Transition to spatially periodic patterns in nematics under oscillatory shear flow: linear analysis

We consider the orientational instabilities, both homogeneous and spatially periodic, developing in a nematic liquid crystal under rectilinear oscillatory Couette flow for director alignment perpendicular to the flow plane. Using numerical and analytical approaches we determine the critical amplitude of oscillatory flow instabilities, the critical wave number and the symmetry of the destabilizing mode. It was found, that by varying of the oscillatory flow frequency the instability changes its temporal symmetry. The mechanism of this transition is coupled with the inertia of the nematic fluid. We also shown that an electric field applied to the nematic layer can induce switching between instabilities with different spatial and temporal symmetries. The complete phase diagram of the flow instabilities is presented