### Abstract

We study the motion of a counter-rotating vortex pair with the circulations ±Γ move in incompressible fluid. The assumption is made that the core is very thin, that is the core radius σ is much smaller than the vortex radius d such that ϵ = σ/d ≤ 1. With this condition, the method of matched asymptotic expansion is employed. The solutions of the Navier-Stokes equations and the Biot-Savart law, regarding the inner and outer solutions respectively, are constructed in the form of a small parameter. An asymptotic expansion of the Biot-Savart law near the vortex core provides with the matching condition for an asymptotic expansion for limiting the Navier-Stokes equations for large radius r. The general formula of an anti-parallel vortex pair is established. At leading order O(ϵ), we apply the special case in inviscid fluid, the Rankine vortex, a circular vortex of uniform vorticity. Furthermore at leading order O(ϵ^{5}) we show the traveling speed of a vortex pair.

Original language | English |
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Title of host publication | International Conference and Workshop on Mathematical Analysis and its Applications, ICWOMAA 2017 |

Editors | Adem Kilicman, Marjono, Ratno Bagus Edy Wibowo, Moch. Aruman Imron |

Publisher | American Institute of Physics Inc. |

ISBN (Electronic) | 9780735416055 |

DOIs | |

Publication status | Published - Dec 5 2017 |

Event | International Conference and Workshop on Mathematical Analysis and its Applications, ICWOMAA 2017 - Malang, Indonesia Duration: Aug 2 2017 → Aug 3 2017 |

### Publication series

Name | AIP Conference Proceedings |
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Volume | 1913 |

ISSN (Print) | 0094-243X |

ISSN (Electronic) | 1551-7616 |

### Other

Other | International Conference and Workshop on Mathematical Analysis and its Applications, ICWOMAA 2017 |
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Country | Indonesia |

City | Malang |

Period | 8/2/17 → 8/3/17 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Physics and Astronomy(all)

### Cite this

*International Conference and Workshop on Mathematical Analysis and its Applications, ICWOMAA 2017*[020022] (AIP Conference Proceedings; Vol. 1913). American Institute of Physics Inc.. https://doi.org/10.1063/1.5016656

**A counter-rotating vortex pair in inviscid fluid.** / Habibah, Ummu; Fukumoto, Yasuhide.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*International Conference and Workshop on Mathematical Analysis and its Applications, ICWOMAA 2017.*, 020022, AIP Conference Proceedings, vol. 1913, American Institute of Physics Inc., International Conference and Workshop on Mathematical Analysis and its Applications, ICWOMAA 2017, Malang, Indonesia, 8/2/17. https://doi.org/10.1063/1.5016656

}

TY - GEN

T1 - A counter-rotating vortex pair in inviscid fluid

AU - Habibah, Ummu

AU - Fukumoto, Yasuhide

PY - 2017/12/5

Y1 - 2017/12/5

N2 - We study the motion of a counter-rotating vortex pair with the circulations ±Γ move in incompressible fluid. The assumption is made that the core is very thin, that is the core radius σ is much smaller than the vortex radius d such that ϵ = σ/d ≤ 1. With this condition, the method of matched asymptotic expansion is employed. The solutions of the Navier-Stokes equations and the Biot-Savart law, regarding the inner and outer solutions respectively, are constructed in the form of a small parameter. An asymptotic expansion of the Biot-Savart law near the vortex core provides with the matching condition for an asymptotic expansion for limiting the Navier-Stokes equations for large radius r. The general formula of an anti-parallel vortex pair is established. At leading order O(ϵ), we apply the special case in inviscid fluid, the Rankine vortex, a circular vortex of uniform vorticity. Furthermore at leading order O(ϵ5) we show the traveling speed of a vortex pair.

AB - We study the motion of a counter-rotating vortex pair with the circulations ±Γ move in incompressible fluid. The assumption is made that the core is very thin, that is the core radius σ is much smaller than the vortex radius d such that ϵ = σ/d ≤ 1. With this condition, the method of matched asymptotic expansion is employed. The solutions of the Navier-Stokes equations and the Biot-Savart law, regarding the inner and outer solutions respectively, are constructed in the form of a small parameter. An asymptotic expansion of the Biot-Savart law near the vortex core provides with the matching condition for an asymptotic expansion for limiting the Navier-Stokes equations for large radius r. The general formula of an anti-parallel vortex pair is established. At leading order O(ϵ), we apply the special case in inviscid fluid, the Rankine vortex, a circular vortex of uniform vorticity. Furthermore at leading order O(ϵ5) we show the traveling speed of a vortex pair.

UR - http://www.scopus.com/inward/record.url?scp=85037826797&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85037826797&partnerID=8YFLogxK

U2 - 10.1063/1.5016656

DO - 10.1063/1.5016656

M3 - Conference contribution

AN - SCOPUS:85037826797

T3 - AIP Conference Proceedings

BT - International Conference and Workshop on Mathematical Analysis and its Applications, ICWOMAA 2017

A2 - Kilicman, Adem

A2 - Marjono, null

A2 - Wibowo, Ratno Bagus Edy

A2 - Imron, Moch. Aruman

PB - American Institute of Physics Inc.

ER -