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### Open Access Journals

PROC

A dynamical system governing the collective interaction of N point-vortexdipoles is derived.Each dipole has an inherent orientation $\psi $ and generates a velocity field that decayslike $O(\mu /2 \pi r^2)$ where $ \mu $ is the dipole strength and$r$ is the distance from the dipole.The system of N-complex ordinary differentialequations plus N-real ordinary differentialequations for the dipole positions and orientationsare derived based on theassumption that each dipole moves with and tries to align itselfwith the local fluid velocity field.

DCDS

The problem of finding and classifying all relative equilibrium configurations of $N$-point vortices in the plane is first described as
a classical variational principle and then formulated as a
problem in
linear algebra. Given a configuration of $N$ points in the plane, one must
understand the structure of
the
$N(N-1)/2 \times N$

*configuration*matrix $A$ obtained by requiring that all interparticle distances remain fixed in time. If the determinant of the square, symmetric $N \times N$*covariance matrix*$A^T A$ is zero, there is a non-trivial nullspace of $A$ and a basis set for this nullspace can be used to determine all vortex strengths $\vec{\Gamma} \in R^N$ for which the configuration remains rigid. Optimal basis sets are obtained by using the singular value decomposition of $A$ which allows one to categorize exact equilibria, approximate equilibria, and the distance between different equilibria in the appropriate vector space, as characterized by the Frobenius norm.
DCDS-B

The Hamiltonian system governing
$N$-interacting particles
constrained to lie on a
closed planar curve are derived.
The problem
is formulated in detail for the case of logarithmic
(point-vortex) interactions.
We show that when the curve is circular with radius
$ R $, the system is completely integrable for
all particle strengths $ \Gamma _ \beta $, with particle $ \Gamma _ \beta $
moving with frequency $ \omega _ \beta = (\Gamma - \Gamma _ \beta )/4 \pi R^2 $,
where $ \Gamma = \sum^{N}_{\alpha=1} \Gamma _ \alpha $
is the sum of the strengths of all the particles.
When all the particles have equal strength, they move
periodically around the circle keeping
their relative distances fixed.
When not all the strengths are equal, two or more
of the particles collide in finite time.
The diffusion of a neutral particle (i.e. the problem of 1D mixing) is examined.
On a circular curve,
a neutral particle moves
uniformly with frequency
$ \Gamma / 4 \pi R^2 $.
When the curve is not perfectly circular,
for example when given a sinusoidal perturbation,
or when the particles
move on concentric circles with different
radii,
the particle dynamics is considerably more complex, as shown numerically
from an examination of power spectra
and collision diagrams. Thus, the circular constraint appears to
be special in that it induces completely integrable dynamics.

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