The Amplitude Equation for the Rosensweig Instability in Magnetic Fluids and Gels

TL;DR

This paper derives a detailed amplitude equation for the Rosensweig instability in magnetic fluids and gels, addressing unique challenges posed by the deformable surface and boundary-driven forces.

Contribution

It introduces new procedures for weakly nonlinear analysis that account for the deformable surface and boundary-only driving forces in the instability.

Findings

Derived the amplitude equation considering surface deformability.

Implemented a dynamic treatment of the surface in the analysis.

Modified bifurcation theory formalism to incorporate boundary effects.

Abstract

The Rosensweig instability has a special character among the frequently discussed instabilities. One distinct property is the necessary presence of a deformable surface, and another very important fact is, that the driving force acts purely via the surface and shows no bulk effect. These properties make it rather difficult to give a correct weakly nonlinear analysis. In this paper we give a detailed derivation of the appropriate amplitude equation based on the hydrodynamic equations emphasizing the conceptually new procedures necessary to deal with the distinct properties mentioned above. First the deformable surface requires a fully dynamic treatment of the instability and the observed stationary case can be interpreted as the limiting case of a frozen-in characteristic mode. Second, the fact that the driving force is manifest in the boundary conditions, only, requires a considerable change in the formalism of weakly nonlinear bifurcation theory. To obtain the amplitude equations a combination of solubility conditions and (normal stress) boundary conditions has to be invoked in all orders of the expansions.