Dynamics of labyrinthine pattern formation in magnetic fluids : a mean-field theory

Abstract: "We are interested in the flow of a droplet of viscous ferrofluid in the Hele-Shaw cell under a transverse magnetic field. The (two-dimensional) phase configuration is observed to evolve into a labyrinthine pattern. We show that the conventional model for this flow has the form of a gradient flux w.r.t. an energy functional, which is the sum of magnetic and surface energy. In particular, we are interested in the behaviour of this flow problem in the regime of large magnetization M┬▓ >> 1. In this regime, the details of the pattern evolution are observed to be highly sensitive to changes in the initial configuration. This is reflected in the linear stability analysis of the circular phase configuration, a stationary point of the dynamics which is more and more unstable in the limit M┬▓ [up arrow] [infinity]. In order to capture the 'generic' behaviour of the dynamical system in this regime, we need a selection principle which dismisses those non-generic solutions. We propose a selection principle for the limit M┬▓ [up arrow] [infinity] which is based on the natural implicit discretization in time of our gradient flux formulation. We prove that this approach leads (in an appropriate scaling) to the equation [delta subscript t]s - [delta]s┬▓ = 0 for s(t,x) [element of] [0,1], the local spatial average of the phase configuration [subscript X](t,x) [element of] [0,1] ([subscript X])(t,x) = 1 if x [element of] R┬▓ lies in the two-dimensional cross-section of the fluid at time t, [subscript X](t,x) = 0 else). Thus this quantity, which contains information on the 'microstructured zone', evolves deterministically, although [subscript X] is essentially unpredictable."