Nonlinear dynamics of a nematic liquid crystal in the presence of a shear flow
In this thesis we describe the complex array of behaviours of a homogeneous thermotropic nematic liquid crystal in the context of a Landau-de Gennes theory. There exist two parameters that control the behaviour of the system: the temperature and the shear rate, and by employing continuation and bifurcation theory we describe the different time dependent states for the two and three dimensional cases. For the two dimensional case we compute the steady state solution branches finding that the flow favours an in-plane nematic state at higher temperatures, while at lower temperatures it favours a nematic state with preferred direction of alignment perpendicular to the shear plane, the so-called log-rolling state. We have found excellent agreement between the numerical calculations and analytical results in the limit of very low and very large values of the shear rate. The existence of a Takens-Bogdanov bifurcation in the underlying bifurcation diagram organises the steady and the time dependent solutions in the state diagram. The periodic orbits can be either of the wagging type, at intermediate values of the shear rate or of the tumbling type at lower shear rates. We complete the analysis of the two dimensional case, by considering a general planar flow and studying the influences of strain and vorticity in the system. We provide a very detailed account of the behaviour of the liquid crystal in the three dimensional case, when the direction of alignment of the molecules that constitute the liquid crystal is allowed out of the shear plane. We establish that the only out-of-plane steady solution of the system is an anomalous continuum of equilibria, and therefore the Landau-de Gennes model that we are employing is structurally unstable. The time dependent solutions of the liquid crystal fall into one of the following categories: in plane periodic orbits, which are the tumbling and wagging solutions and out-of-plane periodic orbits, the so-called kayaking state. The use of bifurcation theory in the context of nematodynamics allows us to give a complete summary of the nonlinear behaviour of a nematic liquid crystal in a shear flow, for the two and three dimensional cases.