Nonlinear Kelvin—Helmholtz instability in magnetic fluids of finite thickness: effects of a tangential magnetic field

1994 ◽  
Vol 51 (3) ◽  
pp. 451-465 ◽  
Author(s):  
Abdel Raouf F. Elhefnawy
Keyword(s):  
Magnetic Field ◽  
Multiple Scales ◽  
Second Harmonic ◽  
Wave Train ◽  
Finite Thickness ◽  
Magnetic Fluids ◽  
Two Dimensions ◽  
Stability And Instability ◽  
Regions Of Stability ◽  
The Stability

The nonlinear stability of a horizontal interface separating two streaming magnetic fluids of finite thickenss is investigated in two dimensions. The fluids are considered to be inviscid and incompressible. The magnetic field is applied along the direction of streaming. The method of multiple scales, in both space and time, is used to examine the stability properties of the system arising from second-harmonic resonance. A pair of partial differential equations for the amplitude of the wave and its second harmonic are derived. These describe the evolution of the wave train up to cubic order, and may be regarded as the counterparts of the single nonlinear Schrödinger equation that occurs in the non-resonant case. The stability condition of this equation is discussed both analytically and numerically, and stability diagrams are obtained. Regions of stability and instability are identified. The nonlinear cut-off wavenumber separating the regions of stability from those of instability is obtained. The equation governing the evolution of the amplitude at the critical point is also obtained, which leads to a nonlinear Klein—Gordon equation.

scholarly journals Stability analysis of magnetic fluids in the presence of an oblique field and mass and heat transfer

2020 ◽  
Vol 330 ◽  
pp. 01035
Author(s):  
Rabah Djeghiour ◽  
Bachir Meziani
Keyword(s):  
Magnetic Field ◽  
Heat Transfer ◽  
Magnetic Fluids ◽  
Physical Parameters ◽  
Stability Diagrams ◽  
Stability And Instability ◽  
Mass And Heat Transfer ◽  
Model Equations ◽  
The Stability ◽  
Oblique Magnetic Field

In this paper, we investigate an analysis of the stability of a basic flow of streaming magnetic fluids in the presence of an oblique magnetic field is made. We have use the linear analysis of modified Kelvin-Helmholtz instability by the addition of the influence of mass transfer and heat across the interface. Problems equations model is presented where nonlinear terms are neglected in model equations as well as the boundary conditions. In the case of a oblique magnetic field, the dispersion relation is obtained and discussed both analytically and numerically and the stability diagrams are also obtained. It is found that the effect of the field depends strongly on the choice of some physical parameters of the system. Regions of stability and instability are identified. It is found that the mass and heat transfer parameter has a destabilizing influence regardless of the mechanism of the field.

Nonlinear hydrodynamic Rayleigh—Taylor instability of viscous magnetic fluids: effect of a tangential magnetic field

1994 ◽  
Vol 51 (1) ◽  
pp. 1-11 ◽  
Author(s):  
Yusry O. El-Dib
Keyword(s):  
Magnetic Field ◽  
Multiple Scales ◽  
Magnetic Fluids ◽  
Taylor Instability ◽  
Stable Region ◽  
Solvability Conditions ◽  
System A ◽  
Rayleigh Taylor Instability ◽  
The Stability ◽  
Complex Coefficients

The nonlinear Rayleigh—Taylor instability of viscous magnetic fluids is considered under the influence of gravity and surface tension in the presence of a constant tangential magnetic field. The method of multiple-scales expansion is employed. A nonlinear Schrödinger equation with complex coefficients is imposed from the solvability conditions and used to analyse the stability of the system. A quadratic dispersion relation with complex coefficients is obtained. The Hurwitz criterion for a quadratic polynomial with complex coefficients is used to control the stability of the system. It is found that an increase in the viscosity increases the extent of the stable region in the presence of a magnetic field. Finally it is shown that the magnetic permeability of the fluid affects the stability conditions.

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

1993 ◽  
Vol 49 (2) ◽  
pp. 317-330 ◽  
Author(s):  
Yusry O. El-Dib
Keyword(s):  
Magnetic Field ◽  
Multiple Scales ◽  
Sufficient Conditions ◽  
Magnetic Fluids ◽  
Nonlinear Wave Propagation ◽  
Field Frequency ◽  
Solvability Conditions ◽  
Periodic Magnetic Field ◽  
The Stability ◽  
Necessary And Sufficient

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 evolution of a wave packet propagating along a hot magnetoplasma column

1987 ◽  
Vol 37 (1) ◽  
pp. 107-115
Author(s):  
B. Ghosh ◽  
K. P. Das
Keyword(s):  
Magnetic Field ◽  
Multiple Scales ◽  
Nonlinear Evolution ◽  
Axial Magnetic Field ◽  
Modulational Instability ◽  
Wave Train ◽  
Dielectric Medium ◽  
Ion Effects ◽  
The Magnetic Field ◽  
Uniform Plasma

The method of multiple scales is used to derive a nonlinear Schrödinger equation, which describes the nonlinear evolution of electron plasma ‘slow waves’ propagating along a hot cylindrical plasma column, surrounded by a dielectric medium and immersed in an essentially infinite axial magnetic field. The temperature is included as well as mobile ion effects for ail possible modes of propagation along the magnetic field. From this equation the condition for modulational instability for a uniform plasma wave train is determined.

scholarly journals Levitation of non-magnetizable droplet inside ferrofluid

2018 ◽  
Vol 857 ◽  
pp. 398-448 ◽  
Author(s):  
Chamkor Singh ◽  
Arup K. Das ◽  
Prasanta K. Das
Keyword(s):  
Magnetic Field ◽  
Steady State ◽  
Viscosity Ratio ◽  
Temporal Dynamics ◽  
Liquid Droplet ◽  
Droplet Surface ◽  
Two Dimensions ◽  
Projection Algorithm ◽  
The Stability ◽  
The Individual

The central theme of this work is that a stable levitation of a denser non-magnetizable liquid droplet, against gravity, inside a relatively lighter ferrofluid – a system barely considered in ferrohydrodynamics – is possible, and exhibits unique interfacial features; the stability of the levitation trajectory, however, is subject to an appropriate magnetic field modulation. We explore the shapes and the temporal dynamics of a plane non-magnetizable droplet levitating inside a ferrofluid against gravity due to a spatially complex, but systematically generated, magnetic field in two dimensions. The coupled set of Maxwell’s magnetostatic equations and the flow dynamic equations is integrated computationally, utilizing a conservative finite-volume-based second-order pressure projection algorithm combined with the front-tracking algorithm for the advection of the interface of the droplet. The dynamics of the droplet is studied under both the constant ferrofluid magnetic permeability assumption as well as for more realistic field-dependent permeability described by Langevin’s nonlinear magnetization model. Due to the non-homogeneous nature of the magnetic field, unique shapes of the droplet during its levitation, and at its steady state, are realized. The complete spatio-temporal response of the droplet is a function of the Laplace number $La$ , the magnetic Laplace number $La_{m}$ and the Galilei number $Ga$ ; through detailed simulations we separate out the individual roles played by these non-dimensional parameters. The effect of the viscosity ratio, the stability of the levitation path and the possibility of existence of multiple stable equilibrium states is investigated. We find, for certain conditions on the viscosity ratio, that there can be developments of cusps and singularities at the droplet surface; we also observe this phenomenon experimentally and compare with the simulations. Our simulations closely replicate the singular projection on the surface of the levitating droplet. Finally, we present a dynamical model for the vertical trajectory of the droplet. This model reveals a condition for the onset of levitation and the relation for the equilibrium levitation height. The linearization of the model around the steady state captures that the nature of the equilibrium point goes under a transition from being a spiral to a node depending upon the control parameters, which essentially means that the temporal route to the equilibrium can be either monotonic or undulating. The analytical model for the droplet trajectory is in close agreement with the detailed simulations.

Hydromagnetic stability of a thin electrically conducting fluid film flowing down the outside surface of a long vertically aligned column

2009 ◽  
Vol 223 (2) ◽  
pp. 415-426 ◽  
Author(s):  
P-J Cheng
Keyword(s):  
Magnetic Field ◽  
Multiple Scales ◽  
Vertical Cylinder ◽  
Conducting Fluid ◽  
Fluid Film ◽  
Electrically Conducting ◽  
Free Film ◽  
Electrically Conducting Fluid ◽  
Non Linear ◽  
The Stability

This article considers the stability of a thin electrically conducting fluid film flowing down the outer surface of a long vertical cylinder in the presence of an applied magnetic field. Using the long-wave perturbation method to solve the generalized non-linear kinematic equations with free film interface, the normal mode approach is first used to compute the linear stability solution. The method of multiple scales is then used to obtain the weak non-linear dynamics. The results indicate that both subcritical instability and supercritical stability conditions are possible. The degree of instability in the film flow is intensified by the lateral curvature of the cylinder. The results also show that increasing the strength of the magnetic field tends to enhance the stability.

Nonlinear electrohydrodynamic Rayleigh–Taylor instability with mass and heat transfer: effect of a normal field

1994 ◽  
Vol 72 (9-10) ◽  
pp. 537-549 ◽  
Author(s):  
Abou El Magd A. Mohamed ◽  
Abdel Raouf F. Elhefnawy ◽  
Y. D. Mahmoud
Keyword(s):  
Heat Transfer ◽  
Multiple Scales ◽  
Landau Equation ◽  
Finite Thickness ◽  
Surface Charges ◽  
Stability Diagrams ◽  
Dielectric Fluids ◽  
Mass And Heat Transfer ◽  
The Stability ◽  
Cubic Nonlinear Schrödinger Equation

The nonlinear electrohydrodynamic stability 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 normal electric field in the absence of surface charges. Using a technique based on the method of multiple scales it is shown that the evolution of the amplitude is governed by a Ginzburg–Landau equation. When the mass and heat transfer are neglected, the cubic nonlinear Schrödinger equation is obtained. Further, it is shown that, near the marginal state, a nonlinear diffusion equation is obtained in the presence of mass and heat transfer. The various stability criteria are discussed both analytically and numerically and the stability diagrams are obtained.

Evolution of wave packets in magnetohydrodynamics

1995 ◽  
Vol 53 (2) ◽  
pp. 145-167 ◽  
Author(s):  
Anju Pusri ◽  
S. K. Malik
Keyword(s):  
Magnetic Field ◽  
Wave Packets ◽  
Stability Diagram ◽  
Envelope Soliton ◽  
Electrically Conducting ◽  
Self Focusing ◽  
Electrically Conducting Fluid ◽  
Regions Of Stability ◽  
Taylor Problem ◽  
The Stability

The propagation of wave packets on the surface of an electrically conducting fluid of uniform depth in the presence of a tangential magnetic field is investigated in (2 + 1) dimensions. The evolution of wave envelope is governed by two coupled partial differential equations with cubic nonlinearity. The stability analysis reveals the existence of different regions of instability. The effect of the applied magnetic field is not only significant but also different for different regions of stability. ‘Envelope soliton’ and ‘waveguide’ solutions of the amplitude equation are also discussed. The self-focusing phenomenon that arises when the amplitude of the wave becomes infinite in finite time is also examined. It is found that in a certain region of the stability diagram it may be easier to observe this phenomenon in the presence of a magnetic field. The Rayleigh-Taylor problem is also studied and various criteria for the existence of instability are obtained.

Parametric Vibrations in a Magneto-Thermo-Elastic Layer of Finite Thickness

1995 ◽  
Author(s):  
Jörg Wauer
Keyword(s):  
Magnetic Field ◽  
Finite Thickness ◽  
Elastic Field ◽  
Steady State Response ◽  
Bias Magnetic Field ◽  
Parametric Vibrations ◽  
State Response ◽  
Parametric Resonances ◽  
The Stability ◽  
The Magnetic Field

Abstract The dynamic behavior of a magneto-thermo-elastic planar layer subjected to a bias magnetic field is examined. The magnetic field acts parallel to the layer surfaces and is composed of a constant and a harmonically oscillating part. In particular, the small coupled magneto-thermo-elastic vibrations superimposed on the basic steady state response due to the stationary magnetic excitation are analyzed. Attention is focused on the influence of the pulsating part of the bias magnetic field on the governing variational equations to prove the stability of the basic state. In general, the stability equations describing the perturbations of the magnetic, thermal and elastic field properties are coupled in a special form and also the parametric excitation acts in a non-classical manner. Hence the variety of possible parametric resonances seems to be limited.

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