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The Dynamics of a Rigid Body in Potential Flow with Circulation

Vankerschaver, J. and Kanso, E. and Marsden, J. E. (2010) The Dynamics of a Rigid Body in Potential Flow with Circulation. Regular and Chaotic Dynamics, 15 (4-5). pp. 606-629. ISSN 1560-3547. doi:10.1134/S1560354710040143.

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We consider the motion of a two-dimensional body of arbitrary shape in a planar irrotational, incompressible fluid with a given amount of circulation around the body. We derive the equations of motion for this system by performing symplectic reduction with respect to the group of volume-preserving diffeomorphisms and obtain the relevant Poisson structures after a further Poisson reduction with respect to the group of translations and rotations. In this way, we recover the equations of motion given for this system by Chaplygin and Lamb, and we give a geometric interpretation for the Kutta–Zhukowski force as a curvature-related effect. In addition, we show that the motion of a rigid body with circulation can be understood as a geodesic flow on a central extension of the special Euclidian group SE(2), and we relate the cocycle in the description of this central extension to a certain curvature tensor.

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Additional Information:© 2010 Pleiades Publishing, Ltd. Received September 24, 2009; accepted November 13, 2009. Special Issue: Valery Vasilievich Kozlov–60 We would like to thank Scott Kelly, Jair Koiller, Tudor Ratiu and Banavara Shashikanth for useful suggestions and interesting discussions. J. Vankerschaver is supported through a postdoctoral fellowship from the Research Foundation – Flanders (FWO-Vlaanderen). Additional financial support from the Fonds Professor Wuytack is gratefully acknowledged. E. Kanso and J. E. Marsden would like to acknowledge the support of the National Science Foundation through the grants CMMI 07-57092 and CMMI 07-57106, respectively.
Funding AgencyGrant Number
Fonds Wetenschappelijk Onderzoek (FWO)UNSPECIFIED
Fonds Professor WuytackUNSPECIFIED
NSFCMMI 07-57092
NSFCMMI 07-57106
Subject Keywords:fluid-structure interactions, potential flow, circulation, symplectic reduction, diffeomorphism groups, oscillator group
Issue or Number:4-5
Classification Code:MSC 2000 numbers: 76B47, 53D20, 74F10
Record Number:CaltechAUTHORS:20101221-102207305
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:21477
Deposited By: Tony Diaz
Deposited On:12 Jan 2011 19:00
Last Modified:09 Nov 2021 15:57

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