Journal article
Non-local dynamics governing the self-induced motion of a planar vortex filament
- Abstract:
- While the Hasimoto planar vortex filament is one of the few exact solutions to the local induction approximation (LIA) approximating the self-induced motion of a vortex filament, it is natural to wonder whether such a vortex filament solution would exist for the non-local Biot-Savart dynamics exactly governing the filament motion, and if so, whether the non-local effects would drastically modify the solution properties. Both helical vortex filaments and vortex rings are known to exist under both the LIA and non-local Biot-Savart dynamics; however, the planar filament is a bit more complicated. In the present paper, we demonstrate that a planar vortex filament solution does exist for the non-local Biot-Savart formulation, provided that a specific non-linear integral equation (governing the spatial structure of such a filament) has a non-trivial solution. By using the Poincaré-Lindstedt method, we are able to obtain an accurate analytical approximation to the solution of this integral equation under physically reasonable assumptions. To obtain these solutions, we approximate local effects near the singularity of the integral equation using the LIA and non-local effects using the Biot-Savart formulation. Mathematically, the results constitute an analytical solution to an interesting nonlinear singular integrodifferential equation in space and time variables. Physically, these results show that planar vortex filaments exist and maintain their forms under the non-local Biot-Savart formulation, as one would hope. Due to the regularization approach utilized, we are able to compare the structure of the planar filaments obtained under both LIA and Biot-Savart formulations in a rather straightforward manner, in order to determine the role of the non-locality on the structure of the planar filament.
- Publication status:
- Published
- Peer review status:
- Peer reviewed
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(Preview, Accepted manuscript, pdf, 230.5KB, Terms of use)
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- Publisher copy:
- 10.1063/1.4922171
Authors
- Publisher:
- American Institute of Physics
- Journal:
- Physics of Fluids More from this journal
- Publication date:
- 2015-06-08
- Acceptance date:
- 2015-05-26
- DOI:
- EISSN:
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1089-7666
- ISSN:
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1070-6631
- Keywords:
- Pubs id:
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pubs:531364
- UUID:
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uuid:6bbd202f-5ad8-4e12-bec2-5d41436c5175
- Local pid:
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pubs:531364
- Source identifiers:
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531364
- Deposit date:
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2016-04-16
- ARK identifier:
Terms of use
- Copyright holder:
- American Institute of Physics
- Copyright date:
- 2015
- Notes:
- This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This is the author accepted manuscript following peer review version of the article. The final version is available online from American Institute of Physics at: http://dx.doi.org/10.1063/1.4922171
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