| A quasi two-dimensional drop of a magnetic fluid (ferrofluid) in a magnetic field is one example of the many systems including thin magnetic films, amphiphilic mono-layers, and type-I superconductors, that form labyrinthine patterns. A simple description that captures the essential physics of this pattern formation is to view these systems as thin, uniformly magnetized domains endowed with tension.; We study the hydrodynamics of shape instabilities in magnetized domains, and develop a generalization of Darcy's law which includes a boundary condition with competing Young-Laplace and Biot-Savart terms. A linear stability analysis results in a spectrum of growth rates which has a complicated wavevector dependence due to the long range forces, and reveals that the fingering arises from a negative effective surface tension. This further suggests a simple model of mode selection in the presence of time dependent magnetic fields, one which yields a power law behavior as a function a magnetic field ramp-rate, in good agreement with experiments. To study the shape evolution far beyond the initial instability, we solve the free boundary problem for the interface motion numerically using conformal mapping methods and find excellent qualitative agreement with experimental results.; We also consider this problem from an energetic point of view using a dissipative dynamical formalism, and apply this model to two other systems. Specializing to the limit of ultra thin domains, we make contact with experiments on the spectrum of thermal fluctuations in Langmuir monolayers, and obtain estimates of the dipole density and line tension. Finally, an interacting current loop model is developed for the intermediate state of a type-I superconductor, offering the first explanation for the complex patterns of flux domains seen in experiments. |
