Stability of an inhomogeneous ferrofluid in a channel, subject to a normal field

The stability of a ferrofluid with a fairly arbitrary non-uniform magnetic susceptibility between two parallel walls, subject to a non-uniform magnetic field acting normal to the walls, is investigated. The susceptibility may depend on position and the field strength, and the stationary state is such that the gradient of the susceptibility with respect to the modulus of the field is negative. Previous work suggests that the configuration may be linearly unstable, as regions of higher susceptibility do not coincide with regions of strongest field, and this is proved here. Adding a constant field in the plane of the layer suppresses parallel instabilities, but has no effect on those orthogonal to it. However, it is demonstrated that stability can be achieved by applying a rapidly rotating field.