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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 crosssection) into concentric circles enables one to solve for the stream function by means of a series. In the first problem, the solution is carried far enough to afford an estimate for convergence. The torque on the inner cylinder and the second order steady streaming are computed, and a necessary re-examination of the meaning of the second concept is given. Various special cases of the second problem are shown to illuminate the concepts of virtual mass and viscous damping. The results for high frequency vibration bear on an experimental paper by Stuart and Woodgate, some of the low frequency results on theoretical papers by Andres and Ingard, and by Stokes. Both problems are of general interest in connection with the design of certain control mechanisms, and with experiments on Helium II.

Application of conformal mapping to viscous flow between moving circular cylinders