinger equation may be approximated by the well-known Korteweg-de Vries equation. The speed of the solitary on the interface is seen to be reduced by the electric field. It is found that there are two critical values of the applied voltage that lead to (i) breaking up of the solitary waves, and (ii) bifurcation of solutions of the governing equations. On the other hand, the complex amplitude of standing wave trains near the marginal state is governed by a similar type of nonlinear Schrödinger equation in which the roles of time and space are interchanged. This equation, under a suitable transformation, is obtained as the Korteweg-de Vries equation with a variable coefficient. It is shown that this type of equations admit a solitary wave type of solutions with variable speed. Using the tangent hyperbolic method, it is observed that the wave speed increases as well as decreases, with the increase of electric field values, according to the chosen wavenumbers range. Finally, the nonlinear stability analysis is discussed in view of the coefficients of nonlinear Schrödinger equation to show the effects of various physical parameters, and also to recover the some limiting cases studied earlier in the literature.

Formation of curvature singularities on the interface between dielectric liquids in a strong vertical electric field

“…IV, this means that the influence of nonlinearity shifts the time of singularity formation by ∼ |S| 1/2 . Note that in the case of such dependence of on the parameter S, even the rough estimate for the surface-slope angles (22) predicts the formation of strong singularities violating the small-angle approximation. In accordance with (29), the boundary shape becomes singular at the moment t = t c : η(x,t c ) = πSδ(x), where δ(x) is the Dirac δ function.…”

“…As a rule, the resulting equations are nonlocal (i.e., they contain integrodifferential operators), which seriously hampers their analysis. In most publications, nonlinear waves on the surface of dielectric or conducting fluids are considered under the assumption that the characteristic wavelength is much larger than the depth of the fluid layer [7][8][9][10][11][12][13][14][15] or that the excited wave packet is spectrally narrow [6,[16][17][18][19][20][21][22][23]. Both approaches make it possible to reduce the original problem to the consideration of relatively simple local partial differential equations.…”

Nonlinear electroviscoelastic potential flow instability theory of two superposed streaming dielectric fluids

“…Assuming that A depends on x 2 , y 2 , and t 2 through the transformation ␥ 2 ϭ k Ϫ1 ͓͑k x x 2 ϩ k y y 2 ͒ Ϫ ͑k x k x ϩ k y k y ͒t 2 ͔ and = t 2 [7], we obtain finally from (34) the following nonlinear Ginzburg-Landau equation with complex coefficients:…”

“…They found that the electric field plays a dual role in the stability criterion in the sense that it stabilized perturbations having relatively small wavenumbers if the dielectric constant of the upper fluid is smaller than that of the lower fluid and vice versa. For recent developments of this topic, see the investigations of El-Sayed and co-workers [7][8][9][10][11].…”

Nonlinear Kelvin-Helmholtz Instability of Two Superposed Dielectric Finite Fluids in Porous Medium Under Vertical Electric Fields

“…It was shown that the complex amplitude of a quasi-monochromatic traveling wave can be described by a nonlinear Schrö dinger equation in a frame of reference moving with the group velocity. For a recent developments of this topic, see the investigations of Papageorgiou and Petropoulos (2004), Shankar and Sharma (2004), Tomar et al (2007), Ugug and Aubry (2008), El-Sayed (2008), and Yecko (2009. It should be noted that in all the above-mentioned studies, the medium has been considered to be nonporous.…”