| The flat interface between a pool of ferromagnetic liquid, or ferrofluid, and non-magnetic surroundings, usually air, turns unstable when the strength of a magnetic field applied perpendicularly to the interface exceeds a threshold. The instability, called normal field instability, results from force competition: magnetostatic force destabilizes a flat interface, whereas surface tension and gravity stabilize the interface. Post-instability static equilibrium interface shapes are patterns of tall peaks, or spikes, and hollows. In small pools axisymmetric patterns form; in large pools two-dimensional patterns form that are often hexagonal or square.; Mechanisms of instabilities and pattern formation at ferrofluid interfaces are the subject of this thesis. Fundamental physical laws of mechanics and electromagnetism govern magnetohydrostatic equilibria of ferrofluids and give rise to nonlinear partial differential equations that are solved for the magnetic field inside the ferrofluid and in the neighboring non-magnetic phase, and for interface shape. The equations are posed in two- or three-dimensional domains with free and often highly irregular boundaries. The equations are solved by powerful methods of modern computer-aided nonlinear analysis. Central are Galerkin's method of weighted residuals, and finite element basis functions; with these the continuum equations are discretized. Efficient methods of parametric continuation and bifurcation and stability analyses are employed to determine the structure of the solution space of the equations; that is, sensitivity of equilibrium states to changes in parameters, existence and multiplicity of states and response of states to small disturbances.; Theoretical analysis reveals that many patterns can potentially form at a ferrofluid interface at onset of the normal field instability: hexagonal, triangular, square, roll and ring. Pattern selection is determined from the stability of patterned states to disturbances that are patterned differently. Hexagonal patterns, the most often observed in experiments, are found to be stable only at finite-amplitude interface distortions. Spikes of finite height form abruptly at onset of instability; they grow as field strength is raised and subside with hysteresis as field strength is subsequently lowered. Experimental and theoretical analyses establish the significance of pool finiteness, i.e. of the presence of confining walls, in pattern selection. |
