The problem considered is that of the two-dimensional motion of the fluid in a cylinder of finite radius after the outer portion of the fluid is given an initial uniform velocity. The primary purpose of the investigation is the study of the changes in the energy distribution in the fluid as the initial motion decays. The appropriate flow equations are developed and then approximated by finite-difference equations. Numerical solutions of these equations are presented, and the energy-transfer processes are discussed in some detail. During the early stages of the flow, it is found that the spatial distribution of energy depends strongly on the Prandtl number. During the later stages, however, there is a net outward flow of energy for the case of a liquid and a net inward flow for a gas.