The stability of a combined current and vortex sheet in a perfectly conducting fluid

Michael, D. H.

doi:10.1017/s0305004100030541

Cited by 71 publications

(29 citation statements)

References 3 publications

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“…The problems of hydromagnetic st,ability of jets and vortex sheets are of importance in the laboratory, terrestrial and astronomical contexts (GERWIN [l]). The magnetic field has been found to have a stabilizing influence for vortex sheets and jets in inviscid, incompressible and infinitely conducting fluids (~KICHAEL [2], LESSEN e t a1 [3] and CHAKRABOHTY [4]).…”

Section: Liitroductioiimentioning

confidence: 99%

Hydromagnetic Stability of a Plane Vortex Sheet in Finitely Conducting Fluids

Chakraborty

Jindia²

1971

Z Angew Math Mech

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Die vorliegende Arbeit untersucht den Einfluß kleiner Störungen auf die Stabilität einer ebenen Wirbelfläche in einer reibungsfreien, inkompressiblen Flüssigkeit von endlicher Leitfähigkeit in Gegenwart eines gleichförmigen Magnetfeldes. Es stellt sich heraus, daß es ausreicht, den Fall zweidimensionaler Störungen, die sich in Richtung des Magnetfeldes ausbreiten, zu betrachten. Jede Instabilität, die bei allgemeineren Störungen auftreten kann, tritt auch bei zweidimensionalen Störungen bei einem geringeren Wert der Leitfähigkeit auf. Die Dispersionsbeziehung wird in den beiden Grenzfällen großer bzw. kleiner Leitfähigkeit diskutiert, und es werden die Ergebnisse für große, aber endliche bzw. kleine, aber nicht verschwindende Leitfähigkeit mit denen für unendliche bzw. verschwindende Leitfähigkeit (d. h. für den klassischen Kelvin‐Helmholtz‐Fall) verglichen.

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Section: Liitroductioiimentioning

confidence: 99%

Hydromagnetic Stability of a Plane Vortex Sheet in Finitely Conducting Fluids

Chakraborty

Jindia²

1971

Z Angew Math Mech

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“…At last, it is worth noting that inequality (6) is redundant since it can be shown to follow from (8) (actually, inequality (6) is the 2D stability condition [10], i.e., the stability condition for the case v ± = (0, 0, v ± 3 ) and H ± = (0, 0, H ± 3 )). The stability condition (8) is always satisfied for current sheets, i.e., for the case…”

Section: Introductionmentioning

confidence: 98%

“…This was done a long time ago by Syrovatskij [12] and Axford [1] (and for the 2D case by Michael [10]) by the normal modes analysis. This condition reads [1,12] (see also [7]) [v] 2 < 2 |H + | 2 + |H − | 2 ,…”

Section: Introductionmentioning

confidence: 99%

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Stability of incompressible current-vortex sheets

Morando

Trakhinin

Trebeschi

2008

Journal of Mathematical Analysis and Applications

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We revisit the study in [Y. Trakhinin, On the existence of incompressible current-vortex sheets: study of a linearized free boundary value problem, Math. Methods Appl. Sci. 28 (2005) 917-945] where an energy a priori estimate for the linearized free boundary value problem for planar current-vortex sheets in ideal incompressible magnetohydrodynamics was proved for a part of the whole stability domain found a long time ago in [S.I. Syrovatskij, The stability of tangential discontinuities in a magnetohydrodynamic medium, Zh. Eksper. Teor. Fiz. 24 (1953) 622-629 (in Russian); W.I. Axford, Note on a problem of magnetohydrodynamic stability, Canad. J. Phys. 40 (1962) 654-655]. In this paper we derive an a priori estimate in the whole stability domain. The crucial point in deriving this estimate is the construction of a symbolic symmetrizer for a nonstandard elliptic problem for the small perturbation of total pressure. This symmetrizer is an analogue of Kreiss' type symmetrizers. As in hyperbolic theory, the failure of the uniform Lopatinski condition, i.e., the fact that current-vortex sheets are only weakly (neutrally) stable yields loss of derivatives in the energy estimate. The result of this paper is a necessary step to prove the local-in-time existence of stable nonplanar incompressible current-vortex sheets by a suitable Nash-Moser type iteration scheme

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“…for 0, which satisfies the fourth-order differential equation (9). and for R.lf = 00 to [6] [0] = 0 = [pIS -(U -c)2IDO]. 4.…”

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confidence: 99%