Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-24T22:25:04.237Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  17 February 2009

R. B. Leipnik
R. B. Leipnik
Affiliation:
Department of Mathematical Physics, University of Adelaide, G.P.O. Box 498, Adelaide. S.A. 5001.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Information
The ANZIAM Journal , Volume 25 , Issue 2 , October 1983 , pp. 175 - 189
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Charwat, A., “Motion of near-neutrally buoyant tracers in vortical flows”, Phys. Fluids 20 (1977), 401–403.CrossRefGoogle Scholar
[2]Fowler, R. H., “Emden's and similar differential equations”, Quart. J. Math. Oxford 92 (1931), 259–290.CrossRefGoogle Scholar
[3]Green, H. S. and Leipnik, R. B., “Diffusion and conductivity of plasmas in strong external fields”, Internat. J. Engrg. Sci. 3 (1965), 491–514.CrossRefGoogle Scholar
[4]Ince, E. L., Ordinary differential equations (Dover, New York, 1944; Longmans Green, London, 1926).Google Scholar
[5]Jordan, D. W. and Smith, P., Nonlinear ordinary differential equations (Oxford University Press, Oxford, 1977).Google Scholar
[6]Leipnik, R. B., Seymour, P. W. and Nicholson, A. F., “Charged particle motion in a time-dependent axially symmetric field”, Austral. J. Phys. 18 (1965), 553–566.Google Scholar
[7]McLachlan, N. W., Theory and application of Mathieu functions (Oxford University Press, Oxford, 1947).Google Scholar