Hamiltonian Structure and Dynamics of a Neutrally Buoyant Rigid Sphere Interacting with Thin Vortex Rings |
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Authors: | Banavara N Shashikanth Artan Sheshmani Scott David Kelly Mingjun Wei |
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Institution: | 1. Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, New Mexico, 88003, USA 2. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois, 61801, USA 3. Department of Mechanical Engineering and Engineering Science, University of North Carolina at Charlotte, Charlotte, NC, 28205, USA
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Abstract: | In a previous paper, we presented a (noncanonical) Hamiltonian model for the dynamic interaction of a neutrally buoyant rigid
body of arbitrary smooth shape with N closed vortex filaments of arbitrary smooth shape, modeled as curves, in an infinite ideal fluid in
\mathbbR3\mathbb{R}^3. The setting of that paper was quite general, and the model abstract enough to make explicit conclusions regarding the dynamic
behavior of such systems difficult to draw. In the present paper, we examine a restricted class of such systems for which
the governing equations can be realized concretely and the dynamics examined computationally. We focus, in particular, on
the case in which the body is a smooth sphere. The equations of motion and Hamiltonian structure of this dynamic system, which
follow from the general model, are presented. Following this, we impose the constraint of axisymmetry on the entire system
and look at the case in which the rings are all circles perpendicular to a common axis of symmetry passing through the center
of the sphere. This axisymmetric model, in our idealized framework, is governed by ordinary differential equations and is,
relatively speaking, easily integrated numerically. Finally, we present some plots of dynamic orbits of the axisymmetric system. |
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