Mixing dynamics in closed flows

Un article de Surface du verre et interfaces.

(Experiments described here were realized at CEA Saclay)

Many industrial applications involve the mixing of viscous fluids. Fields as diverse as chemical engineering, the pharmaceutical and cosmetics industries, and food processing depend on the stirring of initially heterogeneous substances to obtain a product with a sufficient degree of homogeneity. An essential issue for engineers is to predict the rate at which some inhomogeneity (e.g. a dye spot) is mixed by a given stirring protocol.

In this experiment, poorly mixed fluid from the vicinity of the wall is repeatedly injected in the bulk along the cusp of the heart-shaped mixing pattern. This results in slow mixing dynamics.

Here, the mixing pattern is separated from the wall by an unmixed region (note the circular form of the pattern, which has no cusps). Fast exponential mixing dynamics are retrieved in this case.

In recent work, we have demonstrated the importance of the phase portrait in the vicinity of no-slip walls for homogenization dynamics in closed flows.

First, 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. This class of mixers promotes an algebraic rate for mixing, which is slower than the exponentiel decay of fluctuations expected for mixers where stretching has only small fluctuations.

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.

Predicting the topology of the phase portrait for a given stirring protocol is therefore of high importance, as mixing dynamics are drastically different in the two cases...

Back to Chaotic mixing of viscous fluids

Collaborations

Olivier Dauchot, Service de Physique de l'Etat Condensé, CEA Saclay
Jean-Luc Thiffeault, University of Madison, WI USA.

Recent publications

E. Gouillart, O. Dauchot, B. Dubrulle, S. Roux, and J.-L. Thiffeault, Slow decay of concentration variance due to no-slip walls in chaotic mixing, submitted to Physical Review E.
E. Gouillart, N. Kuncio, O. Dauchot, B. Dubrulle, S. Roux, and J.-L. Thiffeault, Walls Inhibit Chaotic Mixing, Physical Review Letters 99, 114501, 2007. Featured as a Physics World and Sciences et Avenir news item.