On Charwat's theory of motion of tracers in planar vortex flows

An adaptation of nonlinear equations for the motion of a charged particle in a cylindrically symmetric magnetic field, solved through use of a Riccati equation and complex numbers, is employed to describe the motion of neutrally buoyant tracers in vortex flows. The method is explored for both the solid body and free vortex conditions. It is shown that particles denser than the fluid will be forced outward in a spiral motion caused by Coriolis forces. The free vortex motion tends toward degeneracy with the method, but is amenable to analysis with an Emden-Fowler equation. Attention is given to two examples studied by Charwat (1977), who devised the unadapted form of the nonlinear model, demonstrating that undamped Mathieu equations can be obtained for a single harmonic in a periodically perturbed flow. Conditions under which the motions of the tracer particles will not be chaotic are defined.