Orientational instability of nematics under oscillatory flow

Generalizing a recently introduced approximation scheme valid when the director relaxation time is large compared to the inverse frequency of an oscillatory flow, we derive equations for the time-averaged torques on the director for a prescribed plane shear flow. Whereas for a linear flow field (simple Couette flow) the average torques vanish, one has for Poiseuille flow (and more general flow fields) torques that tend to orient the director essentially perpendicular to the flow plane. For flow-aligning materials the orientation parallel to the flow is also (weakly) stable. Including the effect of homeotropic surface alignment we estimate the threshold of the oscillation amplitude for the out-of-plane transition. The results are essentially recovered (and improved) by direct numerical simulations