Motivated by many recent experimental studies of nonclassical rotational inertia (NCRI) in superfluid and supersolid samples, we present a study of the hydrodynamics of a superfluid confined in the two-dimensional region (equivalent to a long cylinder) between two concentric arcs of radii b and a (b < a) subtending an angle beta, with 0 <=beta <= 2 pi. The case beta=2 pi corresponds to a blocked ring. We discuss the methodology to compute the NCRI effects and calculate these effects both for small angular velocities, when no vortices are present, and in the presence of a vortex. We find that, for a blocked ring, the NCRI effect is small and that therefore there will be a large discontinuity in the moment of inertia associated with blocking or unblocking circular paths. For blocked wedges (b=0) with beta>pi, we find an unexpected divergence of the velocity at the origin, which implies the presence of either a region of normal fluid or a vortex for any nonzero value of the angular velocity. Implications of our results for experiments on ``supersolid″ behavior in solid He-4 are discussed. A number of mathematical issues are pointed out and resolved.