Shear flow in nematic liquid crystals: Fréedericksz transition as a bifurcation

We numerically investigate the Fréedericksz transition for steady state plane shear flow of nematic liquid crystals between two parallel plates in the presence of external magnetic fields. Three typical configurations with the external field in the plane of the flow and perpendicular to it, in the plane and along the flow, and where it is perpendicular to the plane of the flow are considered. In each case, the Fréedericksz transition is studied as a bifurcation problem. Beginning with a steady state shear flow, solutions corresponding to slowly increasing magnetic fields and those corresponding to fields which are suddenly turned on at a given intensity are studied. For a typical idealized nematic, we show that the symmetric pitch-fork bifurcation in the absence of shear becomes a transcritical bifurcation from the trivial solution in one configuration while in another it resembles a disconnected pitch-fork where the turning point of the disconnected branch is a generic singularity in the absence of symmetry or a trivial solution.