This paper studies the dynamical fluid plus rigid-body system consisting of a two-dimensional rigid cylinder of general cross-sectional shape interacting with N point vortices. We derive the equations of motion for this system and show that, in particular, if the vortex strengths sum to zero and the rigid-body has a circular shape, the equations are Hamiltonian with respect to a Poisson bracket structure that is the sum of the rigid body Lie-Poisson bracket on se(2)*, the dual of the Lie algebra of the Euclidean group on the plane, and the canonical Poisson bracket for the dynamics of N point vortices in an unbounded plane. We then use this Hamiltonian structure to study the linear and nonlinear stability of the moving Föppl equilibrium solutions using the energy-Casimir method.
The vortex dynamics of Euler's equations for a constant density fluid flow in R 4 is studied. Most of the paper focuses on singular Dirac delta distributions of the vorticity two-form ω in R 4 . These distributions are supported on two-dimensional surfaces termed membranes and are the analogs of vortex filaments in R 3 and point vortices in R 2 . The self-induced velocity field of a membrane is shown to be unbounded and is regularized using a local induction approximation. The regularized self-induced velocity field is then shown to be proportional to the mean curvature vector field of the membrane but rotated by 90 • in the plane of normals. Next, the Hamiltonian membrane model is presented. The symplectic structure for this model is derived from a general formula for vorticity distributions due to Marsden and Weinstein ["Coadjoint orbits, vortices and Clebsch variables for incompressible fluids," Physica D 7, 305-323 (1983)]. Finally, the dynamics of the four-form ω ∧ ω is examined. It is shown that Ertel's vorticity theorem in R 3 , for the constant density case, can be viewed as a special case of the dynamics of this four-form. C
We present a (noncanonical) Hamiltonian model for the interaction of a neutrally buoyant, arbitrarily shaped smooth rigid body with N thin closed vortex filaments of arbitrary shape in an infinite ideal fluid in Euclidean three-space. The rings are modeled without cores and, as geometrical objects, viewed as N smooth closed curves in space. The velocity field associated with each ring in the absence of the body is given by the Biot-Savart law with the infinite self-induced velocity assumed to be regularized in some appropriate way. In the presence of the moving rigid body, the velocity field of each ring is modified by the addition of potential fields associated with the image vorticity and with the irrotational flow induced by the motion of the body. The equations of motion for this dynamically coupled body-rings model are obtained using conservation of linear and angular momenta. These equations are shown to possess a Hamiltonian structure when written on an appropriately defined Poisson product manifold equipped with a Poisson bracket which is the sum of the Lie-Poisson bracket from rigid body mechanics and the canonical bracket on the phase space of the vortex filaments. The Hamiltonian function is the total kinetic energy of the system with the self-induced kinetic energy regularized. The Hamiltonian structure is independent of the shape of the body, (and hence) the explicit form of the image field, and the method of regularization, provided the self-induced velocity and kinetic energy are regularized in way that satisfies certain reasonable consistency conditions.
We present two interesting features of vortex rings in incompressible, Newtonian uids that involve their Hamiltonian structure.The ÿrst feature is for the Hamiltonian model of dynamically interacting thin-cored, coaxial, circular vortex rings described, for example, in the works of Dyson (Philos. Trans. Roy. Soc. London Ser. A 184 (1893) 1041) and Hicks (Proc. Roy. Soc. London Ser. A 102 (1922) 111). For this model, the symplectic reduced space associated with the translational symmetry is constructed. Using this construction, it is shown that for periodic motions on this reduced space, the reconstructed dynamics on the momentum level set can be split into a dynamic phase and a geometric phase. This splitting is done relative to a cotangent bundle connection deÿned for abelian isotropy symmetry groups. In this setting, the translational motion of leapfrogging vortex pairs is interpreted as the total phase, which has a dynamic and a geometric component.Second, it is shown that if the rings are modeled as coaxial circular ÿlaments, their dynamics and Hamiltonian structure is derivable from a more general Hamiltonian model for N interacting ÿlament rings of arbitrary shape in R 3 , where the mutual interaction is governed by the Biot-Savart law for ÿlaments and the self-interaction is determined by the local induction approximation. The derivation is done using the ÿxed point set for the action of the group of rotations about the axis of symmetry using methods of discrete reduction theory.
The dynamic interaction of N symmetric pairs of point vortices with a neutrally buoyant two-dimensional rigid circular cylinder in the inviscid Hamiltonian model of Shashikanth et al. ͓Phys. Fluids 14, 1214 ͑2002͔͒ and Shashikanth ͓Reg. Chaotic Dyn. 10, 1 ͑2005͔͒ is examined. The model may be thought of as a section of an inviscid axisymmetric model of a neutrally buoyant sphere interacting with N coaxial circular vortex rings and has possible applications to problems such as fish swimming. The Hamiltonian structure of this half-space model is first presented. The cases N = 1 and N = 2 are then examined in detail. Equilibria and bifurcations are studied, and for both these cases an important bifurcation parameter involving the total linear "momentum" of the system, the strength of the vortex pairs, and the radius of the cylinder emerges. For N = 1, it is shown that there exist the moving Föppl equilibrium and the moving normal line equilibrium ͑in which the vortices in the pair are located on the top and bottom of the moving cylinder͒. For N = 2, when ⌫ 1 = ⌫ 2 and when ⌫ 1 =−⌫ 2 , there exists another set of equilibrium configurations. Linear stability analysis of all these equilibria, within the symmetric class of solutions, is carried out and phase portraits presented. In addition, for N = 1, the velocity and acceleration surfaces for the cylinder are presented.
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