Non-linear dynamics of a nematic liquid crystal in the presence of a shear flow

In this paper we study a Landau-de Gennes model of a nematic liquid crystal in a model shear flow employing a range of analytical and numerical techniques. We use asymptotic methods and numerical bifurcation theory to provide a comprehensive description of the different in-plane modes that exist in terms of the temperature and imposed shear rate, as well as determining their stability to in-plane disturbances. We show that there are two physically distinct solution branches, one of which corresponds to an in-plane director (denoted IPN). The other branch corresponds to the director being aligned perpendicular to the shear plane, i.e. so-called log-rolling states. We find both tumbling and wagging time-dependent modes. The mode structure is organized by a Takens-Bogdanov point, at which a family of Hopf bifurcation points and a family of limit points coincide. At low strain rates tumbling occurs as a transition from the IPN branch. In addition, we show that at larger values of the strain rate the wagging modes emanate from a Hopf bifurcation on the IPN branch and evolve continuously into tumbling modes. Moreover, we show analytically that this boundary between tumbling and wagging corresponds to a state in which the distribution function is cylindrical in the shear plane. This provides confirmation of the tumbling-wagging transition mechanism observed by Tsuji and Rey in numerical calculations relating to a confined geometry