Abstract We report the results of a numerical study of the creation of stagnation points in a rotating cylinder of fluid where both end-walls are rotated. Good agreement is found with previous results where the stagnation points are formed on the core of the primary columnar vortex. Novel phenomena have been uncovered at small aspect ratios where stagnation occurs off-axis directly and the secondary vortex which is created forms a toroid. The case is then considered of a small cylinder placed along the centre of the flow and despite the qualitatively different boundary condition, the phenomena are found to be robust.