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.
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.
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
\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}
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Linking options:
https://www.mathnet.ru/eng/rcd371
https://www.mathnet.ru/eng/rcd/v23/i7/p850
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E. V. Vetchanin, A. R. Valieva, “Analysis of the Force and Torque Arising During the Oscillatory Motion of a Joukowsky Foil in a Fluid”, Rus. J. Nonlin. Dyn., 20:1 (2024), 79–93
V. D. Anisimov, A. G. Egorov, A. N. Nuriev, O. N. Zaitseva, “Propulsive Motion of Cylindrical Vibration-Driven Robot in a Viscous Fluid”, jour, 166:3 (2024), 277
E. V. Vetchanin, I. S. Mamaev, “Periodicheskoe vozmuschenie dvizheniya neuravnoveshennogo krugovogo profilya v prisutstvii tochechnykh vikhrei v idealnoi zhidkosti”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 32:4 (2022), 630–643
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Sergey M. Ramodanov, Sergey V. Sokolov, “Dynamics of a Circular Cylinder and Two Point Vortices
in a Perfect Fluid”, Regul. Chaotic Dyn., 26:6 (2021), 675–691
I. S. Mamaev, I. A. Bizyaev, “Dynamics of an unbalanced circular foil and point vortices in an ideal fluid”, Phys. Fluids, 33:8 (2021), 087119
L. I. Mogilevich, S. V. Ivanov, Yu. A. Blinkov, “Modeling of Nonlinear Waves in Two Coaxial Physically Nonlinear Shells with a Viscous Incompressible Fluid Between Them, Taking into Account the Inertia of its Motion”, Rus. J. Nonlin. Dyn., 16:2 (2020), 275–290
E. M. Artemova, E. V. Vetchanin, “Control of the motion of a circular cylinder in an ideal fluid using a source”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 30:4 (2020), 604–617
E. V. Vetchanin, I. S. Mamaev, “Asymptotic behavior in the dynamics of a smooth body in an ideal fluid”, Acta Mech., 231:11 (2020), 4529–4535
A. V. Borisov, E. V. Vetchanin, I. S. Mamaev, “Motion of a smooth foil in a fluid under the action of external periodic forces. II”, Russ. J. Math. Phys., 27:1 (2020), 1–17
Anton V. Klekovkin, Yury L. Karavaev, Ivan S. Mamaev, Evgeny V. Vetchanin, Valentin A. Tenenev, 2020 International Conference Nonlinearity, Information and Robotics (NIR), 2020, 1
E. V. Vetchanin, “The Motion of a Balanced Circular Cylinder in an Ideal Fluid Under the Action of External Periodic Force and Torque”, Rus. J. Nonlin. Dyn., 15:1 (2019), 41–57
L. I. Mogilevich, S. V. Ivanov, “The Study of Wave Propagation in a Shell with Soft Nonlinearity and with a Viscous Liquid Inside”, Rus. J. Nonlin. Dyn., 15:3 (2019), 233–250
E. V. Vetchanin, E. A. Mikishanina, “Vibrational Stability of Periodic Solutions of the Liouville Equations”, Rus. J. Nonlin. Dyn., 15:3 (2019), 351–363
A. V. Borisov, E. V. Vetchanin, I. S. Mamaev, “Motion of a smooth foil in a fluid under the action of external periodic forces. I”, Russ. J. Math. Phys., 26:4 (2019), 412–427
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
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