Numerical simulation of the behaviour of magnetic liquids

Magnetic liquids or ferrofluids are complex fluids. Some of their physical properties can be influenced by applying an outer magnetic field. One interesting phenomenon is the so-called Rosensweig instability where a magnetic field is applied normal to a flat an horizontal free surface. If the magnetic field strength exceeds a certain threshold a spontaneous surface deformation takes place. The interface between ferrofluid and air forms a regular hexagonal pattern of peaks.

This phenomenon can be described by a coupled systems of nonlinear partial differential equations involving the Young-Laplace equation on the free surface. A decoupling of the whole problem into two subproblems will be presented. Both are solved using finite element methods. For the Young-Laplace equation as one subproblem, a finite element error estimate will be proven.

Finally, numerical simulations of the Rosensweig instability using different discretisation schemes will be given.