Magnetic liquid or ferrofluids are complex fluids which interact with external magnetic field. Many effects can be observed. One of the most spectacular phenomena is the Rosensweig or normal field instability where an external magnetic field is applied perpendicular to a flat and horizontal surface. For small magnetic field strength the surface remains flat. If the magnetic field strength exceeds a critical value then a regular pattern of so-called peaks occurs. This phenomenon can be described by a coupled system of nonlinear partial differential equations. On the one hand we have to consider the Maxwell‘s equations in the fluid and the surrounding. On the other hand we have to deal with the Navier-Stokes equations in the time-dependent domain which is occupied by the magnetic liquid. Finally, the force balance at the free surface which is given by the Young-Laplace equation has to be taken into account. If a constant magnetic field is applied then the magnetic fluid reaches a stationary state where the liquid is in rest. For the static case we present a decoupling strategy which is based on the subproblems. An error estimate for a finite element discretization of the Young-Laplace equation is given and proven. We use the ALE approach to deal with the time-dependent domain in the dynamic case. The whole problem is decoupled in a similar way as in the static case. The simulations in both the static and the dynamic case show that the used finite element methods are able to calculate the peaks shapes and the critical value for the magnetic field strength. Moreover, the differences between the static case and the stationary limit of the dynamic case are very small.

Gunar Matthies was born in Haldensleben, Germany. He received his diploma degree and his Ph.D. degree from the Otto von Guericke University Magdeburg, Germany. He is currently professor of Numerical Analysis at Technische Universität Dresden, Germany. The main working fields of Gunar Matthies are finite element method for convection-dominated problems. His interests cover the development of stabilised methods for convection-diffusion and incompressible flow problems and their analysis.