# Instability of a surface of discontinuity of velocity in a parallel uniform magnetic field

TL;DR: In this paper, the magnetic field was uniform and parallel to the velocity on the two sides of a surface of discontinuity of velocity in an electrically conducting inviscid fluid, and it was found that the flow is unstable for all values of BETA and N.

Abstract: The problem was investigated when the magnetic field was uniform and parallel to the velocity on the two sides of a surface of discontinuity of velocity in an electrically conducting inviscid fluid. The secular equation depends on two parameters BETA and N, where BETA is the ratio of magnetic Reynolds number to dimensionless wave number and N is the ratio of the magnetic to the kinetic energy of the fluid. It was found that the flow is unstable for all values of BETA and N. (auth)

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TL;DR: In this paper, a perturbation theory using asymptotic series is applied for small resistivity and sero viscosity in a plane sheet pinch with shear bulk plasma flow present.

Abstract: Resistive tearing modes are analysed in a plane sheet pinch with shear bulk plasma flow present. A perturbation theory using asymptotic series is applied for small resistivity and sero viscosity. The analyticity problems arising in the vicinity of singular points are taken into account. For a suitably defined solution uniform convergence is proved and the scaling law for the growth rate derived. The instability region is shown to depend on the function fluid kinetic energy magnetic field energy where both stabilization and destabilization are possible effect of the shear flow. The relationship to the Kelvin-Helmholtz instability is discussed.

50 citations

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TL;DR: In this article, the stability of a plane vortex sheet in an inviscid, incompressible and finitely conducting fluid in the presence of a uniform magnetic field, parallel to the fluid velocity directions, is studied against small disturbances.

Abstract: Die vorliegende Arbeit untersucht den Einflus kleiner Storungen auf die Stabilitat einer ebenen Wirbelflache in einer reibungsfreien, inkompressiblen Flussigkeit von endlicher Leitfahigkeit in Gegenwart eines gleichformigen Magnetfeldes. Es stellt sich heraus, das es ausreicht, den Fall zweidimensionaler Storungen, die sich in Richtung des Magnetfeldes ausbreiten, zu betrachten. Jede Instabilitat, die bei allgemeineren Storungen auftreten kann, tritt auch bei zweidimensionalen Storungen bei einem geringeren Wert der Leitfahigkeit auf. Die Dispersionsbeziehung wird in den beiden Grenzfallen groser bzw. kleiner Leitfahigkeit diskutiert, und es werden die Ergebnisse fur grose, aber endliche bzw. kleine, aber nicht verschwindende Leitfahigkeit mit denen fur unendliche bzw. verschwindende Leitfahigkeit (d. h. fur den klassischen Kelvin-Helmholtz-Fall) verglichen.
The stability of a plane vortex sheet in an inviscid, incompressible and finitely conducting fluid in the presence of a uniform magnetic field, parallel to the fluid velocity directions is studied against small disturbances. It is found that it is sufficient to consider the stability against two -dimensional disturbances propagating along the magnetic field. Any instability which may be present for more general disturbances is also present for two-dimensional disturbances at a lower value of the conductivity. The dispersion relation is discussed in the limiting cases of large and small conductivities. The effect of a large but finite conductivity and that of a small but nonzero conductivity on the infinite conductivity and zero conductivity (classical Kelvin-Helmholtz case) results, respectively, are discussed.

5 citations

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TL;DR: In this article, the hydromagnetic Kelvin-Helmholtz instability of two superposed fluids of different densities is studied and the influence of β on stable and unstable regions as compared to the case when β is unity has been investigated.

Abstract: The hydromagnetic Kelvin-Helmholtz instability of two superposed fluids of different densities is studied. One of the fluids is assumed to be static with finite-resistivity and another fluid is streaming and nonconducting. The equations of the problem are linearized and the dispersion relation using relevant boundary conditions has been derived. It is found that the ratio of densities of the fluids (β) modifies the condition of ideal-plasma modes. The influence of β on stable and unstable regions as compared to the case when β is unity has been investigated and illustrated. Further, the combined effect of small finite-resistivity and different densities of the fluids is analyzed. It has been found that β merely changes the constant of proportionality of the growth rate, which is obtained for the fluids of the same densities.

3 citations

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TL;DR: In this paper, an eigenvalue equation for the complex wave velocity is obtained from which the stability conditions of a vortex sheet are obtained. But the eigenvalues of the wave velocity were not obtained for the case of background vorticity.

Abstract: The electrohydrodynamic Kelvin–Helmholtz instability of a vortex sheet under the effect of background vorticity and a tangential electric field is described theoretically by an inviscid linear stability analysis. An eigenvalue equation for the complex wave velocity is obtained from which we can obtain the stability conditions. It is found, when the electric field is absent, that the background vorticity increases the growth rate of the deformed vortex sheet whose vorticity is of the same sign as the background vorticity. In the presence of a tangential electric field, it is found that the electric field has a stabilizing effect and the vortex sheet is neutrally stable if the wavenumber and the electric field values are less than some critical values depending on the background vorticity.

3 citations

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TL;DR: In this paper, the effect of a uniform axial magnetic field on the sinuous unstable mode of an axisymmetric jet or wake is investigated, and it is shown that the magnetic field always decreases the amplification rate.

Abstract: Theoretical calculations on the effect of a uniform axial magnetic field on the sinuous unstable mode of an axisymmetric jet or wake are presented. The fluid is incompressible. The viscous Reynolds number is assumed infinite, and the magnetic Reynolds number is low. The profile studied is trapezoidal with thin shear layers. The calculations are done for small axial wavenumber, α. In the nonmagnetic case, the approximate amplification rate as a function of α agrees with numerical calculations up to the α for maximum amplification rate, when properly scaled. In the magnetic case, a new effect occurs. Due to the helical nature of the instability, the electric field must be nonzero to make the electric current source‐free. It is shown that the magnetic field always decreases the amplification rate. The effect is small, of order α2, because the electric field for small α cancels the current generated by fluid crossing field lines.

3 citations

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TL;DR: In this article, the hydromagnetic stability of a basic two-dimensional parallel flow of an incompressible conducting fluid in a uniform magnetic field parallel to the flow is considered, and it is shown that any given small wave disturbance can be stabilized by a sufficiently strong magnetic field if the Reynolds number is finite and the magnetic Reynolds number small.

Abstract: The hydromagnetic stability of a basic two-dimensional parallel flow of an incompressible conducting fluid in a uniform magnetic field parallel to the flow is considered. By use of the generalization of the Orr-Sommerfeld equation for an electrically conducting fluid, it is shown that any given small wave disturbance can be stabilized by a sufficiently strong magnetic field if the Reynolds number is finite and the magnetic Reynolds number small. Stability of velocity profiles with a point of inflexion at small magnetic Reynolds number and infinite Reynolds number is considered in detail. Perturbation methods are developed to find stability characteristics in two cases, when the magnetic field is weak, and when the disturbance is a long wave. These methods are applied to the jet and the half-jet, which are both found to be unstable to long-wave disturbances, however strong the magnetic field. Nonetheless, these two flows can be stabilized for any given harmonic disturbance of finite wavelength. The analysis of the jet reveals the surprising result that the magnetic field makes inviscid long-wave disturbances more unstable.

29 citations

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