Mathematical analysis of the oscillations of a liquid metal drop submitted to low frequency fields

K. Spragg^1,2 - A. Sneyd² - Y. Fautrelle¹

¹ SIMAP/EPM Laboratory of Polytechnic Institute of Grenoble, France
² Mathematical Department of University of Waikato, Hamilton, New Zealand

Abstract
We investigate the free-surface deformation of a circular liquid-metal pool under the influence of a vertical low frequency alternating magnetic field. We develop a heuristic mathematical model based on a Lagrangian approach. The system reduces to a set of two differential equations governing the vertical deformation as well as the amplitude of a single horizontal mode. The theory is applied in the case of an elongated drop. It is shown that the horizontal deformation is governed by a Mathieu-type equation giving birth to a parametric instability. Thanks to the model we retrieve qualitatively the main phenomena observed in previous experiments. Figs 3, Refs 7.