The development of Long's vortex

This paper describes the solution of Long's problem for steady rotationally symmetric swirling jets in a uniform viscous fluid. Long found these vortices in 1958 by assuming a similarity form of solution, and in 1961 solved the consequent problem in the boundary-layer limit, finding dual solutions. The overall pattern of the solutions to the problem for general values of the Reynolds number is described. The linear spatial stability of the flows to small steady disturbances is analysed and a few results presented. In particular, details of the solutions and their stability are given asymptotically for small and large values of the Reynolds number. The asymptotic results for the basic flow are linked by direct numerical integration of the flow at several finite positive values of the Reynolds number.