The stability of a Batchelor flow enclosed between a stationary and a rotating disk with a central hub is investigated by extensive experimental visualizations and direct numerical simulations. The first instability appears as circular rolls (CR), which propagate along the stator towards the rotation axis. Above a second threshold, spiral rolls (SRI) appear at the periphery of the cavity due to the destabilization of the boundary layer along the external cylinder. These spirals coexist with the circular rolls in the experiment. For the first time, the presence of an inner rotating hub, which is of major industrial importance, measured by the influence of the curvature parameter has been studied experimentally. This parameter strongly modifies the wave number of the instabilities as well as the angle of the spirals. Finally, for a given geometry, a very good agreement on the characteristics of these instabilities is obtained between the experimental and numerical approaches.