Analytic methods for treating unsteady axisymmetric jets

The steady axisymmetric jet and Hill's spherical vortex are two rare examples of closed form solutions of the Navier-Stokes equation. The work of Proudman and Pearson showed an infinite series based on Legendre polynomials which leads to a simplification of the steady Navier-Stokes equation. These earlier discoveries are combined by reformulating the axisymmetric jet solution into an infinite series based on Legendre polynomials. This new solution form is then used as a guide in developing an unsteady ansatz or template to which the steady solution will satisfy in the steady limit. This ansatz when inserted into the unsteady Navier-Stokes equation reduces the equation into a single linear ordinary differential equation. This new differential equation not only describes the axisymmetric jet of Landau and Squire but also Hill's spherical vortex and an infinite number of other jets. Through use of this new ordinary differential equation, the unsteady axisymmetric jet is solved to second order. In addition, a complete formulation of Hill's spherical vortex is developed to first order.