Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory
The paper deals with the Hele-shaw cell; it is made of two horizontal sheets of glass which have a very small spacing \(b\). Part of the gap is filled with a ferroliquid (a colloidal suspension of iron particles), the rest is filled with an ordinary fluid. The ferroliquid fills a cylindrical domain \(\Omega \times(0,b)\) so the phase configuration is described by the two-dimensional cross section \(\Omega\). The effect of a strong magnetic field perpendicular to the cell is applied. The dynamics is governed by two competing forces: surface tension and magnetostatic repulsion. The paper starts from a model for the evolution proposed by Jackson et al.. The conventional analysis does not bridge the gap between the early stages (captured by the linearization) and the late stages of the evolution (described by a variational problem). The numerical analysis only describes the early stages. The author presents a new type of analysis for the intermediate stage. An evolution equation is derived for the local volume fraction of the gap filled by the ferroliquid. The analysis is based on the discovery that this type of strongly overdamped flow problems can be considered as a gradient flux of the relevant energy functionals.
scientific article; zbMATH DE number 1154551