Inertialess incompressible flow through a rapidly rotating, variable-area conduit of circular cross-section is treated. For the practical case of an expansion (or contraction) placed between two pipes, the flow is strongly asymmetrical and involves regions of weak reverse flow in the expansion and downstream pipe, while the disturbance to the fully developed pipe flows persists for large, O(E−½) distances upstream and downstream, where E is the (small) Ekman number. The flow in the two pipes depends only on the ratio of their radii and is independent of the shape and length of the expansion. The startling implication of the disturbance's persistence is that, in practice, fully developed flow will almost never be realized in rapidly rotating pipes.