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

TL;DR

This study investigates the edge pinch instability of oblate liquid metal drops in a transverse AC magnetic field, deriving an energy principle, analyzing stability, and modeling deformed shapes to understand the instability mechanisms.

Contribution

It introduces a new energy variation principle for modeling liquid metal drops in magnetic fields and analyzes their stability and deformed shapes under high-frequency AC magnetic influence.

Findings

Edge deformations with azimuthal wavenumbers m=2,3,4... occur above a critical magnetic Bond number.

The most unstable mode is m=2, leading to elliptical deformations.

Deformed equilibrium shapes exhibit two- or threefold rotational symmetry depending on magnetic field strength.

Abstract

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 Bm_c=3pi(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.