## Chaos and Integrability in Ideal Body-Fluid Interactions

Publication: Research › Ph.D. thesis – Annual report year: 2011

We consider interaction of a rigid body with a surrounding ideal
uid containing
a number of point vortices. The
uid is assumed to be planar and unbounded and
the body is assumed to be free to move in response to the
uid forces. Except that
the body should be simply connected and rigid, no assumptions are made on the body
shape or on its internal mass distribution. There may also be an arbitrary and constant
circulation around the body. The governing equations reduce to an autonomous set of
coupled ODEs for the vortex positions and the body position and orientation. The
form of these equations are derived by combining the classical equations for free body
motion in ideal
uid of G. Kirchho and Lord Kelvin with C. C. Lin's bounded domain
generalisation of H. Helmholtz's celebrated point vortex equations. The Hamiltonian
nature of the coupled body-vortex ODEs is demonstrated and the existence of additional
conserved quantities is discussed.
A survey of the integrable motions of the system is given. Integrability is demonstrated
explicitly by exploiting conservation laws to devise reduced phase space coordinates
in which the orbits of the system are the contours of an energy landscape.
The existence of relative equilibria, their stability, and the qualitatively dierent kinds
of motion is studied analytically and numerically. We then perform small parametric
perturbations destroying the symmetry or conservation law that makes the system integrable.
The emergence of chaos in the system is diagnosed by generating Poincare
sections from numerically obtained solutions. By identifying the chaotic solutions and
studying the body and vortex orbits, we obtain a better mechanistic understanding of
the causes of chaotic behavior. As is well-known from dynamical system theory, the
chaos can often be traced back to unstable relative equilibria of the perturbed integrable
system.
By this methodology we demonstrate that, even when there are no vortices in the
uid, a freely moving elongated body, whose motion is dominated by rotation, may
have an atmosphere of
uid particles following it through the
uid. This atmosphere
contains both regular and chaotic regions, and may be understood from KAM theory.
We also discover two separate chaotic regimes in the interaction of a body and one
point vortex when the body is either noncircular or has asymmetric internal mass distribution.
For one of these chaotic regimes the eect of chaos seems to be largest on
the vortex, while for the other, the chaos primarily expresses itself in the body motion.
Finally we brie
y demonstrate the occurrence of chaos in the interaction of a body with
two point vortices.

Original language | English |
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Number of pages | 121 |
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State | Published - 2011 |

### Keywords

- Rigid Body, Invariants, Body-Vortex Interactions, Nonlinear, Chaos, Hamiltonian, 2D Ideal Fluid, Point Vortex, Integrability, Fluid-Structure Dynamics

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