Stability of Couette flow in nematic liquid crystals

We study the stability of the cylindrical Couette flow in nematics when the director is parallel to the rotation axis. The contribution of the inertial coupling of velocity fluctuations (responsible for the Taylor instability in isotropic liquids) is shown to be destabilizing when the inner cylinder rotates faster than the outer one. However, the instability remains driven by the mechanisms first discovered by Pieranski & Guyon for the plane shear case and quite specific to nematics. This mechanism couples the different orientation fluctuations via viscous torques and the corresponding threshold is given by \[ s\tau_0\sim 1, \] where τ₀ is the time constant for the diffusion of orientation fluctuations. The contribution of inertia terms is measured by 2ω_m τ_v, where τ_v is the time constant for the diffusion of velocity fluctuations. In usual nematics one has τ_v/τ₀ ∼ 10⁻⁵ so that corrections due to rotation are small in general. At different stages of the discussion differences between the case of nematics and that of isotropic liquids are pointed out. We also study the possibility of an oscillatory instability when α₃ is positive and large, where no stationary instability can occur.