scholarly journals Edge pinch instability of oblate liquid metal drops in a transverse AC magnetic field

2011 ◽  
Vol 676 ◽  
pp. 218-236 ◽  
Author(s):  
JĀNIS PRIEDE
Keyword(s):  
Magnetic Field ◽  
Liquid Metal ◽  
Rotational Symmetry ◽  
Initial Perturbation ◽  
Magnetic Potential ◽  
Surface Integral ◽  
Ac Magnetic Field ◽  
Scalar Magnetic Potential ◽  
Equilibrium Shapes ◽  
The Stability

This paper considers the stability of liquid metal drops subject to a high-frequency AC magnetic field. An energy variation principle is derived in terms of the surface integral of the scalar magnetic potential. This principle is applied to a thin perfectly conducting liquid disk, which is used to model the drops constrained in a horizontal gap between two parallel insulating plates. Firstly, the stability of a circular disk is analysed with respect to small-amplitude harmonic edge perturbations. Analytical solution shows that the edge deformations with the azimuthal wavenumbers m = 2, 3, 4, . . . start to develop as the magnetic Bond number exceeds the critical threshold Bmc = 3π(m + 1)/2. The most unstable is m = 2 mode, which corresponds to an elliptical deformation. Secondly, strongly deformed equilibrium shapes are modelled numerically by minimising the associated energy in combination with the solution of a surface integral equation for the scalar magnetic potential on an unstructured triangular mesh. The edge instability is found to result in the equilibrium shapes of either two- or threefold rotational symmetry depending on the magnetic field strength and the initial perturbation. The shapes of higher rotational symmetries are unstable and fall back to one of these two basic states. The developed method is both efficient and accurate enough for modelling of strongly deformed drop shapes.

scholarly journals Methods for Numerical Modeling of Two-Dimensional Capillary Surfaces

2004 ◽  
Vol 4 (1) ◽  
pp. 66-93 ◽  
Author(s):  
Victor K. Polevikov
Keyword(s):  
Magnetic Field ◽  
Physical Phenomenon ◽  
Iterative Algorithms ◽  
Axially Symmetric ◽  
Capillary Surface ◽  
Capillary Surfaces ◽  
Equilibrium Shapes ◽  
Grid Nodes ◽  
The Stability ◽  
Simply Connected

Abstract Certain methods for numerical solving plane and axially symmetric problems on equilibrium shapes of a capillary surface are presented. The methods possess a high order of approximation on a nonuniform grid. They are easy to realize, fairly universal and suitable for constructing not only simply connected but also doubly connected and disconnected surfaces, including strongly curved ones. It is shown that the iterative algorithms constructed are absolutely stable at each iteration. The condition for convergence of iterations is obtained within the framework of a linear theory. To describe peak-shaped configurations of a magnetic uid in a high magnetic field, an algorithm of generation of adaptive grid nodes in accordance with the surface curvature is proposed. The methods have been tested for the well-known problems of capillary hydrostatics on equilibrium shapes of a drop adjacent to the horizontal rotating plate under gravity, and of an isolated magneticuid drop in a high uniform magnetic field. It has been established that they adequately respond to the physical phenomenon of a crisis of equilibrium shapes, i.e., they can be adopted to investigate the stability of equilibrium states of a capillary surface.

scholarly journals Computation of the magnetic potential induced by a collection of spherical particles using series expansions

2020 ◽  
Vol 54 (4) ◽  
pp. 1073-1109
Author(s):  
Stéphane Balac ◽  
Laurent Chupin ◽  
Sébastien Martin
Keyword(s):  
Magnetic Field ◽  
Magnetic Potential ◽  
Point Of View ◽  
Spherical Particles ◽  
Main Magnetic Field ◽  
Minimal Distance ◽  
Metallic Particles ◽  
Scalar Magnetic Potential ◽  
The Magnetic Field ◽  
Made In

In Magnetic Resonance Imaging there are several situations where, for simulation purposes, one wants to compute the magnetic field induced by a cluster of small metallic particles. Given the difficulty of the problem from a numerical point of view, the simplifying assumption that the field due to each particle interacts only with the main magnetic field but does not interact with the fields due to the other particles is usually made. In this paper we investigate from a mathematical point of view the relevancy of this assumption and provide error estimates for the scalar magnetic potential in terms of the key parameter that is the minimal distance between the particles. A special attention is paid to obtain explicit and relevant constants in the estimates. When the “non-interacting assumption” is deficient, we propose to compute a better approximation of the magnetic potential by taking into account pairwise magnetic field interactions between particles that enters in a general framework for computing the scalar magnetic potential as a series expansion.

Stability of the compressible viscous fluid around the plane Couette flow in the presence of a transverse uniform magnetic field

2018 ◽  
Vol 17 (01) ◽  
pp. 57-84
Author(s):  
Xingwei Zhang ◽  
Guojing Zhang ◽  
Hai-Liang Li
Keyword(s):  
Magnetic Field ◽  
Viscous Fluid ◽  
Couette Flow ◽  
Transverse Magnetic Field ◽  
Initial Perturbation ◽  
Three Dimensional ◽  
Transverse Magnetic ◽  
Plane Couette Flow ◽  
Compressible Viscous Fluid ◽  
The Stability

In this paper, we consider the stability of three-dimensional compressible viscous fluid around the plane Couette flow in the presence of a uniform transverse magnetic field and show that the uniform transverse magnetic field has a stabilizing effect on the plane Couette flow. Namely, for a sufficiently large Hartmann number, the compressible viscous plane Couette flow is nonlinear stable for small Mach number and arbitrary Reynolds number so long as the initial perturbation is small enough.

scholarly journals Stability of eigenvalues of quantum graphs with respect to magnetic perturbation and the nodal count of the eigenfunctions

2014 ◽  
Vol 372 (2007) ◽  
pp. 20120522 ◽  
Author(s):  
Gregory Berkolaiko ◽  
Tracy Weyand
Keyword(s):  
Magnetic Field ◽  
Morse Index ◽  
Magnetic Potential ◽  
Zero Magnetic Field ◽  
Quantum Graphs ◽  
Nodal Count ◽  
Magnetic Perturbation ◽  
Number Of Zeros ◽  
Stability Of Eigenvalues ◽  
The Stability

We prove an analogue of the magnetic nodal theorem on quantum graphs: the number of zeros ϕ of the n th eigenfunction of the Schrödinger operator on a quantum graph is related to the stability of the n th eigenvalue of the perturbation of the operator by magnetic potential. More precisely, we consider the n th eigenvalue as a function of the magnetic perturbation and show that its Morse index at zero magnetic field is equal to ϕ −( n −1).

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