We present a lattice Boltzmann algorithm for the hydrodynamics of liquid cr
ystals in the isotropic and nematic phases. The coupling between the tensor
order parameter and the flow is treated consistently allowing investigatio
n of a wide range of non-Newtonian flow behaviour. To illustrate the use of
the method we present results for the effect of hydrodynamics on defect co
alescence and demonstrate the occurrence of shear-banding and shear-thinnin
g.