Scaling behaviour of the maximal growth rate in the Rosensweig instability

Abstract

PACS. 47.20.Ma -Hydrodynamic stability and instability. PACS. 75.50.Mm -Magnetic liquids.Abstract. -The dependence of the maximal growth rate of the modes of the Rosensweig instability on the properties of the magnetic fluid and the external magnetic induction is studied. An expansion and a fit procedure are applied in the appropriate ranges of the supercritical inductionB. With increasingB the scaling of the maximal growth rate changes from linear to a combination of linear and square-root-like scaling. The scaling of the corresponding wave number alternates from quadratic to primarily linear. For very smallB the dependence of the maximal growth rate on the viscosity is given. Suggestions are made for experiments to test the predicted scaling behaviours.Introduction. -The investigation of instabilities in magnetic fluids has a long history where the most prominent instability being the normal field or Rosensweig instability [1,2]. Above a threshold of the induction, the initially flat surface exhibits a stationary array of peaks. Despite its long history, some aspects of the Rosensweig instability have been addressed only recently: the hexagon-square transition [3] or the wave number selection problem [3,4]. The wave number is the absolute value of the wave vector, q = |q|, which characterizes small disturbances. The ground state of a pattern forming system is subjected to such small disturbances in order to study its stability. The corresponding growth rate ω may depend on system parameters {P}, e.g. the viscosity of the fluid, and control parameters {R}, e.g. an external magnetic field, ω = ω(q; {P}, {R}). Generally it is assumed that in the linear stage of the pattern forming process the wave number with the largest growth rate will prevail. Therefore this mode is called linearly most unstable mode. Due to its role in the pattern formation it is of particular interest to examine the maximal growth rate ω m , and its dependence on the different parameters.This dependence has received only limited attention in classical hydrodynamic systems. For the Küppers-Lortz (KL) instability in Rayleigh-Bénard convection rotated about a vertical axis, the growth rate of different KL angles were calculated for one fixed rotation rate and different temperature differences [5]. In surface-tension-driven Bénard convection the growth rates are calculated for two fixed values of heat loss and different temperature differences [6]. But in both systems it was not analysed how ω m depends on the control parameters. In the problem of convection for autocatalytic reaction fronts, the maximal growth rate was analysed

Rosensweig instability of ferrogel thin films or membranes

“…Note that the critical wave vector is independent of M ′ 0 , dominated by the bending elastic coefficient, and rather similar to equation (16). The threshold field is inversely proportional to the magnitude of the intrinsic permanent magnetization.…”

“…If the dissipation in the film or membrane can be neglected, the growth rate is given by the bulk fluid viscosity, only, λ = κ 1 (B 2 − B 2 c )/(2η) as in the case of a bulk ferrofluid or ferrogel, and the most unstable mode is the critical one, k u = k c in linear order [16].…”

Electrospray from an Ionic Liquid Ferrofluid utilizing the Rosensweig Instability

“…The onset and spacing of the Rosensweig instability Equation (1). 23 When the applied magnetic field is strong enough to create the Rosensweig instability, and the external field is uniform, such as inside a Helmholtz configuration, the spacing betwe governed by the capillary length of the fluid, as given in Equation (2). magnetic field is non uniform, such as in the case of a permanent magnet placed behind the ferrofluid, the internal force of the magnetic field gradient can be stronger than tip spacing between the peaks is no longer a function of the capillary length, but strength and gradient of the applied magnetic field and fluid properties, such as given in Equation (3).…”

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

“…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.…”

“…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…”