On the surface stability of liquid conductors in electromagnetic shaping

1995 ◽  
Vol 302 ◽  
pp. 1-28 ◽  
Author(s):  
Thomas P. Felici
Keyword(s):  
Circular Cross Section ◽  
Energy Functional ◽  
Exact Nature ◽  
Surface Stability ◽  
Metal Shield ◽  
High Frequency Electromagnetic Field ◽  
The Stability ◽  
The Second Variation ◽  
Electromagnetic Shaping ◽  
The Magnetic Field

In a process involving electromagnetic shaping, a high-frequency electromagnetic field is used to deform a liquid conductor into a required shape. This is particularly relevant to processes such as levitation melting. In this paper the stability of such configurations are investigated. The second variation of an appropriate energy functional is derived whose minimum states correspond to stable configurations, thus providing a stability criterion. As an example, this is applied to the shaping of a levitated cylinder of circular cross-section and to an almost spherical axisymmetric shape. In both cases we find that these shapes are unstable. We then consider enclosing the entire shaping device in a metal shield, thus preventing the escape of the magnetic field. It is then shown that in general the shield has a stabilizing effect, whose exact nature depends on the topology of the liquid shape and on the field structure on its surface. This differing behaviour is discussed for two-dimensional spherical and toroidal shapes.

SOME RESULTS ON HARMONIC MAPS FOR FINSLER MANIFOLDS

2005 ◽  
Vol 16 (09) ◽  
pp. 1017-1031 ◽  
Author(s):  
QUN HE ◽  
YI-BING SHEN
Keyword(s):  
Riemannian Manifold ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Functional ◽  
Second Variation ◽  
Finsler Manifolds ◽  
Totally Geodesic ◽  
The Stability ◽  
The Second Variation ◽  
Weitzenböck Formula

By simplifying the first and the second variation formulas of the energy functional and generalizing the Weitzenböck formula, we study the stability and the rigidity of harmonic maps between Finsler manifolds. It is proved that any nondegenerate harmonic map from a compact Einstein Riemannian manifold with nonnegative scalar curvature to a Berwald manifold with nonpositive flag curvature is totally geodesic and there is no nondegenerate stable harmonic map from a Riemannian unit sphere Sn (n > 2) to any Finsler manifold.

scholarly journals Variational formulations of steady rotational equatorial waves

2020 ◽  
Vol 10 (1) ◽  
pp. 534-547
Author(s):  
Jifeng Chu ◽  
Joachim Escher
Keyword(s):  
Linear Stability ◽  
Stream Function ◽  
Water Waves ◽  
Variational Formulation ◽  
Energy Functional ◽  
Equatorial Waves ◽  
Second Variation ◽  
Governing Equations ◽  
The Second Variation ◽  
Variational Formulations

Abstract When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f-plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.

scholarly journals No-z Model for Magnetic Fields of Different Astrophysical Objects and Stability of the Solutions

Data ◽  
2021 ◽  
Vol 6 (1) ◽  
pp. 4
Author(s):  
Evgeny Mikhailov ◽  
Daniela Boneva ◽  
Maria Pashentseva
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Large Scale ◽  
Point Of View ◽  
Accretion Discs ◽  
Wide Range ◽  
Astrophysical Objects ◽  
Alpha Effect ◽  
The Stability ◽  
The Magnetic Field

A wide range of astrophysical objects, such as the Sun, galaxies, stars, planets, accretion discs etc., have large-scale magnetic fields. Their generation is often based on the dynamo mechanism, which is connected with joint action of the alpha-effect and differential rotation. They compete with the turbulent diffusion. If the dynamo is intensive enough, the magnetic field grows, else it decays. The magnetic field evolution is described by Steenbeck—Krause—Raedler equations, which are quite difficult to be solved. So, for different objects, specific two-dimensional models are used. As for thin discs (this shape corresponds to galaxies and accretion discs), usually, no-z approximation is used. Some of the partial derivatives are changed by the algebraic expressions, and the solenoidality condition is taken into account as well. The field generation is restricted by the equipartition value and saturates if the field becomes comparable with it. From the point of view of mathematical physics, they can be characterized as stable points of the equations. The field can come to these values monotonously or have oscillations. It depends on the type of the stability of these points, whether it is a node or focus. Here, we study the stability of such points and give examples for astrophysical applications.

The energy and helicity of knotted magnetic flux tubes

1995 ◽  
Vol 451 (1943) ◽  
pp. 609-629 ◽  
Keyword(s):  
Magnetic Field ◽  
Magnetic Energy ◽  
Minimum Energy ◽  
Circular Cross Section ◽  
Parametric Representation ◽  
Energy Functional ◽  
Helical Axis ◽  
Torus Knots ◽  
Flux Tubes ◽  
Kink Mode

Magnetic relaxation of a magnetic field embedded in a perfectly conducting incompressible fluid to minimum energy magnetostatic equilibrium states is considered. It is supposed that the magnetic field is confined to a single flux tube which may be knotted. A local non-orthogonal coordinate system, zero-framed with respect to the knot, is introduced, and the field is decomposed into toroidal and poloidal ingredients with respect to this system. The helicity of the field is then determined; this vanishes for a field that is either purely toroidal or purely poloidal. The magnetic energy functional is calculated under the simplifying assumptions that the tube is axially uniform and of circular cross-section. The case of a tube with helical axis is first considered, and new results concerning kink mode instability and associated bifurcations are obtained. The case of flux tubes in the form of torus knots is then considered and the ‘ground-state’ energy function ͞m ( h ) (where h is an internal twist parameter) is obtained; as expected, ͞m ( h ), which is a topological invariant of the knot, increases with increasing knot complexity. The function ͞m ( h ) provides an upper bound on the corresponding function m ( h ) that applies when the above constraints on tube structure are removed. The technique is applicable to any knot admitting a parametric representation, on condition that points of vanishing curvature are excluded.

Magnetorheological fluid based on amorphous Fe-Si-B alloy magnetic particles

2021 ◽  
pp. 1045389X2110643
Author(s):  
Chuncheng Yang ◽  
Zhong Liu ◽  
Xiangyu Pei ◽  
Cuiling Jin ◽  
Mengchun Yu ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Field Strength ◽  
Rheological Property ◽  
Magnetic Particles ◽  
Annealing Treatment ◽  
Magnetorheological Fluid ◽  
The Stability ◽  
The Magnetic Field ◽  
Amorphous Particle ◽  
Better Than

Magnetorheological fluids (MRFs) based on amorphous Fe-Si-B alloy magnetic particles were prepared. The influence of annealing treatment on stability and rheological property of MRFs was investigated. The saturation magnetization ( Ms) of amorphous Fe-Si-B particles after annealing at 550°C is 131.5 emu/g, which is higher than that of amorphous Fe-Si-B particles without annealing. Moreover, the stability of MRF with annealed amorphous Fe-Si-B particles is better than that of MRF without annealed amorphous Fe-Si-B particles. Stearic acid at 3 wt% was added to the MRF2 to enhance the fluid stability to greater than 90%. In addition, the rheological properties demonstrate that the prepared amorphous particle MRF shows relatively strong magnetic responsiveness, especially when the magnetic field strength reaches 365 kA/m. As the magnetic field intensified, the yield stress increased dramatically and followed the Herschel-Bulkley model.

The stability of viscous flow in a curved channel in the presence of a magnetic field

1961 ◽  
Vol 264 (1317) ◽  
pp. 155-164 ◽  
Keyword(s):  
Magnetic Field ◽  
Viscous Flow ◽  
Uniform Magnetic Field ◽  
Axial Direction ◽  
Electrical Conductor ◽  
Curved Channel ◽  
Coaxial Cylinders ◽  
Transverse Pressure ◽  
The Stability ◽  
The Magnetic Field

The stability of viscous flow between two coaxial cylinders maintained by a constant transverse pressure gradient is considered when the fluid is an electrical conductor and a uniform magnetic field is impressed in the axial direction. The problem is solved and the dependence of the critical number for the onset of instability on the strength of the magnetic field and the coefficient of electrical conductivity of the fluid is determined.

The stability of viscous flow between rotating cylinders in the presence of a magnetic field

1953 ◽  
Vol 216 (1126) ◽  
pp. 293-309 ◽  
Keyword(s):  
Magnetic Field ◽  
Electrical Conductivity ◽  
Viscous Flow ◽  
Thermal Instability ◽  
Horizontal Layer ◽  
Rotational Stability ◽  
Marginal Stability ◽  
Inhibiting Effect ◽  
The Stability ◽  
The Magnetic Field

In this paper the theory of the stability of viscous flow between two rotating coaxial cylinders which has been developed by Taylor, Jeffreys and Meksyn is extended to the case when the fluid considered is an electrical conductor and a magnetic field along the axis of the cylinders is present. A differential equation of order eight is derived which governs the situation in marginal stability; and a significant set of boundary conditions for the problem is formulated. The case when the two cylinders are rotating in the same direction and the difference ( d ) in their radii is small compared to their mean (R 0 ) is investigated in detail. A variational procedure for solving the underlying characteristic value problem and determining the critical Taylor numbers for the onset of instability is described. As in the case of thermal instability of a horizontal layer of fluid heated below, the effect of the magnetic field is to inhibit the onset of instability, the inhibiting effect being the greater, the greater the strength of the field and the value of the electrical conductivity. In both cases, the inhibiting effect of the magnetic field depends on the strength of the field ( H ), the density ( ρ ) and the coefficients of electrical conductivity ( σ ), kinematic viscosity ( v ) and magnetic permeability ( μ ) through the same non-dimensional combination Q =μ 2 H 2 d 2 σ/ pv ; however, the effect on rotational stability is more pronounced than on thermal instability. A table of the critical Taylor numbers for various values of Q is provided.

Quantum Hall effects

Quantum 20/20 ◽  
2019 ◽  
pp. 303-322
Author(s):  
Ian R. Kenyon
Keyword(s):  
Magnetic Field ◽  
Hall Effect ◽  
Energy Gap ◽  
Quantum Hall ◽  
Free Electrons ◽  
Electron Gases ◽  
Hall Conductance ◽  
Hall Effects ◽  
The Stability ◽  
The Magnetic Field

It is explained how plateaux are seen in the Hall conductance of two dimensional electron gases, at cryogenic temperatures, when the magnetic field is scanned from zero to ~10T. On a Hall plateau σ‎xy = ne 2/h, where n is integral, while the longitudinal conductance vanishes. This is the integral quantum Hall effect. Free electrons in such devices are shown to occupy quantized Landau levels, analogous to classical cyclotron orbits. The stability of the IQHE is shown to be associated with a mobility gap rather than an energy gap. The analysis showing the topological origin of the IQHE is reproduced. Next the fractional QHE is described: Laughlin’s explanation in terms of an IQHE of quasiparticles is presented. In the absence of any magnetic field, the quantum spin Hall effect is observed, and described here. Time reversal invariance and Kramer pairs are seen to be underlying requirements. It’s topological origin is outlined.

scholarly journals Dipolar stability in spherical simulations: The impact of an inner stable zone

2019 ◽  
Vol 15 (S354) ◽  
pp. 185-188
Author(s):  
Bonnie Zaire ◽  
Laurène Jouve
Keyword(s):  
Magnetic Fields ◽  
Rossby Number ◽  
Constant Ratio ◽  
Stellar Rotation ◽  
Convective Motions ◽  
The Stability ◽  
The One ◽  
The Impact ◽  
The Magnetic Field ◽  
Stable Zone

AbstractMagnetic fields vary in complexity for different stars. The stability of dipolar magnetic fields is known to depend on different quantities, e.g., the stellar rotation, the stratification, and the intensity of convective motions. Here, we study the dipolar stability in a system with an inner stable zone. We present preliminary results of dynamo simulations using the Rayleigh number as a control parameter. The stiffness of the stable zone is accordingly varied to keep a constant ratio of the Brunt-Väisälä frequency to the angular velocity. Similarly to the completely convective spherical shell, we find that a transition exists between a regime where the magnetic field is dipolar to a multipolar regime when the Rossby number is increased. The value of the Rossby number at the transition is very close to the one of the fully convective case.

Sign in / Sign up

Export Citation Format

Share Document