Chaotic Mixing

Olivier Dauchot

Many industrial applications would benefit greatly of further insight into the mixing mechanisms of viscous fluids. The study of fluid mixing enhanced by closed flows goes back to the early 80’ s and the introduction of chaotic advection. The interest for mixing in open flows is more recent. Transient mixing created by an open flow whose time dependency is restricted to a bounded region – the mixing region – can be studied in the same dynamical systems framework as for bounded flows, since for typical open flows, Lagrangian trajectories have a transient chaotic behavior inside the mixing region. Nevertheless, quantitative analysis of homogenization realized by such flows is still lacking. We have characterized transient mixing in open flows and in parallel, we carried complementary investigations in closed flows as a guide on the less-beaten track of open flows.

Our study, experimental, numerical and theoretical, is based on two types of chaotic mixing experiments, in a closed vat and in an open channel. For closed flows, we have first proposed a topological description of mixing by the entanglement of periodic orbits that we called ”ghost rods”. The experimental study of the concentration field of a dye has then revealed the role of the walls of the domain where mixing takes place, in closed and open flows as well. For closed flows, the chaotic or regular nature of trajectories initialized close to the wall determines the evolution of the concentration field, even far from the walls. In particular, we have observed slow (algebraic) dynamics of homogenization when the chaotic region extends to no-slip walls. In open flows, we have reported on the evolution of the concentration field in the mixing and downstream regions, resulting from the injection of a dye blob. We have described the poorly mixed elements that escape quickly, as well as the asymptotic onset of a permanent (self-similar) pattern determined by the periodic orbits inside the mixing region. Finally, various models derived from the baker’s map have allowed us to understand most observed mechanisms.

The results for closed flows can be summarized as follows. First, we have demonstrated the importance of the phase portrait in the vicinity of a noslip wall for homogenization dynamics in a closed flow. We have described a universal mixing scenario for mixers where the chaotic region extends to the solid wall. No-slip hydrodynamics in the wall region impose that poorly mixed fluid is slowly reinjected in the bulk along the unstable manifold of a parabolic point. Mixing dynamics are then controlled by the slow stretching at the wall, which contaminates the whole mixing pattern up to its core. A second universality class correspond to a phase portrait separated into a central chaotic region, and a regular region encircling the wall. This is a trick to retrieve “slip” boundary conditions for the chaotic region, and experiments realized for this class indeed yield an exponential decay of fluctuations, consistent with the observation of a strange eigenmode.

Predicting the topology of the phase portrait for a given stirring protocol is therefore of high importance.

**Open flows**

Periodic orbits are also paramount in open flows: even in the case of a global advection, an infinity of unstable periodic orbits survive in the vicinity of the stirring rods, forming the so-called "chaotic saddle". Fluid enters the mixing region along the stable manifold of the chaotic saddle, and leaves it along its unstable manifold. In particular, particles initialized regions not covered by the stable manifold escape directly downstream without being caught by the rods. Using the symmetries of the flow, we have described a simple geometric method that allows to predict whether particles in the upstream region will escape directly, or be caught in the mixing region. This partition into two sets of initial conditions also allows to estimate independently the mean residence time.

We also have concentrated on the long-time concentration field. We have first focused on the butterfly protocol, where stirring rods help the global advection along the channel sides, but travel in the opposite sense in the center. In this case, the chaotic saddle, is well shielded from the side walls by a layer of free trajectories. For all stirring frequencies where no elliptical islands where visible, we have observed an open-flow strange eigenmode, that is a permanent concentration field which repeats self-similarly every period, although its amplitude decays exponentially. We have observed that the relative homogeneity of the pattern increases for greater stirring frequencies, where thinner slices of upstream fluid are injected inside the mixing region at each period.

We have then took on the analysis of breaststroke protocols, where the rods accelerate the main flow in the center, and conversely go against it along the channel sides. Due to the latter competition, a stagnation parabolic point is located on each side wall, at the frontier between the upstream region and the chaotic saddle. The chaotic region – i.e. the chaotic saddle – goes to the no-slip walls, where poorly stretched fluid is stored for long times, and then escapes along the unstable manifolds of the parabolic points. We have also observed some departure from an exponential evolution of the moments.

#993 - Last update : 11/04 2009

• Statistical physics and complex systems › Statistical physics and complex systems

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