The two-dimensional flow of a viscous fluid contained in a cylinder which is subjected to rotational oscillatory motion is considered. The instantaneous flow relative to the cylinder is driven by a Poincar\'e-type force which provides a uniform rate of production of vorticity. The problem may be linearised when the viscosity is sufficiently large and the amplitude of oscillations is sufficiently small; the Reynolds number of the instantaneous flow relative to the cylinder is then small, and the Strouhal number is large. If the cross-section of the cylinder has any sharp corners, the nature of the flow near these corners may be analysed through comparison of the `driven' component of the flow, and the eigenfunction ingredients of the corresponding homogeneous problem which are inevitably present. A sufficient condition for the appearance of oscillatory eddy structures emerging from the corners is obtained and confirmed numerically for various geometries of the boundary.