Three-dimensional instabilities of ferromagnetic liquid bridges

The cross section of a ferromagnetic liquid drop held in equilibrium between horizontal plates in a magnetic field loses its circular symmetry past a critical value of the applied field strength. This is caused by instabilities that give way to non-circular cross sectional shapes which, in turn, produce three-dimensional magnetic field distribution inside and outside the drop. Theoretical predictions of equilibrium non-circular shapes and their stability are drawn from the equations governing the magnetohydrostatic equilibrium of the drop. The computational problem is three-dimensional, nonlinear and free boundary and it is solved with the Galerkin/finite element method. Entire branches of circular solutions and non-circular ones are traced by continuation in multi-parameter space. Circular, elliptical and dumbbell-shaped drops have been found. The relative stability of the various shapes is computed by means of computer-implemented bifurcation theory.