Steady vortex flows driven by a radial convection of angular momentum are considered. The incompressible Navier-Stokes equations are linearized by considering perturbations about both simple, nonrotating flows and strongly rotating flows, i.e., the equations are expanded for large and small Rossby numbers. Axial variations in a swirl superimposed upon a stagnation-point flow (with radial inflow) are considered for large Rossby numbers. An analytic solution is found for the axial decay of a given swirl. By considering flows for small Rossby numbers, it is found that, in flows dominated by rotation, the fluid motion is forced to be two-dimensional except in thin shear regions where necessary adjustments imposed by boundary conditions are made. The properties of these different shear layers are dependent on the gradient of the basic circulation of the flow as well as the Reynolds number.