The instability of streaming fluids with fine dust in porous medium

The instability of the plane interface between two uniform, superposed, and streaming fluids permeated with suspended particles through porous medium is considered. The effect of a uniform horizontal magnetic field on the problem is also studied. In the absence of surface tension, perturbations transverse to the direction of streaming are found to be unaffected by the presence of streaming if perturbations in the direction of streaming are ignored, whereas for perturbations in all other directions there exists instability for a certain wavenumber range. The instability of the system is postponed by the presence of magnetic field. The magnetic field and surface tension are able to suppress this Kelvin-Helmholtz instability for small wavelength perturbations and the medium porosity reduces the stability range given in terms of a difference in streaming velocities and the Alfvén velocity. The suspended particles do not affect the above results.     

Hydromagnetic instability of streaming fluids in porous medium

The instability of the plane interface between two uniform, superposed, electrically conducting and counter-streaming fluids through a porous medium is considered in the presence of a horizontal magnetic field. In the absence of surface tension, perturbations transverse to the direction of streaming are found to be unaffected by the presence of streaming if perturbations in the direction of streaming are ignored. For perturbations in all other directions there exists instability for a certain wavenumber range. The instability of this system is postponed by the presence of magnetic field. The magnetic field and surface tension are able to suppress this Kelvin-Helmholtz instability for small wavelength perturbations and the medium porosity reduces the stability range given in terms of a difference between the streaming velocities and the Alfvén velocity.This research forms a part of the research project awarded to the first author (R.C.S.) by the University Grants Commission.     

Hydromagnetic transverse instability of two highly viscous fluid-particle flows with finite ion Larmor radius corrections

A linear analysis of the combined effect of viscosity, finite ion Larmor radius and suspended particles on Kelvin-Helmholtz instability of two superposed incompressible fluids in the presence of a uniform magnetic field is carried out. The magnetic field is assumed to be transverse to the direction of streaming. A general dispersion relation for such a configuration has been obtained using appropriate boundary conditions. The stability analysis is discussed analytically, and the obtained results are numerically confirmed. Some special cases are recovered and corrected. The limiting cases of absence of suspended particles (or fluid velocities) and finite Larmor radius, absence of suspended particles are discussed in detail. In both cases, all other physical parameters are found to have stabilizing as well as destabilizing effects on the considered system. In the former case, the kinematic viscosity is found to has a stabilizing effect, while in the later case, the finite Larmor radius is found to has a stabilizing influence for a vortex sheet. It is shown also that both finite Larmor radius and kinematic viscosity stabilizations for interchange perturbations are similar to the stabilization effect due to a magnetic field for non-interchange perturbations. Received 13 January 2003 Published online 24 April 2003 RID="a" ID="a"Also at: Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt. e-mail: m.elsayed@uaeu.ac.ae     

Nonlinear evolution of two-magnetofluid instability

The nonlinear surface instability of a horizontal interface separating two magnetic fluids of different densities, magnetic permeabilities, and velocities, including surface tension effects, is investigated. The magnetic field is applied along the direction of streaming. It is shown that the evolution of the amplitude is governed by a nonlinear Ginzburg-Landau equation with the use of the multiple scale method. When the influence of streaming is neglected, the nonlinear diffusion equation is obtained. Further, it is shown that a nonlinear Schrödinger equation is obtained in the absence of gravity. The various stability criteria are discussed from these equations, of both Rayleigh-Taylor and Kelvin-Helmholtz problems, both analytically and numerically and the stability diagrams are obtained. Obtained also are the stability properties of solitary solutions to the Ginzburg-Landau equation in the case of constant surface tension.     

Stability of magnetopause boundary and generation of long period geomagnetic pulsations

The effect of irrotational electric field and tensorial plasma conductivity on the growth rate of Kelvin-Helmholtz instability has been investigated. It is shown that the presence of irrotational electric field alters the growth rate. The dependence of Pedersen conductivity on the growth rate has been shown. The Kelvin-Helmholtz perturbations generate a surface wave in the frozen-in plasma. The propagation of these waves gives rise to polarization of the transverse hydromagnetic pulsations. It is shown that the modified K-H spectrum would result in a corresponding change in polarization features of the hydromagnetic pulsations.     

Intervals of an unsteady electrohydrodynamic Kelvin-Helmoltz stability

An analysis is made of the stability of a basic flow of streaming fluids in the presence of an oblique periodic electric field. The particular profile investigated is the classical Kelvin-Helmholtz profile modified by the addition of the influence of mass and heat transfer across the interface. The intervals of electrohydrodynamic Kelvin-Helmholtz instability are considered. It is shown that a linear model of the interface is governed by Hill's differential equation. Characteristic values and intervals of stability are discussed. The special case of the Mathieu differential equation type is obtained. From the latter equation, the various criteria are discussed for both Rayleigh-Taylor and Kelvin-Helmholtz problems in the presence of an oblique periodic electric field, with and without mass and heat transfer across the interface.     

The Effect of Finite Larmor Radius on Gravitational Instability of a Finitely Conducting Magnetised Hall Medium in Presence of Suspended Particles

The problem of stability of a self-gravitating, infinite homogeneous gas in the presence of suspended particles is investigated. The medium is assumed conducting and effect of external magnetic field, Hall current and finite Larmor radius corrections are also considered. The equations of the problem are linearized and from linearized equations a general dispersion relation for a dusty gas-particle medium is obtained. The dispersion relation is reduced for two special cases of wave propagations: Parallel and perpendicular to the direction of uniform magnetic field. The effect of suspended particles on the medium is investigated in both the cases. It is found that in the presence of finite Larmor radius corrections and suspended particles the condition of instability is determined by Jeans's criterion for a self gravitating finitely conducting magnetised Hall medium.     

Effect of Conductivity on Magneto-Gravitational Instability of a Medium with Finite Larmor Radius and Suspended Particles

The self-gravitational instability of an infinite homogeneous magnetised and finitely conducting gas-particle medium is considered to include the finite Larmor radius effect in the presence of suspended particles. The equations of the problem are linearized and from linearized equations a general dispersion relation for dusty-gas is obtained. The dispersion relations are also obtained for propagation, parallel and perpendicular to the direction of uniform magnetic field. The Jeans, criterion is discussed for these two different directions of wave propagation. It is found that in the presence of finite Larmor radius corrections and suspended particles the condition of instability is determined by Jeans' criterion for a self gravitating, finitely conducting, magnetized gas-particle medium.     

On the stability of capillary waves on the surface of a charged jet moving relative to the environment

A dispersion relation is derived for capillary waves with arbitrary symmetry (arbitrary azimuthal numbers) on the surface of a charged cylindrical jet of an ideal incompressible conducting liquid moving relative to an ideal incompressible dielectric medium. It is shown that a tangential discontinuity in the velocity field on the surface of the jet leads to periodic instability of waves (similar to the Kelvin-Helmholtz instability) at the interface and destabilizes both axisymmetric and flexural waves. The wavenumber range for unstable waves and the instability growth rate increase with the field strength and relative speed of motion, varying as the square of these parameters. In the case of the neutral jet, the flexural instability is of the threshold character and sets in starting from a certain finite value of the speed rather than at an arbitrary small speed.     

On the stability of a cylindrical jet moving in a material medium along an external electrostatic field

A dispersion relation is derived for capillary waves with arbitrary symmetry (with arbitrary azimuthal numbers) on the surface of a jet of an ideal incompressible dielectric liquid moving in an ideal incompressible dielectric medium along an external uniform electrostatic field. A tangential discontinuity in the velocity field on the jet surface is shown to cause Kelvin-Helmholtz periodical instability at the interface and destabilize axisymmetric, flexural, and flexural-deformational waves. Both the flexural and flexural-deformational instabilities have a threshold and are observed not at an arbitrarily small velocity of the jet but starting from a certain finite value. It is shown that the instability of waves generated by the tangential discontinuity of the velocity field is periodic only formally (from the pure mathematical point of view). Actually, the jet disintegrates within the time of instability development, which is shorter than the half-cycle of the wave.     

Suspended Particles and the Gravitational Instability of Rotating Magnetised Medium

The effect of uniform rotation on the self gravitational instability of an infinite homogeneous magnetised gas particle medium in the presence of suspended particles is investigated. The equations of the problem are linearized and the general dispersion relation for such system is obtained. The rotation is assumed along two different directions and separate dispersion relation for each case is obtained. The dispersion relation for propagation parallel and perpendicular to the uniform magnetic field along with rotation is derived. The effect of suspended particles on the different modes of propagation is investigated. It is found that in presence of suspended particles, magnetic field, rotation and viscosity, Jeans' criterion determines the condition of gravitational instability of gas-particle medium.     

Instability of a charged plane interface between two liquids with respect to the tangential discontinuity of a time-dependent velocity field

The differential equation that describes the evolution of perturbations of a charged plane boundary between immiscible liquids when the upper liquid moves relative to the lower one with a time-dependent velocity parallel to the boundary is the Hill equation. In this system, the interface can exhibit instabilities of three types at various values of physical parameters: the Kelvin-Helmholtz, Tonks-Frenkel, and parametric instability. When physical parameters have certain values, the interface that is unstable with respect to surface charge and the tangential discontinuity of the velocity field across the interface can be parametrically stabilized.     

Dissipative instability of charged aerosol flows in the mesosphere

We consider the possible mechanism of generation of charged-particle density irregularities and electric field in the middle atmosphere based on the development of the dissipative instability of a flow of large charged aerosols. A dispersion equation describing the properties of the spectral component of a quasi-static electric field with allowance for the aerosol charging inertia is obtained. This equation is used to study characteristics of the instability threshold. It is shown that the charging inertia and the presence of photoelectrons lead to an increase and a decrease in the threshold plasma frequency of the aerosols, respectively. It is found that there exist optimal combinations of such parameters as the radius of spherical aerosols and the mass of heavy ion clusters for which the instability threshold is minimum. It is also shown that the instability threshold is lower for the particles stretched along the motion direction. Quantitative estimates are given for medium parameters necessary for the excitation of instability in the region of existence of polar mesospheric summer echo as well as for spatial scales of unstable perturbations. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 49, No. 11, pp. 942–957, November 2006.     

The instability of streaming Walters' viscoelastic fluid B′ in porous medium

The instability of streaming Walters' elastico-viscous fluid B in porous medium is considered. The case of two uniform streaming fluids separated by a horizontal boundary is considered. It is found that for the special case when perturbations in the direction of streaming are ignored, the system can be stable or unstable, depending upon kinematic viscoelasticity, medium porosity and medium permeability, for both potential unstable and potentially stable configurations. In every other direction, a minimum value of wave-number has been found and the system is unstable for all wave-numbers greater than this minimum wave number.     

Rayleigh-taylor instability of viscoelastic fluids with suspended particles in porous medium in hydromagnetics

The instability of the plane interface between two viscoelastic (Oldroydian) superposed conducting fluids permeated with suspended particles in porous medium is studied when the whole system is immersed in a uniform magnetic field. The dispersion relation for the Oldroydian viscoelastic fluid is obtained which also yields dispersion relations for Maxwellian and Newtonian fluids in special cases, in the presence of suspended particles in porous medium in hydromagnelics. The system is found to be stable for potentially stable case. The presence of magnetic field stabilizes certain wave number band whereas the system was unstable for all wave numbers in the absence of magnetic field, for the potentially unstable configuration. The growth rates increase (for certain wave numbers) and decrease (for other wave numbers) with the increase in stress relaxation time, strain retardation time, suspended particles number density and medium permeability.     

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.     

Electrogravitational stability of oscillating streaming fluid cylinder

The electrogravitational instability of on oscillating streaming fluid cylinder under the action of the selfgravitating, capillary and electrodynamic forces has been discussed. The model is governed by the Mathieu second order integro-differential equation. Some limiting cases are recovering from the present general one. The capillary force is destabilizing in a small axisymmetric domain 0<x<1 and stabilizing otherwise. In the absence of electric fields, we found that the model is unstable in a small domain while it is selfgravitating stable in all other domains. The presence of the electric field led to the presence of a great number of stable waves. The electric field has a strong stabilizing influence on the selfgravitating instability of the model. The capillary force has a strong destabilizing influence on the selfgravitating instability of the model.Generally, the uniform stream supports the unstable waves, while the oscillating streaming has stability tendency.     

黏性各向异性磁流体Kelvin-Helmholtz不稳定性:二维数值研究

利用corner transport upwind和constrained transport算法求解非理想磁流体动力学方程组,对匀强平行磁场作用下,黏性各向异性等离子体自由剪切层中的Kelvin-Helmholtz不稳定性进行了数值模拟.从流动结构、涡结构演化、磁场分布、横向磁压力、抗弯磁张力等角度对各向同性和各向异性黏性算例结果进行了讨论,分析了黏性各向异性对Kelvin-Helmholtz不稳定性的影响.结果表明,黏性各向异性比黏性各向同性更利于流动的稳定.其稳定性作用是由于磁感线方向上剪切速率降低导致界面卷起程度和圈数的降低,并使卷起结构中小涡产生增殖、合并,破坏了涡的常规增长,从而导致流动的稳定.黏性各向异性对横向磁压力的影响比对抗弯磁张力更大.     

Simulating liquid-vapor phase separation under shear with lattice Boltzmann method

We study liquid-vapor phase separation under shear via the Shan-Chen lattice Boltzmann model. Besides the rheological characteristics, we analyze the Kelvin-Helmholtz (K-H) instability resulting from the tangential velocity difference of the fluids on two sides of the interface. We discuss also the growth behavior of droplets. The domains being close to the walls are lamellar-ordered, where the hydrodynamic effects dominate. The patterns in the bulk of the system are nearly isotropic, where the domain growt...     

Kelvin Helmholtz Instability in a Magnetoplasma

Two dimensional transverse Kelvin-Helmholtz (K-H) instability has been studied at the interface between the two fluids (plasma medium) of finite thickness in relative motion to each other. The perturbations on the interface are assumed to be electromagnetic and a dispersion relation is obtained. The interface (boundary) has been found to be unstable for a wide range of perturbation wavelengths (wave numbers kx, ky). It is shown that the modification introduced by electromagnetic (quasi-electrostatic) perturbations in comparison to electrostatic one is to reduce the growth rate of perturbations. The growth rate maximizes when kx = ky. The applications of this study have been discussed to explain some of the observed ionospheric (auroral arc formation) and magnetospheric (unstable magnetopause boundary, hydromagnetic pulsations) phenomena.