Chaotic mixing of viscous fluids

Un article de Surface du verre et interfaces.

Emmanuelle Gouillart, Franck Pigeonneau

Very viscous fluids are not easily mixed together: trying to homogenize intimately honey and sugar syrup, for instance, requires much effort! One of the reasons for this challenge is that turbulence is inexistent for such viscous flows, and the very regular velocity field doesn't display important spatial fluctuations over a wide range of scales, that mix very efficiently in turbulent flows.

The best strategy for mixing efficiently viscous fluids is to use flows that create chaotic, that is complicated, Lagrangian trajectories. In such flows, neighboring fluid particles separate exponentially with time, hence they visit rapidly different regions of the fluid - a requirement for efficient mixing. This phenomenon is known since the 80's as chaotic advection. Flows promoting chaotic advection allows visualization of the beautifully elongated structures of chaos directly in dye spreading experiment (see picture below). Besides all industrial processes benefiting from a better understanding of mixing, this aesthetic facet might explain partly all the enthusiasm generated by chaotic advection since 20 years!

 

Dye pattern obtained while moving a rod on a figure-eight trajectory. The regular pattern of dye filaments is typical of chaotic advection. (Experiments realized at CEA Saclay with O. Dauchot)

 

Poincaré section showing the position of particles stroboscoped at each stirring period for a rod moving on a figure-eight. A large chaotic region (all blue points correspond to a single trajectory that spans the chaotic region) is visible, together with two small regular islands (elliptic red trajectories).

Our work on chaotic mixing stems mostly from the following question : " How fast can you mix ? ", that is, what are the temporal dynamics of homogenizing the concentration field of a diffusive species inside a viscous fluid stirred by chaotic advection ? Needless to say, this question is of central importance for industrial processes, where one would like to predict the time needed to achieve a given quality of mixing.

Our contribution to the study of mixing goes along the following lines:

• Topological mixing
• Mixing dynamics in closed flows
• Mixing in open flows
• Current issues on mixing

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.