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The Motion of an Unbalanced Circular Disk in the Field of a Point Source

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Abstract

Describing the phenomena of the surrounding world is an interesting task that has long attracted the attention of scientists. However, even in seemingly simple phenomena, complex dynamics can be revealed. In particular, leaves on the surface of various bodies of water exhibit complex behavior. This paper addresses an idealized description of the mentioned phenomenon. Namely, the problem of the plane-parallel motion of an unbalanced circular disk moving in a stream of simple structure created by a point source (sink) is considered. Note that using point sources, it is possible to approximately simulate the work of skimmers used for cleaning swimming pools. Equations of coupled motion of the unbalanced circular disk and the point source are derived. It is shown that in the case of a fixed-position source of constant intensity the equations of motion of the disk are Hamiltonian. In addition, in the case of a balanced circular disk the equations of motion are integrable. A bifurcation analysis of the integrable case is carried out. Using a scattering map, it is shown that the equations of motion of the unbalanced disk are nonintegrable. The nonintegrability found here can explain the complex motion of leaves in surface streams of bodies of water.

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Notes

  1. Usually, when it comes to considering the plane-parallel motion of a rigid body in a fluid, it is called the motion of a smooth (in particular, circular) foil or a foil with a sharp edge. However, in this paper we use the term “circular disk” instead of “circular foil”.

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ACKNOWLEDGMENTS

The authors extend their gratitude to Prof. Oliver O’Reilly for useful comments. The authors are grateful to I. A. Bizyaev, I. S. Mamaev and A. A. Kilin for useful discussions.

Funding

The work of Elizaveta M. Artemova (Sections 2 and 4) was carried out within the framework of the state assignment of the Ministry of Education and Science of Russia (FEWS-2020-0009), and was supported in part by the Moebius Contest Foundation for Young Scientists. The work of Evgeny V. Vetchanin (Sections 1 and 3) is supported by the RFBR under grant 18-29-10050-mk.

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Authors and Affiliations

  1. Ural Mathematical Center, Udmurt State University , ul. Universitetskaya 1, 426034, Izhevsk, Russia

    Elizaveta M. Artemova

  2. Kalashnikov Izhevsk State Technical University, ul. Studencheskaya 7, 426069, Izhevsk, Russia

    Evgeny V. Vetchanin

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  1. Elizaveta M. Artemova
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  2. Evgeny V. Vetchanin
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Contributions

Elizaveta M. Artemova has derived equations of motions and investigated their nonintegrability. Evgeny V. Vetchanin has made literature review and investigated intagrable case of the equations of motion. All authors participated in discussing the results.

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Correspondence to Elizaveta M. Artemova or Evgeny V. Vetchanin.

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The authors declare that they have no conflicts of interest.

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MSC2010

76Bxx,70Exx,37Jxx

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Artemova, E.M., Vetchanin, E.V. The Motion of an Unbalanced Circular Disk in the Field of a Point Source. Regul. Chaot. Dyn. 27, 24–42 (2022). https://doi.org/10.1134/S1560354722010051

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