Yusry O. El-Dib

Nonlinear Mathieu equation and coupled resonance mechanism

Abstract The method of multiple-scales is used to determine a third-order solution for a cubic nonlinear Mathieu equation. The perturbation solutions are imposed on the so-called solvability conditions. Solvability conditions in the non-resonance case yield the standard Landau equation. Several types of a parametric Landau equation are derived in the neighborhood of five different resonance cases. These parametric Landau equations contain a parametric complex conjugate term or a parametric second-order complex conjugate term or a parametric complex conjugate term as well as a parametric second-order term. Necessary and sufficient conditions for stability are performed in each resonance case. Stability criteria correspond to each parametric Landau equation and are derived by linear perturbation. Stability criteria for the non-trivial steady-state response are discussed. The analysis leads to simultaneous resonance. Transition curves are performed in each case. Numerical calculations are made for some transition curves to illustrate the coupled resonance regions, where the induced stability tongues within the instability tongues are observed. The amplitude of the periodic coefficient of Mathieu equation plays a dual role in the stability criteria for nonlinear Mathieu equation.

Nonlinear stability of Kelvin-Helmholtz waves in magnetic fluids stressed by a time-dependent acceleration and a tangential magnetic field

The nonlinear stability of surface waves propagating between two superposed streaming magnetic fluids is investigated. The fluids are stressed by a constant tangential magnetic field and a vertical periodic acceleration. The solution employs the method of multiple scales. Owing to the periodicity, resonant cases appear. Two parametrically nonlinear Schrodinger equations are derived for the resonant cases to describe the elevation of weakly nonlinear capillary waves. The standard nonlinear Schrodinger equation is satisfied for the non-resonant case. Necessary and sufficient conditions for stability are obtained. A formula for the surface elevation is obtained in each case. It is found that the magnetic field, the velocities and the frequency of the applied periodic force play dual roles in the resonant region. Investigation of the stability criterion by nonlinear perturbation shows that an increase in the acceleration frequency has a stabilizing effect. The stabilizing role of the frequency is due to the destabilizing effect of the amplitude of the periodic acceleration.

Nonlinear stability of surface waves in magnetic fluids: effect of a periodic tangential magnetic field

Nonlinear wave propagation on the surface between two superposed magnetic fluids stressed by a tangential periodic magnetic field is investigated using the method of multiple scales. A stability analysis reveals the existence of both nonresonant and resonant cases. From the solvability conditions, three types of nonlinear Schrodinger equation are obtained. The necessary and sufficient conditions for stability are obtained in each case. Formulae for the surface elevation are also obtained in both the non-resonant and the resonant cases. It is found from the numerical calculation that the tangential periodic magnetic field plays a dual role in the stability criterion, while the field frequency has a destabilizing influence.

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.

Instability of parametrically second- and third-subharmonic resonances governed by nonlinear Shrödinger equations with complex coefficients

Abstract A theoretical analysis of the parametric harmonic response of two resonant modes is made based on a cubic nonlinear system. The analysis based on the method of multiple scales. Two types of the modified nonlinear Schrodinger equations with complex coefficients are derived to govern the resonance wave. One of these equations contains the first derivatives in space for a complex-conjugate type as well as a linear complex-conjugate term that is valid in the second-harmonic resonance cases. The second parametric equation contains a complex-conjugate type which is valid at the third-subharmonic resonance case. Estimates of nonlinear coefficients are made. The resulting equations have an interesting in many dynamical and physical cases. Temporal modulational method is confirmed to discuss the stability behavior at both parametric second- and third-harmonic resonance cases. Furthermore, the Benjamin–Feir instability is discussed for the sideband perturbation. The instability behavior at the sharp resonance is examined and the existence of the instability is found.

Nonlinear Electrohydrodynamic Stability of a Fluid Layer: Effect of a Tangential Electric Field

The weakly nonlinear electrohydrodynamic stability of fluid layer sandwiched between two semi-infinite fluids is investigated. The nonlinear theory of perturbation is applied for symmetric and anti-symmetric modes. The method of multiple scales is used to expand the various perturbation quantities to yield the linear and successive nonlinear partial differential equations of the various orders. The solutions of these equations are obtained. The application of the boundary conditions leads to two nonlinear Schrodinger equations. It is found that the presence of the tangential field plays a stabilizing role and can be used to suppress the instability of the system at a given wavenumber which is unstable linear stability. Numerical calculations show a global stability for certain wavenumbers. A local instability is also observed in the graphs. The field plays a dual role. It is observed that the change of the layer thickness redistributes the stable areas.

Electroviscoelastic Rayleigh-Taylor instability of Maxwell fluids: I. Effect of a constant tangential electric field

The stability of the Rayleigh-Taylor model for an electroviscoelastic Maxwell fluid are investigated. The method of multiple scales is used in order to obtain the stability conditions. A transcendental dispersion relation is obtained at zero-order. The special case, when the two fluids have the same kinematic viscosity, is considered to relax the complexity of the transcendental dispersion relation. The solvability conditions introduce a first-order differential equation. It is found that the increase in the relaxation time lambda has a destabilizing influence. Also the increase in the kinematic viscosity in the presence of the parameter lambda yields a destabilizing effect. The increase in the kinematic viscosity in the absence of elasticity (pure viscous fluids) has a stabilizing effect.

Nonlinear interfacial stability for magnetic fluids in porous media

Abstract The weakly nonlinear stability is employed to analyze the interfacial phenomenon of two magnetic fluids in porous media. The effect of an oblique magnetic field to the separation face of two fluids is taken into account. The solutions of equations of motion under nonlinear boundary conditions lead to deriving a nonlinear equation in terms of the interfacial displacement. This equation is accomplished by utilizing the cubic nonlinearity. The method of multiple scale expansion is employed in order to obtain a dispersion relation for the first-order problem and nonlinear Ginzburg–Landau equation, for the higher-order problem, describing the behaviour of the system in a nonlinear approach. Regions of stability and instability are identified for the magnetic field intensity versus the wave number. It is found that the oblique magnetic filed has a stabilizing influence under some certain conditions for the directions of the magnetic fields. The resistance coefficient has a destabilizing influence in the linear description. Further, in the nonlinear scope, the increase of the resistance parameters plays both stabilizing and destabilizing role in the stability criteria.

The stability of a rigidly rotating magnetic fluid column effect of a periodic azimuthal magnetic field

The stability of an infinitely long magnetic fluid column of weak viscous effects is investigated. The column is subjected to a periodic azimuthal magnetic field and a rigid-body rotation. Non-axisymmetric two-dimensional perturbations are considered in this investigation. Linear analysis leads to a Mathieu equation with complex coefficients. The analytical results show that the constant magnetic field plays a stabilizing role and can be used to suppress the instability due to the rotation. When the field has been oscillating, the stabilizing role of the amplitude of the magnetic field decreases somewhat due to the applied frequency . The oscillating magnetic field plays a dual role in the stability criterion. The increase of the azimuthal wavenumber decreases the unstable region due to the increase of the column radius. A small viscosity plays a destabilizing effect due to the influence of the angular velocity in the presence of a constant or an oscillating magnetic field. A magnetic column can be stabilized at a given azimuthal wavenumber by a suitable choice of the angular velocity, the density and viscosity for the outer fluid being greater than the corresponding parameters for the inside fluid.

Nonlinear gravitational stability of streaming in an electrified viscous flow through porous media

Abstract Weakly nonlinear Kelvin–Helmholtz instability for two viscous fluids streaming through porous media is investigated. The electro-gravitational stability of the horizontal plane interface is examined. A vertical or a horizontal electric field stresses the system. The linear form of equation of motion is solved in the light of the nonlinear boundary conditions. The present boundary value problem leads to construct nonlinear characteristic equation. This nonlinear characteristic equation has complex coefficients for the elevation function. The nonlinearity is kept to the third order. The method of multiple scales, in both space and time, is used. The use of the Taylor expansion through the multiple scale scheme leads to the derivation of the well-known nonlinear Schrodinger equation with complex coefficients from the nonlinear characteristic equation. This equation describes the evolution of the wave train up to cubic order, and may be regarded as the counterparts of the single nonlinear Schrodinger equation that occurs in the non-resonance case. The relation between the stratified kinematic viscosity and the porous permeability is performed in order to control the marginal state representation. This marginality is utilised in order to relax the complexity of the linear dispersion relation. Stability conditions are discussed both analytically and numerically, and stability diagrams are obtained. Regions of stability and instability are identified. It is found that the porosity of the media increases the destabilizing influence for the fluid density. In nonlinear scope, a destabilizing influence for the upper porous permeability is recorded, while a stabilizing influence is found for the lower porous permeability. Both the horizontal and vertical electric fields are still playing the same roles in linear and nonlinear examinations as in the non-porous media. Two opposite roles are presented for the variation of the stratified fluid velocity V . A stabilizing influence for V ⩽1 and a destabilizing effect for V >1 are illustrated in this examination. A dual role in the nonlinear examination is recorded for values of the wave-frequency.