Kelvin--Helmholtz instability of a horizontal interface between a finite subsonic gas and a finite magnetic liquid

1998 ◽  
Vol 76 (5) ◽  
pp. 361-374 ◽  
Author(s):  
K Zakaria
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Solid Surfaces ◽  
Wave Number ◽  
Magnetic Liquid ◽  
Helmholtz Instability ◽  
Perturbed System ◽  
Progressive Waves ◽  
The Stability ◽  
Horizontal Interface

The nonlinear Kelvin-Helmholtz instability of a horizontal interface between a magnetic inviscid incompressible liquid and an inviscid laminar subsonic gas is investigated. The gas and the liquid are assumed to have finite thicknesses. The applied magnetic field is parallel to the solid surfaces of the considered system. The method of multiple scales is used to obtain two nonlinear Schrodinger equations describing the behaviour of the perturbed system. The stability of the progressive waves is discussed theoretically. The nonlinear cutoff wave number is obtained, where the stability conditions of the standing waves are obtained. A numerical example is applied to discuss the stability diagrams.PACS Nos.: 51.60 and 47.20

The Rayleigh–Taylor instability in magnetic fluids and nonlinear interfacial waves

1992 ◽  
Vol 70 (8) ◽  
pp. 603-609 ◽  
Author(s):  
Abdel Raouf F. Elhefnawy
Keyword(s):  
Nonlinear Evolution ◽  
Multiple Scale ◽  
Magnetic Fluids ◽  
Wave Number ◽  
Perturbed System ◽  
Stability Diagrams ◽  
Rayleigh Taylor Instability ◽  
The Stability ◽  
Normal Magnetic Field ◽  
Horizontal Interface

The nonlinear evolution of a horizontal interface separating two magnetic fluids of different densities, including surface tension effects, is investigated. The fluids are considered incompressible and inviscid, being stressed by the force of gravity, the normal magnetic field, and a constant acceleration in a direction normal to the interface. The method of multiple-scale perturbations is used to obtain two nonlinear Schrödinger equations describing the behavior of the perturbed system. The stability of the perturbed system is discussed both analytically and numerically, and the stability diagrams are obtained. We also obtain the nonlinear cutoff wave number, which separates the region of stability from that of instability.

The effect of magnetic fields on the nonlinear instability of two superposed magnetic streaming fluids, each of a finite depth

1995 ◽  
Vol 73 (3-4) ◽  
pp. 163-173 ◽  
Author(s):  
Abdel Raouf F. Elhefnawy
Keyword(s):  
Magnetic Field ◽  
Multiple Scale ◽  
Finite Depth ◽  
Nonlinear Instability ◽  
Helmholtz Instability ◽  
Perturbed System ◽  
Stability Diagrams ◽  
The Stability ◽  
Normal Magnetic Field ◽  
Horizontal Interface

The nonlinear Kelvin–Helmholtz instability of a horizontal interface separating two flowing superposed magnetic fluids of finite depths is described in the presence of a normal magnetic field. The fluids are taken to be incompressible and inviscid and the motion is assumed to be irrotational. The method of multiple-scale perturbations is used to obtain two nonlinear Schrödinger equations describing the behaviour of the perturbed system. The stability of the system is discussed both theoretically and numerically and the stability diagrams are obtained. The nonlinear cutoff magnetic field that separates the regions of instability from those of stability is also obtained.

On the modulation of water waves on shear flows

1976 ◽  
Vol 347 (1651) ◽  
pp. 537-546 ◽  
Keyword(s):  
Water Waves ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Vries Equation ◽  
Harmonic Wave ◽  
Short Wave ◽  
Long Wave ◽  
Stokes Waves ◽  
The Stability ◽  
Wave Limit

The method of multiple scales is used to examine the slow modulation of a harmonic wave moving over the surface of a two dimensional channel. The flow is assumed inviscid and incompressible, but the basic flow takes the form of an arbitrary shear. The appropriate nonlinear Schrödinger equation is derived with coefficients that depend, in a complicated way, on the shear. It is shown that this equation agrees with previous work for the case of no shear; it also agrees in the long wave limit with the appropriate short wave limit of the Korteweg-de Vries equation, the shear being arbitrary. Finally, it is remarked that the stability of Stokes waves over any shear can be examined by using the results derived here.

scholarly journals Stability and Hopf Bifurcation Analysis of an Epidemic Model by Using the Method of Multiple Scales

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Wanyong Wang ◽  
Lijuan Chen
Keyword(s):  
Periodic Solution ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Epidemic Model ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Value Of Time ◽  
Nonlinear Incidence Rate ◽  
The Stability ◽  
Bifurcating Periodic Solution

A delayed epidemic model with nonlinear incidence rate which depends on the ratio of the numbers of susceptible and infectious individuals is considered. By analyzing the corresponding characteristic equations, the effects of time delay on the stability of the equilibria are studied. By choosing time delay as bifurcation parameter, the critical value of time delay at which a Hopf bifurcation occurs is obtained. In order to derive the normal form of the Hopf bifurcation, an extended method of multiple scales is developed and used. Then, the amplitude of bifurcating periodic solution and the conditions which determine the stability of the bifurcating periodic solution are obtained. The validity of analytical results is shown by their consistency with numerical simulations.

Parametric Stability of Axially Accelerating Viscoelastic Beams With the Recognition of Longitudinally Varying Tensions

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
Li-Qun Chen ◽  
You-Qi Tang
Keyword(s):  
Governing Equation ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Finite Support ◽  
Governing Equations ◽  
Parametric Stability ◽  
Stability Boundaries ◽  
Axial Acceleration ◽  
The Stability ◽  
Longitudinally Varying Tension

In this paper, the parametric stability of axially accelerating viscoelastic beams is revisited. The effects of the longitudinally varying tension due to the axial acceleration are highlighted, while the tension was approximately assumed to be longitudinally uniform in previous studies. The dependence of the tension on the finite support rigidity is also considered. The generalized Hamilton principle and the Kelvin viscoelastic constitutive relation are applied to establish the governing equations and the associated boundary conditions for coupled planar motion of the beam. The governing equations are linearized into the governing equation in the transverse direction and the expression of the longitudinally varying tension. The method of multiple scales is employed to analyze the parametric stability of transverse motion. The stability boundaries are derived from the solvability conditions and the Routh-Hurwitz criterion for principal and sum resonances. In terms of stability boundaries, the governing equations with or without the longitudinal variance of tension are compared and the effects of the finite support rigidity are also examined. Some numerical examples are presented to demonstrate the effects of the stiffness, the viscosity, and the mean axial speed on the stability boundaries. The differential quadrature scheme is developed to numerically solve the governing equation, and the computational results confirm the outcomes of the method of multiple scales.

ANALYSIS OF NONLINEAR MAGNETOELASTIC DYNAMICS AND STABILITY OF CONDUCTIVE CYLINDRICAL SHELLS

2010 ◽  
Vol 10 (01) ◽  
pp. 153-164
Author(s):  
YUDA HU ◽  
JIANG ZHAO ◽  
PI JUN ◽  
GUANGHUI QING
Keyword(s):  
Cylindrical Shell ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Approximate Analytical Solution ◽  
Transverse Magnetic ◽  
Field Equations ◽  
Thin Cylindrical Shell ◽  
The Stability ◽  
Magnetic Induction Intensity ◽  
Steady State Solutions

The nonlinear magnetoelastic vibration equations and electromagnetic field equations of a conductive thin cylindrical shell in magnetic fields are derived. The nonlinear principal resonances and dynamic stabilities of the cylindrical shell simply supported in a transverse magnetic field are investigated. Approximate analytical solution and bifurcation equations of the system with principal resonances are obtained by using the method of multiple scales. The stabilities and singularities of the steady-state solutions are analyzed and the stability criterion is given. The transition sets and bifurcation figures of unfolding parameters are also obtained. The variations of the resonance amplitudes with respect to the detuning parameter, the magnetic induction intensity, and the amplitude of excitations are presented. The corresponding phase trajectories in moving phase planes are given. The stabilities of solutions, characteristics of singular points, and bifurcation are analyzed. The impacts of electromagnetic and mechanical parameters on dynamic behaviors are discussed in detail.

He’s Multiple-Scale Solution for the Three-dimensional Nonlinear KH Instability of Rotating Magnetic Fluids

2019 ◽  
Vol 9 (1) ◽  
pp. 52-69 ◽  
Author(s):  
Yusry Osman El-Dib ◽  
Amal A Mady
Keyword(s):  
Multiple Scales ◽  
Velocity Ratio ◽  
Three Dimensional ◽  
Multiple Scale ◽  
Stability Criteria ◽  
Helmholtz Instability ◽  
Vertical Field ◽  
Horizontal Flux ◽  
The Stability ◽  
Transition Curves

This paper elucidates a trend in solving nonlinear oscillators of the rotating Kelvin-Helmholtz instability. The system is constituted by the vertical flux or the horizontal flux. He’s multiple-scales perturbation methodology has been applied and therefore the system is represented by a generalized homotopy equation. This approach ends up in a periodic answer to a nonlinear oscillator with high nonlinearity. The cubic-quintic nonlinear Duffing equation is obligatory as a condition to uniformly answer. This equation is employed to derive the stability criteria. The transition curves are plotted to investigate the stability image. It's shown that the angular velocity suppresses the instability. The tangential flux plays a helpful role, whereas the vertical field encompasses a destabilizing influence. Within the existence of the rotation, the velocity ratio reduces stability configuration.

Stability of Nonparallel Developing Flow in a Pipe to Nonaxisymmetric Disturbances

1983 ◽  
Vol 50 (1) ◽  
pp. 210-214 ◽  
Author(s):  
V. K. Garg
Keyword(s):  
Experimental Data ◽  
Critical Reynolds Number ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Main Flow ◽  
Wave Number ◽  
Developing Flow ◽  
Azimuthal Wave Number ◽  
Flow Velocity Profile ◽  
Axisymmetric Disturbances

Linear spatial stability of the nonparallel developing flow in a rigid circular pipe has been studied at several axial locations for nonaxisymmetric disturbances. The main flow velocity profile is obtained by Hornbeck’s finite-difference method assuming uniform flow at entry to the pipe. The method of multiple scales is used to account for all the nonparallel effects. It is found that the nonparallel developing flow is most unstable to nonaxisymmetric disturbances with azimuthal wave number n equal to unity. Axisymmetric disturbances are, however, more unstable than nonaxisymmetric disturbances with n ≥ 2 except in the near-entry region. The results show that the parallel flow theory overpredicts the critical Reynolds number by as much as 136.5 percent in the near entry region for the n = 1 disturbance. The present results compare well with the available experimental data.

The method of multiple scales and non-linear dispersive waves

1971 ◽  
Vol 48 (3) ◽  
pp. 463-475 ◽  
Author(s):  
Ali Hasan Nayfeh ◽  
Sayed D. Hassan
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Electron Plasma ◽  
Wave Number ◽  
Hot Electron ◽  
Dispersive Waves ◽  
Physical Systems ◽  
Non Linear ◽  
Temporal And Spatial ◽  
Number Case

The method of multiple scales is used to analyze three non-linear physical systems which support dispersive waves. These systems are (i) waves on the interface between a liquid layer and a subsonic gas flowing parallel to the undisturbed interface, (ii) waves on the surface of a circular jet of liquid, and (iii) waves in a hot electron plasma. It is found that the partial differential equations that govern the temporal and spatial variations of the wave-numbers, amplitudes, and phases have the same form for all of these systems. The results show that the non-linear motion affects only the phase. For the constant wave-number case, the general solution for the amplitude and the phase can be obtained.

Three-Dimensional Viscous Potential Electrohydrodynamic Kelvin–Helmholtz Instability Through Vertical Cylindrical Porous Inclusions With Permeable Boundaries

2013 ◽  
Vol 136 (2) ◽  
Author(s):  
Galal M. Moatimid ◽  
Mohamed A. Hassan
Keyword(s):  
Dispersion Relation ◽  
Three Dimensional ◽  
Wave Number ◽  
Helmholtz Instability ◽  
Large Wave ◽  
Porous Layers ◽  
Azimuthal Wave ◽  
Azimuthal Wave Number ◽  
Streaming Velocity ◽  
The Stability

In this paper, the electrohydrodynamic three-dimensional Kelvin–Helmholtz instability of a cylindrical interface with heat and mass transfer between liquid and vapor phases is studied. The liquid and the vapor are saturated, two coaxial cylindrical porous layers, and the suction/injection velocities for the fluids at the permeable boundaries are also taken into account. The dispersion relation is derived and the stability analysis is discussed for various parameters. It is found that the streaming velocity has a destabilizing effect, while the axial electric field has a stabilizing one. The suction for both the liquid and the steam has a destabilizing effect in contrast with the injection at both boundaries. The flow through porous structure is more stable than the pure flow. The case of the axisymmetric (for zero value of the azimuthal wave number m) and asymmetric (for nonzero value of the azimuthal wave number m) disturbances at large wavelength (at the wave number k→0) are always stable. Meanwhile, it is the same dispersion relation for the plane geometry at large wave number. Finally, our results are corroborated by comparing them with the previous published results.

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