The orientational order of a nematic liquid crystal in a spatially inhomogeneous flow situation is best described by a Q-tensor field because of the defects that inevitably occur. The evolution is determined by two equations. The flow is governed by a generalised Stokes equation in which the divergence of the stress tensor also depends on Q and its time derivative. The evolution of Q is governed by a convection-diffusion type equation that contains terms nonlinear in Q that stem from a Landau-de Gennes potential.
In this talk, I will show how the most general evolution equations can be derived from a dissipation principle. Based on this, I will identify a specific model with three viscosity coefficients that allows the contribution of the orientation to the viscous stress to be cast in the form of a Q-dependent body force. This leads to a convenient time-discretised strategy for solving the flow-orientation problem using two alternating steps. First, the flow field of the Stokes flow is computed for a given orientation field. Second, with the given flow field, one time step of the orientation equation is carried out. The new orientation field is then used to compute a new body force which is again used in the Stokes equation and so forth.
For some simple test applications at low Reynolds numbers, it is found that the non-homogeneous orientation of the nematic liquid crystal leads to non-linear flow effects similar to those known from Newtonian flow at high Reynolds numbers.