The motion of small, near neutrally buoyant tracers in vortex flows of several types is obtained on the basis of Charwat's mathematical model, which is highly non-linear.
The solution method in the non-degenerate case expresses the squared orbital radius r2 as a product AA*, where the complex number A satisfies a second-order linear differential ‘factor equation’, generally with variable coefficients. The angular coordinate is expressed in terms of log(A*/A). Solid-type rotation and sinusoidally perturbed solid-type rotation correspond respectively to constant coefficients and sinusoidal coefficients. The former exactly yields a scalloped spiral tracer motion; the latter yields unstable tracer motion as t → ∞ except when the perturbing frequency and amplitude are rather specially related to the flow and tracer parameters. Free vortex motion is somewhat degenerate for this solution method but can be partially analyzed in terms of solutions of a generalized Emden–Fowler equation. The method can be used for other planar flow problems with a symmetry axis.