Numerical study of the Rosensweig instability in a magnetic fluid subject to diffusion of magnetic particles

Abstract

The present study is devoted to the classical problem on stability of a magnetic fluid layer under the influence of gravity and a uniform magnetic field. A periodical peak-shaped stable structure is formed on the fluid surface when the applied magnetic field exceeds a critical value. The mathematical model describes a single peak in the pattern assuming axial symmetry of the peak shape. The field configuration in the whole space, the magnetic particle concentration inside the fluid and the free surface structure are unknown quantities in this model. The unknown free surface is treated explicitly, using a parametric representation with respect to the arc length. The nonlinear problem is discretized by means of a finite element method for the Maxwell's equations and a finite-difference method for the free surface equations. Numerical modeling allows to get overcritical equilibrium free surface shapes in a wide range of applied field intensities. Our numerical results show a significant influence of the particle diffusion on the overcritical shapes