This paper presents applications of a control-volume finite-difference method to flow problems in cylindrical geometries. The method is an extension of the method known as ETUDE, which is an Euler-explicit in time, transportive-upwind convection, second-order diffusion, finite-difference estimate. The primary purpose of this paper is to present an interesting new method that comes about from control-volume considerations, i.e., from a proper extension of ETUDE, to solve problems in cylindrical coordinates. One- and two-dimensional test problems are computed to illustrate the properties of this new method. The predicted results of the two-dimensional test problem are compared with similar calculations of the spin-up of homogeneous fluids reported in the literature. With the properties of the method established, it was applied to investigate the spin-up of a two-layered, stably stratified fluid initially at rest in a cylindrical container. The effect of the stable, density structure on spin-up is discussed.