Abstract
The dynamical behavior of a heavy circular cylinder and a point vortex in an unbounded volume of ideal liquid is considered. The liquid is assumed to be irrotational and at rest at infinity. The circulation about the cylinder is different from zero. The governing equations are Hamiltonian and admit an evident autonomous integral of motion — the horizontal component of the linear momentum. Using the integral we reduce the order and thereby obtain a system with two degrees of freedom. The stability of equilibrium solutions is investigated and some remarkable types of partial solutions of the system are presented.
Similar content being viewed by others
References
Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., The Bifurcation Analysis and the Conley Index in Mechanics, Regul. Chaotic Dyn., 2012, vol. 17, no. 5, pp. 457–478.
Borisov, A. V., Kozlov, V. V., and Mamaev, I. S., Asymptotic Stability and Associated Problems of Failing Rigid Body, Regul. Chaotic Dyn., 2007, vol. 12, no. 5, pp. 531–565.
Borisov, A. V. and Mamaev, I. S., An Integrability of the Problem on Motion of Cylinder and Vortex in the Ideal Fluid, Regul. Chaotic Dyn., 2003, vol. 8, no. 2, pp. 163–166.
Borisov, A.V. and Mamaev, I. S., On the Motion of a Heavy Rigid Body in an Ideal Fluid with Circulation, Chaos, 2006, vol. 16, no. 1, 013118, 7 pp.
Borisov, A. V., Mamaev, I. S., and Ramodanov, S. M., Dynamics of a Circular Cylinder Interacting with Point Vortices, Discrete Contin. Dyn. Syst. Ser. B, 2005, vol. 5, no. 1, pp. 35–50.
Chaplygin, S.A., On the Motion of Heavy Bodies in an Incompressible Fluid, in Complete Works: Vol. 1, Leningrad: Izd. Akad. Nauk SSSR, 1933, pp. 133–150 (This is a Chaplygin’s student paper which he wrote in 1890 and published only in 1933).
Jones, M. A. and Shelly, M. J., Falling Cards, J. Fluid Mech., 2005, vol. 540, pp. 393–425.
Kadtke, J.B. and Novikov, E.A., Chaotic Capture of Vortices by a Moving Body: 1.The Single Point Vortex Case, Chaos, 1993, vol. 3, no. 4, pp. 543–553.
Kirchhoff, G. R., Vorlesungen über mathematische Physik: Bd. 1: Mechanik, Leipzig: Teubner, 1874.
Kozlov, V.V., On the Problem of Fall of a Rigid Body in a Resisting Medium, Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., 1990, no. 1, pp. 79–86 [Mosc. Univ. Mech. Bull., 1990, vol. 45, no. 1, pp. 30–36].
Kozlov, V.V., On a Heavy Cylindrical Body Falling in a Fluid, Izv. Ross. Akad. Nauk Mekh. Tverd. Tela, 1993, no. 4, pp. 113–117 (Russian).
Maxwell, J.K., On a Particular Case of Descent of a Heavy Body in a Resisting Medium, Cambridge and Dublin Math. Journ., 1854, vol. 9, pp. 145–148.
Michelin, S. and Llewellyn Smith, S.G., Falling Cards and Flapping Flags: Understanding Fluid-Solid Interactions Using an Unsteady Point Vortex Model, Theor. Comp. Fluid Dyn., 2010, vol. 24, pp. 195–200.
Ramodanov, S. M., On the Influence of Circulation on the Behavior of a Rigid Body Falling in a Fluid, Izv. Ross. Akad. Nauk Mekh. Tverd. Tela, 1996, no. 5, pp. 19–24 (Russian).
Ramodanov, S. M., Motion of a Circular Cylinder and a Vortex in an Ideal Fluid, Regul. Chaotic Dyn., 2001, vol. 6, no. 1, pp. 33–38.
Shashikanth, B.N., Marsden, J.E., Burdick, J.W., and Kelly, S.D., The Hamiltonian Structure of a 2D Rigid Circular Cylinder Interacting Dynamically with N Point Vortices, Phys. Fluids, 2002, vol. 14, pp. 1214–1227.
Tanabe, Y. and Kaneko, K., Behavior of a Falling Paper, Phys. Rev. Lett., 1994, vol. 73, no. 10, pp. 1372–1375.
Zhukovskii, N. E., On the Falling in the Air of Light Oblong Bodies Rotating About Their Longitudinal Axis, Paper I, Collected Papers: Vol. 5, Moscow-Leningrad: Gostekhizdat, 1937, pp. 72–80.
Zhukovskii, N. E., On the Falling in the Air of Light Oblong Bodies Rotating About Their Longitudinal Axis, Paper II, Collected Papers: Vol. 5, Moscow-Leningrad: Gostekhizdat, 1937, pp. 100–115.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sokolov, S.V., Ramodanov, S.M. Falling motion of a circular cylinder interacting dynamically with a point vortex. Regul. Chaot. Dyn. 18, 184–193 (2013). https://doi.org/10.1134/S1560354713010139
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1560354713010139