This work shows that conformal mapping provides an effective way to solve certain unsteady two-dimensional perturbation problems involving the flow of a viscous incompressible fluid, in particular flow between moving circular cylinders. If the outer cylinder is considered fixed, the principal motions treated are the slow rotation of a slightly eccentric inner cylinder, and the vibration of an inner cylinder about a slightly eccentric point. Mapping the given circular boundaries (of a cross-section) into concentric circles enables one to solve for the stream function by means of a series.