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Mixing in a laminar 3D Navier-Stokes flow

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:
\frac{\partial \mathbf{u}}{\partial t} + \mathbf{\omega}\tim...
 ...athbf{u}^2 =
-\mathbf{\nabla}(p/\rho) + \nu
 \nabla^2\mathbf{u}\end{displaymath} (1)

\mathbf{\nabla\cdot u}= 0, \end{displaymath}

where $\mathbf{\omega}=\nabla\times\mathbf{u}$ is the vorticity, $\mathbf{\omega}\times\mathbf{u}$ is the Lamb vector, p is the pressure, $\rho$ is the density, and $\nabla^2\mathbf{u}$ is the dissipation field. Solving these Navier-Stokes equations we obtain the velocity field $\mathbf{u}(\mathbf{x},t)$. The dynamical systems setting for the study of the transport and mixing of a fluid particle is the study of the trajectories of
\dot{\mathbf{x}} = \mathbf{u}(\mathbf{x},t),
\end{displaymath} (2)
where each initial condition corresponds to a different fluid particle, a trajectory is the path the fluid particle takes through the fluid, and the phase space is physical space. Remark: It is useful and important to regard Equations (1) as constraints on the the dynamical systems problem (2).

In order to investigate the mixing in a fully three-dimensonal laminar flow, the first hurdle is to obtain $\mathbf{u}(\mathbf{x},t)$. 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:
\mathbf{u\cdot\nabla \omega}- \mathbf{\omega \cdot\nabla u}= \nu \nabla^2\mathbf{\omega}.
\end{displaymath} (3)
The left-hand side is the Lie bracket of the velocity and vorticity field. For an Euler fluid the right-hand side is identically zero and then the Lie bracket vanishes, meaning that $\mathbf{\omega}$ is an infinitesimal generator of a volume-preserving, spatial symmetry group for $\mathbf{u}$, and the integrability of both is implied if they are nowhere parallel. For a Navier-Stokes fluid we interpret this point-wise: we expect that regions in the flow where the right-hand side is close to zero will be associated with near-integrable streamlines.

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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