The modern approach to the study of mixing in laminar fluid flows applies
dynamical systems theory and concepts to the Lagrangian description of
fluid flow. The *Lagrangian* description uses coordinates
that move with a particle. When we speak of a *fluid particle*
we mean an element of fluid of negligible dimensions. The equations
of fluid motion are obtained by applying the principles of mechanics
to a fluid particle. In the case of an incompressible viscous
(Newtonian) fluid of uniform density the following equations are
obtained:

(1) |

(2) |

In order to investigate the mixing in a fully three-dimensonal laminar flow, the first hurdle is to obtain . Because this is a difficult task, the great majority of studies have restricted to time-dependent forcing of 2D velocity fields (c.f. [PhyF,O89,PhyD]) and 3D flows based on qualitative or kinematic models which mimic a velocity field but do not satisfy Equations (1) [FKP88,BM90,SNS91,HK91,M93,JY94,MW94]. Only by studying the kinematics of velocity fields satisfying the Navier-Stokes equations can an understanding be achieved of how inertia and viscosity influence the mixing properties throughout the fluid - a question of both fundamental and practical interest.

The fundamental issues were what motivated us to make our investigations of particle paths in nonaxisymmetric Taylor-Couette flows (wavy vortices, twisted vortices, ribbons, spirals, eccentric Taylor-vortices) [AK95,AMK95,AK97]. Practical issues were the motivating factor for Rudman, who investigated mixing and dispersion of fluid particles and particles with inertia in numerically computed wavy vortex flows in the Taylor-Couette system [R98]. (See Rudman's paper and references therein for many recent examples of the use of the Taylor-Couette system as a mixing vessel.)

The studies by Ashwin and King and Rudman
revealed both chaotic and near-integrable regions in the flow.
Although the presence of near-integrable regions is not a surprise
from the dynamical systems viewpoint, it is beyond the scope of that
theory to explain where in the fluid they appear, and the physical
conditions for their appearance and/or disappearance as the flow state
and Reynolds number changes. To determine this one might expect to
have to undertake the daunting task of a full particle path
study for each flow solution. An alternative to this approach was
introduced by Yannacopoulos *et al.* [YMRK98] who argued that
because integrable behaviour is associated with the presence of a
symmetry, chaotic and near-integrable regions of particle paths could
be distinguished through a suitable point-wise symmetry test using
quantities which are local functions of the velocity and its
gradients. The theoretical basis for these *Eulerian
diagnostics* comes from work by Arnold [A65] and Mezic and
Wiggins [MW94] on spatial symmetries of volume-preserving fluid
flows.
In [YMRK98] we
argued that a suitable diagnostic for Taylor-Couette flows in the
concentric geometry was the product of the magnitude of measures of
the rotational and translational symmetry (geometric symmetries) and a
measure of
the magnitude of the dissipation field (dynamical
symmetry). To understand the concept of *dynamical symmetry* consider the vorticity equation for steady flow:

(3) |

We applied the diagnostics to the velocity fields derived by Davey, DiPrima and Stuart [DDS68] and DiPrima and Stuart [DS75] and found that regions of the fluid where remnants of one or more of the symmetries persisted correlated well with near-integrable particle paths, and regions far from any of these symmetries correlated well with strongly chaotic particle paths. This is a highly significant result since it is well-known how difficult it is to relate Lagrangian behaviour to the Eulerian description of a fluid.

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