Maximal growth rate at the Rosensweig instability: theory, experiment, and numerics

Abstract

We investigate the growth of a pattern of liquid crests emerging in a layer of magnetic liquid when subjected to a magnetic field oriented normally to the fluid surface. After a step like increase of the magnetic field, the temporal evolution of the pattern amplitude is measured by means of a Hall-sensor array. The extracted growth rate is compared with predictions from linear stability analysis by taking into account the nonlinear magnetization curve M (H). The remaining discrepancy can be resolved by numerical calculations via the finite element method. By starting with a finite surface perturbation it can reproduce the temporal evolution of the pattern amplitude and the growth rate. "In the beginning was the word, . . . " [1], which reads in the greek original Ò ÖÕ Ò Ð Ó , . . . . The word Ð Ó , however, has a plethora of different meanings, including also "way, manner" which is "modus" in latin, "mode" in english. We may therefore translate John, 1.1: "In the beginning was the mode, . . . ", and indeed at the beginning of an evolving pattern stands an unstable mode [2]. As long as the amplitude of the mode is small, its wave number and growth rate can be calculated by linear stability analysis. In this way the early stage of pattern formation has been investigated in many different systems [2]. In the following we examine the growth of the Rosensweig or normal field instability [3]. It is observed in a layer of magnetic fluid [4], when a critical value B c of the vertical magnetic induction is surpassed. For a sudden increase of the magnetic induction B the growth rate of the fastest growing modeω 2,m was recently calculated in detail [5,6] to follow the equation(1) HereB = (B − B c )/B c denotes the scaled overcritical induction, and c 1 = 1.24 and c 2 = 0.94 the calculated parameters taking into account the measured nonlinear magnetization curve M (H) of the fluid. In the following we report an experimental and numerical test of those predictions.In order to measure the temporal evolution of the growing amplitudes of the surface pattern we utilize a linear array of Hall sensors [7], mounted beneath the bottom of the vessel. Details of the experimental setup and the method of measuring can be found in Ref. [6]. On the basis of the time recorded magnetic profiles, we determine the amplitude from the root-mean-square