In this paper we develop a new class of analytical solutions of the Navier-Stokes equations and suggest ways to predict and control complex swirling flows. We consider vortex sinks on curved axisymmetric surfaces with an axial flow and obtain a five-parameter solution family that describes a large variety of flow patterns and models fluid motion in a cylindrical can, whirlpools, tornadoes, and cosmic swirling jets. The singularity of these solutions on the flow axis is removed by matching them with swirling jets. The resulting composite solutions describe flows, consisting of up to seven separation regions (recirculatory "bubbles" and vortex rings), and model flows in the Ranque-Hilsch tube, in the meniscus of electrosprays, in vortex breakdown, and in an industrial vortex burner. The analytical solutions allow a clear understanding of how different control parameters affect the flow and guide selection of optimal parameter values for desired flow features. The approach permits extension to swirling flows with heat transfer and chemical reaction, and have the potential of being significantly useful for further detailed investigation by direct or large-eddy numerical simulations as well as laboratory experimentation.