Dispersion Relation in a Ferrofluid Layer of Any Thickness and Viscosity in a Normal Magnetic Field; Asymptotic Regimes

Issue		J. Phys. II France Volume 7, Number 8, August 1997


Page(s)		1159 - 1171
DOI		10.1051/jp2:1997178

DOI: 10.1051/jp2:1997178
J. Phys. II France 7 (1997) 1159-1171

Dispersion Relation in a Ferrofluid Layer of Any Thickness and Viscosity in a Normal Magnetic Field; Asymptotic Regimes

B. Abou¹, G. Néron de Surgy² and J.E. Wesfreid¹

¹ Laboratoire de Physique et Mécanique des Milieux Hétérogènes PMMH-Unité CNRS 857, École Supérieure de Physique et Chimie Industrielles de Paris (ESPCI), 10 rue Vauquelin, 75231 Paris Cedex 05, France
² Laboratoire de Génie Électrique de Paris (LGEP), Plateau du Moulon, 91191 Gif sur Yvette Cedex, France

(Received 7 November 1996, revised 4 February 1997, accepted 24 April 1997)

Abstract
We have calculated the general dispersion relationship for surface waves on a ferrofluid layer of any thickness and viscosity, under the influence of a uniform vertical magnetic field. The amplification of these waves can induce an instability called peaks instability (Rosensweig instability). The expression of the dispersion relationship requires that the critical magnetic field and the critical wavenumber of the instability depend on the thickness of the ferrofluid layer. The dispersion relationship has been simplified into four asymptotic regimes: thick or thin layer and viscous or inertial behaviour. The corresponding critical values are presented. We show that a typical parameter of the ferrofluid enables one to know in which regime, viscous or inertial, the ferrofluid will be near the onset of instability.

Résumé
Nous avons calculé la relation de dispersion des ondes de surface dans une couche de ferrofluide d'épaisseur et de viscosité quelconques, soumise à un champ magnétique normal à sa surface (instabilité de pics de Rosensweig). Cette relation montre que le champ magnétique critique et le vecteur d'onde critique de l'instabilité dépendent de l'épaisseur de la couche de fluide. La relation de dispersion a été simplifiée pour quatre régimes asymptotiques: couche épaisse ou mince et comportement visqueux ou inertiel. Nous avons calculé les valeurs critiques de l'instabilité dans ces quatre cas. Nous montrons qu'un paramètre typique du ferrofluide permet de savoir dans quel régime, visqueux ou inertiel, se situe le ferrofluide près du seuil de l'instabilité.

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