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Regul. Chaotic Dyn., 2013, Volume 18, Issue 1-2, Pages 33–62 (Mi rcd94)  

This article is cited in 14 scientific papers (total in 14 papers)

The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem

Alexey V. Borisovabc, Alexander A. Kilinbac, Ivan S. Mamaevcab

a Institute of Computer Science, Udmurt State University, Universitetskaya 1, Izhevsk, 426034 Russia
b A.A. Blagonravov Mechanical Engineering Research Institute of RAS, Bardina str. 4, Moscow, 117334 Russia
c Institute of Mathematics and Mechanics of the Ural Branch of RAS, S. Kovalevskaja str. 16, Ekaterinburg, 620990 Russia

Abstract: We consider the problem of motion of axisymmetric vortex rings in an ideal incompressible fluid. Using the topological approach, we present a method for complete qualitative analysis of the dynamics of a system of two vortex rings. In particular, we completely solve the problem of describing the conditions for the onset of leapfrogging motion of vortex rings. In addition, for the system of two vortex rings we find new families of motions where the relative distances remain finite (we call them pseudo-leapfrogging). We also find solutions for the problem of three vortex rings, which describe both the regular and chaotic leapfrogging motion of vortex rings.

Keywords: ideal fluid, vortex ring, leapfrogging motion of vortex rings, bifurcation complex, periodic solution, integrability, chaotic dynamics

Funding Agency Grant Number
Russian Foundation for Basic Research 11-01-91056-NTsNI_a
Ministry of Education and Science of the Russian Federation 1.2953.2011
This research was supported by the RFBR grant 11-01-91056-NTsNI_a, the Target Programme “Development of Scientific Potential of Higher Schools” (State contract 1.2953.2011, 2012–2014); the Federal Target Programme “Scientific and Scientific-Pedagogical Personnel of Innovative Russia” (State contract 14.B37.21.1935, 2009–2013); grant of leading scientific schools NSh-2519.2012.1. A. A. Kilin’s research was supported by the grant of the President of the Russian Federation for the Support of Young Russian Scientists–Candidates of Science (MK-8428.2010.1).


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Document Type: Article
MSC: 76B47
Received: 19.09.2012
Language: English

Citation: Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem”, Regul. Chaotic Dyn., 18:1-2 (2013), 33–62

Citation in format AMSBIB
\by Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev
\paper The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem
\jour Regul. Chaotic Dyn.
\yr 2013
\vol 18
\issue 1-2
\pages 33--62

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    This publication is cited in the following articles:
    1. Phanindra Tallapragada, Scott David Kelly, “Dynamics and Self-Propulsion of a Spherical Body Shedding Coaxial Vortex Rings in an Ideal Fluid”, Regul. Chaotic Dyn., 18:1-2 (2013), 21–32  mathnet  crossref  mathscinet  zmath
    2. A. V. Borisov, A. A. Kilin, I. S. Mamaev, V. A. Tenenev, “The dynamics of vortex rings: leapfrogging in an ideal and viscous fluid”, Fluid Dyn. Res., 46:3 (2014), 031415  crossref  mathscinet  zmath  isi  scopus
    3. Adecarlos C. Carvalho, Hildeberto E. Cabral, “Lyapunov Orbits in the $n$-Vortex Problem on the Sphere”, Regul. Chaotic Dyn., 20:3 (2015), 234–246  mathnet  crossref  mathscinet  zmath  adsnasa
    4. M. Cheng, J. Lou, T. T. Lim, “Leapfrogging of multiple coaxial viscous vortex rings”, Phys. Fluids, 27:3 (2015), 031702  crossref  isi  scopus
    5. N. Hietala, R. Hanninen, H. Salman, C. F. Barenghi, “Leapfrogging Kelvin waves”, Phys. Rev. Fluids, 1:8 (2016), 084501  crossref  isi  scopus
    6. S. S. Gubser, B. Horn, S. Parikh, “Perturbations of vortex ring pairs”, Phys. Rev. D, 93:4 (2016), 046001  crossref  mathscinet  isi  scopus
    7. S.-P. Kou, “Kelvin wave and knot dynamics on entangled vortices”, Int. J. Mod. Phys. B, 31:31 (2017), 1750241  crossref  mathscinet  isi  scopus
    8. V. Sadri, P. S. Krueger, “Formation and behavior of counter-rotating vortex rings”, Theor. Comput. Fluid Dyn., 31:4 (2017), 369–390  crossref  isi  scopus
    9. Evgeny V. Vetchanin, Ivan S. Mamaev, “Dynamics of Two Point Vortices in an External Compressible Shear Flow”, Regul. Chaotic Dyn., 22:8 (2017), 893–908  mathnet  crossref
    10. S.-P. Kou, “Knot physics on entangled vortex-membranes: classification, dynamics and effective theory”, Int. J. Mod. Phys. B, 32:8 (2018), 1850090  crossref  mathscinet  isi  scopus
    11. S. Qin, H. Liu, Ya. Xiang, “Lagrangian flow visualization of multiple co-axial co-rotating vortex rings”, J. Vis., 21:1 (2018), 63–71  crossref  isi  scopus
    12. S. Qin, H. Liu, Ya. Xiang, “On the formation modes in vortex interaction for multiple co-axial co-rotating vortex rings”, Phys. Fluids, 30:1 (2018), 011901  crossref  isi  scopus
    13. Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev, “Three Vortices in Spaces of Constant Curvature: Reduction, Poisson Geometry, and Stability”, Regul. Chaotic Dyn., 23:5 (2018), 613–636  mathnet  crossref
    14. Alexander A. Kilin, Lizaveta M. Artemova, “Integrability and Chaos in Vortex Lattice Dynamics”, Regul. Chaotic Dyn., 24:1 (2019), 101–113  mathnet  crossref
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