Abdel Raouf F. Elhefnawy

Nonlinear electrohydrodynamic Kelvin-Helmholtz instability under the influence of an oblique electric field

The electrohydrodynamic stability of a horizontal interface separating two dielectric streaming fluids stressed by an oblique electric field is studied. The method of multiple scale perturbations is used in order to obtain uniformly valid expansions near the cutoff wavenumber separating stable and unstable deformations. Two nonlinear Schrodinger equations are obtained, one of which leads to the determination of the cutoff wavenumber. The other Schrodinger equation is used to analyze the stability of the system. These equations are derived when the difference between the velocities of the two fluids is less than or equal to the critical velocity. It is shown that, for subcritical values of the velocity difference, the wave train of constant amplitude is unstable against modulations whereas for the supercritical values the instability is governed by a nonlinear Klein-Gordon equation. From the latter equation, the various stability criteria are obtained.

Nonlinear electrohydrodynamic stability of a finitely conducting jet under an axial electric field

The nonlinear analysis of the electrohydrodynamic Rayleigh–Taylor instability of a cylindrical interface separating two conducting fluids of circular cross section is investigated in the absence of gravity. The fluids are assumed to be stressed by a uniform axial electric field. The analysis is based on the method of multiple scale perturbation. The linear dispersion relation, nonlinear Schrodinger equations and a nonlinear Klein–Gordon equation are obtained. The stability conditions of the perturbed system are discussed both analytically and numerically and stability diagrams are obtained. The stability diagrams are discussed in terms of various parameters of the problem. Regions of stability and instability are identified.

Nonlinear electrohydrodynamic Kelvin-Helmholtz instability with mass and heat transfer. Effect of a tangential field

The nonlinear electrohydrodynamic Kelvin-Helmholtz instability of two superposed dielectric fluids with interfacial transfer of mass and heat is presented for layers of finite thickness. The fluids are subjected to a tangential electric field. The method of multiple scale perturbations is used to obtain a dispersion relation for the first-order problem and a Ginzburg-Landau equation, for the higher-order problem, describing the behaviour of the system. The stability criterion is expressible in terms of various competing parameters representing the equilibrium heat flux, latent heat of evaporation, gravity, surface tension, densities of the fluids, dielectric constants of the fluids, thicknesses of the layers and thermal properties of the fluids. The stability of this system is discussed both analytically and numerically and the stability diagrams are obtained.

Nonlinear electrohydrodynamic Rayleigh-Taylor instability with mass and heat transfer subject to a vertical oscillating force and a horizontal electric field

Weakly nonlinear stability of interfacial waves propagating between two electrified inviscid fluids influenced by a vertical periodic forcing and a constant horizontal electric field is studied. Based on the method of multiple-scale expansion for a small-amplitude periodic force, two parametric nonlinear Schrödinger equations with complex coefficients are derived in the resonance cases. A standard nonlinear Schrödinger equation with complex coefficients is derived in the nonresonance case. A temporal solution is carried out for the parametric nonlinear Schrödinger equation. The stability analysis is discussed both analytically and numerically.

The effect of an axial electric field on the stability of cylindrical flows in the presence of mass and heat transfer and absence of gravity

The effect of an axial electric field on the Kelvin-Helmholtz instability of a cylindrical interface, which admits mass and heat transfer, between two fluids is studied. The fluids are in the form of coaxial cylindrical shells of different thicknesses. The gravitational effects are neglected. A dispersion relation that accounts for the growth of asymmetric waves is derived. It is found that the uniform axial electric field has a strong stabilizing influence on the cylindrical interface for all short and long wavelengths in all symmetric and asymmetric modes of perturbation. In contrast, the streaming shows a strong destabilizing influence. It is also found that the surface tension may be stabilized or destabilized according to certain conditions. The present analysis indicates that the instability criterion of the system is independent of the mass and heat transfer coefficient, but it is different from that in the same problem without mass and heat transfer: A comparison of the results, in the two cases, shows the destabilizing effect of the mass and heat transfer. In the absence of mass and heat transfer, it is demonstrated that there is a critical Weber number, below which the nonaxisymmetric disturbance becomes unstable. The Weber number, herein, is defined as the ratio of surface tension force to the inertial force. The critical Weber number depends on the wavelength of the disturbances, the density ratio between the two fluids and the Alfven wave velocity.

The effect of magnetic fields on the stability of a cylindrical flow with mass and heat transfer

Abstract The effect of axial and radial magnetic fields on the Kelvin-Helmholtz stability of a cylindrical interface between the vapor and liquid phases of a fluid is studied when the vapor is hotter than the liquid and the two phases are enclosed between two cylindrical surfaces coaxial with the interface, and when there is mass and heat transfer across the interface. Both axisymmetric and asymmetric disturbances are considered. The linear dispersion relations are obtained and discussed. It is found that a uniform axial magnetic field has a stabilizing effect on the interface, while the effect of a radial magnetic field depends strongly on the choice of some physical parameters of the system. It is also found that the instability criterion is independent of heat and mass transfer coefficient, but it is different fromthat in the same problem without heat and mass transfer. Finally, the heat and mass transfer has a destabilizing influence on the system.

Nonlinear Rayleigh-Taylor instability in magnetic fluids with uniform horizontal and vertical magnetic fields

A weakly nonlinear evolution of two dimensional wave packets on the surface of a magnetic fluid in the presence of an uniform magnetic field is presented, taking into account the surface tension. The method used is that of multiple scales to derive two partial differential equations. These differential equations can be combined to yield two alternate nonlinear Schrödinger equations. The first equation is valid near the cutoff wavenumber while the second equation is used to show that stability of uniform wave trains depends on the wavenumber, the density, the surface tension and the magnetic field. At the critical point, a generalized formulation of the evolution equation governing the amplitude is developed which leads to the nonlinear Klein-Gordon equation. From the latter equation, the various stability crteria are obtained.

Nonlinear electrohydrodynamic instability of two liquid layers

The nonlinear instability of two superposed dielectric fluids is studied under the influence of an oscillating external electric field. In addition, a small constant and normal electric field is superimposed on the system. With the use of the method of multiple scaling, a generalized derivation of the amplitude evolution is obtained in marginally unstable regions of parameter space. A Melnikov function is formulated for such an instability and it is shown that there exist transverse homoclinic orbits leading to chaotic motions.

Nonlinear Rayleigh-Taylor instability in magnetic fluids between two parallel plates

A nonlinear stage of the two-dimensional Rayleigh-Taylor instability for two magnetic fluids of finite thickness is studied by including the effect of surface tension between the two fluids. The system is subjected to a tangential magnetic field. The method of multiple scale perturbations is used in order to obtain uniformly valid expansions near the cutoff wavenumber separating stable and unstable deformations. Two nonlinear Schrödinger equations are obtained, one of which leads to the determination of the cutoff wavenumber. The other Schrödinger equation is used to analyze the stability of the system. It is found that if a finite-amplitude disturbance is stable, then a small modulation to the wave is also stable. It is also found that the tangential magnetic field plays a dual role in the stability criterion. Finally, the magnetic permeability constants of the fluid affect the stability conditions.

Nonlinear instability of superposed magnetic fluids in the presence of an oblique magnetic field

The nonlinear evolution of interfacial waves separating two magnetic fluids subjected to an oblique magnetic field is studied in two dimensions, with the use of the method of multiple scales. It is shown that the evolution of the envelope is governed by two partial differential equations. These equations can be combined to yield two alternate Schrödinger equations with cubic nonlinearity; one of them leads to the determination of the cutoff wave number separating stable from unstable deformations while the other Schrödinger equation is used to analyze the stability of the system. The stability of the system is discussed both theoretically and computationally, and the stability diagrams are obtained. It is found in the linear theory that the oblique magnetic field has a stabilizing influence if 0 ≤ θ1 + θ2 < π/2, or 3π/2 < θ1 + θ2 ≤ 2π and a destabilizing influence if π/2 < θ1 + θ2 < 3π/2, where 0 ≤ θj≤ π, (j=1, 2) and θ, is the angle between the field and the horizontal axis.In the nonlinear theory, the stability analysis reveals that there exist different regions of stability and instability. It is reported that the oblique magnetic field plays a dual role in the stability criterion and the angles θ1 and θ2 play a distinctive role in this analysis besides the effect of the variation of the magnetic permeabilities.