Nonlinear Kelvin-Helmholtz instability of two miscible ferrofluids in porous media

Moatimid, Galal M.

doi:10.1007/s00033-005-2067-1

Cited by 20 publications

(23 citation statements)

References 30 publications

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“…Meanwhile, the equations govern the hydrodynamic motion through out porous media [15], are given by @v @t…”

Section: ð2:2þmentioning

confidence: 99%

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Electrohydrodynamic linear stability of finitely conducting flows through porous fluids with mass and heat transfer

Moatimid¹,

Allah²

2010

Applied Mathematical Modelling

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a b s t r a c tIn this work, a linear stability analysis is used to investigate a capillary surface waves between two horizontal finite fluid layers. The system is acted upon by a vertical periodic electric field. The problem examines few representatives of porous media. It is also includes finite conductivity, mass and heat transfer. It is assumed that the basic flow is two-dimensional streaming flow. A general dispersion relation governing the linear stability is derived. In contrast with our previous work [23], the present problem shows that the stability criterion depends on the mass and heat transfer parameter. The present study recovers some special cases upon appropriate data choices. The presence of finite conductivity's together with the dielectric permeability's make the uniform electric field plays a dual role in the stability criterion. This shows some analogy with the nonlinear stability theory. In addition, the mass and heat transfer parameter as well as the Darcy's coefficients play a stabilizing role in the stability picture. In case of the Rayleigh-Taylor instability, by means of the Whittaker technique, the parametric excitation of the electrohydrodynamic surface waves is obtained. The transition curve equations are calculated up to the fourth order for a small dimensionless parameter. The analytical results are numerically confirmed.Ó 2010 Elsevier Inc. All rights reserved. The introductionThe instability of a plane interface between two superposed fluids of different densities, when it occurs, is called the Rayleigh-Taylor instability (RTI). This instability was first investigated by Rayleigh [1]. Taylor [2] studied the stability of a heterogeneous fluid accelerated in a direction perpendicular to the plane of stratification. The RTI has been addressed in several studies owing to its importance in stratified system, among which planetary and stellar atmosphere are of two examples. The effect of the external forces has an importance, mainly, in planetary and stellar systems. Coriolis and centrifugal forces are more common in these systems. They play an important role in determining many phenomena including the RTI. Different properties of fluids have been included in the RTI through theoretical investigation. Chandrasekhar [3] has given a detailed account of these investigations. The Kelvin-Helmholtz instability (KHI) arises when two superposed fluids are in relative motion. The model of the classical KHI involves a horizontal interface between two fluids with different parallel, uniform, and horizontal velocities. In the KHI problem, the effect of streaming is destabilizing in the sense as given by Chandrasekhar [3]. This instability, which arises as a consequence of a relative drift velocity of two fluids along the surface of discontinuity, has a great relevance to various physical phenomena such as commentary tails and the magnetospherical boundary. The KHI has attracted the attention of many researchers, due to shear flow in stratified fluids, and its dominant 0307-904X/$ -see front matter Ó

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“…Meanwhile, the equations govern the hydrodynamic motion through out porous media [15], are given by @v @t…”

Section: ð2:2þmentioning

confidence: 99%

“…The KHI for flow in porous media was studied by El-Sayed [14]. Moatimid [15] studied the nonlinear KHI of two-fluid layers in porous media. He analyzed linear and nonlinear stability aspects for flow of parallel streaming magnetic fluids, through a porous media and confined between two parallel walls.…”

Section: The Introductionmentioning

confidence: 99%

Electrohydrodynamic linear stability of finitely conducting flows through porous fluids with mass and heat transfer

Moatimid¹,

Allah²

2010

Applied Mathematical Modelling

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“…Moatimid [63] derived the well known stability criteria for the above nonlinear Schrödinger equation having complex coefficients by assuming that µ = µ r + iµ i and ν * = ν * r + iν * i . Thus, the stability conditions take the following form…”

Section: Numerical Stability Discussionmentioning

confidence: 99%

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

El-Sayed

2007

Arch Appl Mech

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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.

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“…Through the multiple scale method they obtained a generalized division of the amplitude equation to obtain the marginally unstable regions of the parameter space. Moatimid 14 employed the method of multiple scale expansion to analyze the nonlinear instability phenomenon of two ferrofluids through porous media while taking into account the effect of magnetic field, mass and heat transfer. The author illustrated that for the linearized problem, both the tangential magnetic field and the surface tension have a stabilizing effect.…”

Section: Introductionmentioning

confidence: 99%

Kelvin-Helmholtz instability for flow in porous media under the influence of oblique magnetic fields: A viscous potential flow analysis

2013

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In this paper, the Kelvin-Helmholtz instability of viscous incompressible magnetic fluid fully saturated porous media is achieved through the viscous potential theory. The flow is considered to be through semi-permeable boundaries above and below the fluids through which the fluid may either be blown in or sucked out, in a direction normal to the main streaming direction of the fluid flow. An oblique magnetic field, mass, heat transfer, and surface tension are present across the interface. Through the linear stability analysis, a general dispersion relation is derived and the natural curves are plotted. Therefore, the linear stability condition is discussed in some depth. In view of the multiple time scale technique, the Ginzburg–Landau equation, which describes the behavior of the system in the nonlinear approach, is obtained. The effects of the orientation of the magnetic fields on the stability configuration in linear, as well as nonlinear approaches, are discussed. It is found that the Darcy's coefficient for the porous layers plays a stabilizing role. The injection of the fluids at both boundaries has a stabilizing effect, in contrast with the suction at both boundaries.

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Nonlinear Kelvin-Helmholtz instability of two miscible ferrofluids in porous media

Cited by 20 publications

References 30 publications

Electrohydrodynamic linear stability of finitely conducting flows through porous fluids with mass and heat transfer

Electrohydrodynamic linear stability of finitely conducting flows through porous fluids with mass and heat transfer

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

Kelvin-Helmholtz instability for flow in porous media under the influence of oblique magnetic fields: A viscous potential flow analysis

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