The Dynamics of a Rigid Body in Potential Flow with Circulation

Abstract

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.

28 pages, 2 figures; v2: typos corrected

References in corpus (3)

• On the motion of a heavy rigid body in an ideal fluid with circulation
• The Geometry and Dynamics of Interacting Rigid Bodies and Point Vortices
• Geodesics on extensions of Lie groups and stability; the superconductivity equation

Cited by in corpus (7)

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• Nonholonomic LL systems on central extensions and the hydrodynamic Chaplygin sleigh with circulation
• Fluid-structure interaction in the Lagrange-Poincare formalism: the Navier-Stokes and inviscid regimes
• Persistence of stationary motion under explicit symmetry breaking perturbation