Competition between the Bénard-Marangoni and the Rosensweig Instability in Magnetic Fluids

Weilepp, Jochen; Brand, Helmut R.

doi:10.1051/jp2:1996189

Cited by 37 publications

(71 citation statements)

References 5 publications

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“…The absolute value of the imaginary part of −iω, |ω 1 |, gives the angular frequency of the oscillations if it is different form zero. The linear stability analysis leads to the dispersion relation [13,14,15] …”

Section: System and Basic Equationsmentioning

confidence: 99%

Linear and nonlinear approach to the Rosensweig instability

2007

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Key words magnetic fluids, ferrofluid, interfacial instability, Rosensweig instability, pattern formation, hydrodynamic stability, solitons, finite elements.We report on recent efforts to improve the understanding of the Rosensweig instability in the linear as well as in the nonlinear regime. In the linear regime we focus on the wavenumber of maximal growth and the oscillatory decay of metastable magnetic liquid ridges, accessible via a new pulse technique. We compare the measurements with the predictions of the linear stability analysis. In the nonlinear regime the fully developed Rosensweig pattern was successfully estimated by the method of finite elements, taking into account the nonlinear magnetization law. For a comparison with these results the three-dimensional surface profile is recorded by a radioscopic measurement technique. The bifurcation diagram measured in this way can be fitted by the roots of an amplitude equation. Eventually we investigate ferrosolitons, which were recently uncovered in the bistability interval of the Rosensweig instability.

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Section: System and Basic Equationsmentioning

confidence: 99%

Linear and nonlinear approach to the Rosensweig instability

2007

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“…Therefore ω is commonly called the growth rate, which is in fact true only for its imaginary part in the chosen normal mode ansatz. The linear stability analysis leads to the dispersion relation [9][10][11] …”

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confidence: 99%

Scaling behaviour of the maximal growth rate in the Rosensweig instability

Lange¹

2001

Europhys. Lett.

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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

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“…͑b͒ The normal component of the magnetic induction B = 0 ͓H + M͔ and the tangential component of the magnetic field H have to be continuous across the top and the bottom boundaries: 5,10,13,16,17 H z + M z = H e sin , and H x = H e cos . ͑8b͒…”

Section: The Boundary Conditionsmentioning

confidence: 99%

“…For EMG 901, the magnetization at saturation is M sat = 4.8ϫ 10 4 A/m. 17 Our study is valid only in the framework of a strong magnetic field ͑4a͒, so that we must consider a magnetic field larger than M sat to obey the strong field assumption. Then, should we take an exterior magnetic field larger than 4.8ϫ 10 4 A / m, our choice of parameters ͑see Table I͒ shows ⑀ H to be less than 0.001.…”

Section: Experimental Values Of the Dimensionless Parametersmentioning

confidence: 99%

Rayleigh–Marangoni–Bénard instability of a ferrofluid layer in a vertical magnetic field

Hennenberg

Weyssow

Slavtchev

et al. 2005

Journal of Magnetism and Magnetic Materials

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A horizontal ferrofluid layer is submitted to a lateral heating and to a strong oblique magnetic field. The problem, combining the momentum and heat balance equations with the Maxwell equations, introduces two Rayleigh numbers, Ra the gravitational one and Ra m the magnetic one, to represent the buoyancy and the Kelvin forces, which induce motion, versus the momentum viscous diffusion and heat diffusion. Whatever the inclination of the magnetic field, the steady solution of the problem is presented as a power series of a small parameter ⑀ H measuring the ratio of variation of the magnetization across the layer divided by the magnitude of the external imposed field. For cases of physical relevance, comparisons between analytical and numerical studies have lead to a major statement: in the strong field region ͑⑀ H 1͒ the zero order solution is the product of the Birikh solution that corresponds to the usual Newtonian fluid submitted to a lateral gradient, multiplied by a modulating factor accounting for inclination and both Rayleigh numbers. Physically, this simplified solution is valid for microgravity conditions where the magnetic field competes enough with microgravity effects to invert the laminar flow and thus suppress the motion for two specific values of the inclination angle.

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Competition between the Bénard-Marangoni and the Rosensweig Instability in Magnetic Fluids

Cited by 37 publications

References 5 publications

Linear and nonlinear approach to the Rosensweig instability

Linear and nonlinear approach to the Rosensweig instability

Scaling behaviour of the maximal growth rate in the Rosensweig instability

Rayleigh–Marangoni–Bénard instability of a ferrofluid layer in a vertical magnetic field

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