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 Regul. Chaotic Dyn., 2018, Volume 23, Issue 7-8, Pages 850–874 (Mi rcd371)

Self-propulsion of a Smooth Body in a Viscous Fluid Under Periodic Oscillations of a Rotor and Circulation

Alexey V. Borisova, Ivan S. Mamaevb, Evgeny V. Vetchaninc

a Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, 141700 Russia
b Izhevsk State Technical University, ul. Studencheskaya 7, Izhevsk, 426069 Russia
c Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia

Abstract: This paper addresses the problem of the self-propulsion of a smooth body in a fluid by periodic oscillations of the internal rotor and circulation. In the case of zero dissipation and constant circulation, it is shown using methods of KAM theory that the kinetic energy of the system is a bounded function of time. In the case of constant nonzero circulation, the trajectories of the center of mass of the system lie in a bounded region of the plane. The method of expansion by a small parameter is used to approximately construct a solution corresponding to directed motion of a circular foil in the presence of dissipation and variable circulation. Analysis of this approximate solution has shown that a speed-up is possible in the system in the presence of variable circulation and in the absence of resistance to translational motion. It is shown that, in the case of an elliptic foil, directed motion is also possible. To explore the dynamics of the system in the general case, bifurcation diagrams, a chart of dynamical regimes and a chart of the largest Lyapunov exponent are plotted. It is shown that the transition to chaos occurs through a cascade of period-doubling bifurcations.

Keywords: self-propulsion in a fluid, smooth body, viscous fluid, periodic oscillation of circulation, control of a rotor

 Funding Agency Grant Number Russian Foundation for Basic Research 18-29-10050 mk18-08-00995 A Russian Science Foundation 18-71-00111 Ministry of Education and Science of the Russian Federation 1.2405.2017/4.6 The work of A.V. Borisov was supported by the RFBR grant No. 18-29-10050 mk. The work of E.V. Vetchanin (Sections 3, 4, and 5) was supported by the Russian Science Foundation under grant 18-71-00111. The work of I.S. Mamaev was carried out within the framework of the state assignment to the Izhevsk State Technical University 1.2405.2017/4.6 and was supported by the RFBR grant No 18-08-00995 A.

DOI: https://doi.org/10.1134/S1560354718070043

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MSC: 37Mxx, 70Exx, 76Dxx
Accepted:03.12.2018
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Citation: Alexey V. Borisov, Ivan S. Mamaev, Evgeny V. Vetchanin, “Self-propulsion of a Smooth Body in a Viscous Fluid Under Periodic Oscillations of a Rotor and Circulation”, Regul. Chaotic Dyn., 23:7-8 (2018), 850–874

Citation in format AMSBIB
\Bibitem{BorMamVet18} \by Alexey V. Borisov, Ivan S. Mamaev, Evgeny V. Vetchanin \paper Self-propulsion of a Smooth Body in a Viscous Fluid Under Periodic Oscillations of a Rotor and Circulation \jour Regul. Chaotic Dyn. \yr 2018 \vol 23 \issue 7-8 \pages 850--874 \mathnet{http://mi.mathnet.ru/rcd371} \crossref{https://doi.org/10.1134/S1560354718070043} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000458183900004} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85061296452} 

• http://mi.mathnet.ru/eng/rcd371
• http://mi.mathnet.ru/eng/rcd/v23/i7/p850

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Ivan S. Mamaev, Evgeny V. Vetchanin, “The Self-propulsion of a Foil with a Sharp Edge in a Viscous Fluid Under the Action of a Periodically Oscillating Rotor”, Regul. Chaotic Dyn., 23:7-8 (2018), 875–886