Bifurcating Instability of the Free Surface of a Ferrofluid

Abstract

Consider a slab of ferrofluid bounded below by a fixed boundary and above by a vacuum. If the fluid is subjected to a vertically directed magnetic field of sufficient strength, surface waves appear. The equations which describe this phenomenon are derived. In the physical space no natural Banach space structure is available due to the free surface. In order to use the available bifurcation theory, a transformation of coordinates is made, mapping the surface flat. In the new coordinate system the equations define an operator between Banach spaces. The minimum eigenvalue of the linearized operator is the critical magnetic field strength where the planar surface loses stability. Using a generalized inverse of the Fréchet derivative of the operator and the implicit function theorem, the existence of a nontrivial branch of solutions is proved. A local stability criterion is also obtained and applied to three periodic structures: rolls, squares and hexagons.