A theory is developed for the dynamics of pattern formation in quasi-two-dimensional domains of magnetic fluids (ferrofluids) in transverse magnetic fields. The pattern formation is treated as a dissipative dynamical process, with the motion derived variationally from a static energy functional using minimal assumptions. This dynamics is one instance of a general formalism applicable to any system that can be modeled as a closed curve in a plane. In applying the formalism to ferrofluids, we present a calculation of the energy of a two-dimensional dipolar domain as a functional of the shape of its boundary. A detailed linear stability analysis of nearly circular shapes is presented, and pattern formation in the nonlinear regime, far from the onset of instability, is studied by numerical solution of the nonlinear, nonlocal evolution equations. The highly branched patterns obtained numerically bear a qualitative resemblance to those found experimentally. The time evolution exhibits sensitive dependence on initial conditions, suggesting the existence of many local minima in the space of accessible shapes. The analysis also provides a deterministic starting point for a theory of pattern formation in dipolar monolayers at the air-water interface, in which thermal fluctuations play a more dominant role.

• Received 23 April 1992

DOI:https://doi.org/10.1103/PhysRevA.46.4894