Spontaneous rotation of disks by active nematics

We consider a two-dimensional active nematic liquid crystal between two concentric circular boundaries. The anchoring conditions are hybrid, meaning that the nematic directors make different constant angles with the inner and outer boundaries. When the activity is zero, the system takes on the "magic spiral" geometry which was proposed by Robert Meyer to elucidate the balance of torques in nematic liquid crystals in equilibrium. When the activity is nonzero and the inner circle bounds a disk which is free to rotate, active flows spin the disk at a speed that depends on the activity, viscosity, liquid crystal parameters, and the size of the gap between the two circles. We calculate the rotation speed analytically using the frozen nematic approximation in which the effect of the flow on the nematic is disregarded, but the viscous flow is accounted for. We examine the accuracy of this approximation by numerically solving the full hydrodynamic equations.