Topological mixing
Un article de Surface du verre et interfaces.
Predicting the mixing properties of a given stirring protocol is in general a tricky problem; however, simple topological characteristics of the protocol may reveal precious information about the stirring flow.
The picture below shows a mixing device consisting of rods moving in a two-dimensional viscous fluid. If the position of the rods is plotted in a three-dimensional graph, with time the vertical axis, one obtains a “spaghetti plot” of the trajectories of the rods. The resulting graph thus describes a braid, in the mathematical sense of a bundle of strands that are not allowed to cross each other.
The braid above is called topologically complex, as its strands are intimately tangled and cannot be separated by deforming smoothly its ends. The existence of a topologically-complex braid formed by the rods has many important implications:
- first, it ensures that fluid particles are chaotic at least in some part of the fluid domain.
- furthermore, the "rate of entanglement" of the braid, called its topological entropy, gives a lower bound on the rate at which material lines are stretched by the rods. This stems from the fact that material lines (such as the frontier of a dye spot) cannot be cut by the rods, and must therefore stretch to follow rods which braid around each other.
Using topological concepts to characterize the mixing of fluids is a recent approach, as using stirring rods to generate topological chaos was proposed for the first time by Boyland, Aref and Stremeler in a 2000 paper (P. L. Boyland, H. Aref, and M. A. Stremler, Topological fluid mechanics of stirring, J. Fluid Mech., 403 (2000), pp. 277–304.)
Moreover, other structures than rods can be used to form complex braids. The mixing pattern on the left shows a large chaotic region where dye filaments are stretched, as well as two "holes" in the pattern. The holes are invariant elliptical islands that don't mix with the chaotic region. The stirring rod follows a figure-eight trajectory that loops around the two islands. As the braid formed by the islands and the rod is complex, we know that topological chaos is at play in the mixer just by looking at the filamentary pattern! We have given islands and other periodic structures the nickname "ghost rods", as they may contribute to topological mixing by braiding and stretching material lines.
Topology therefore provides powerful tools to predict some measures of mixing such as the stretching of material lines. Nevertheless, other mixing indices might be more relevant for comparing mixing devices. Current work about topological mixing therefore focuses on investigating what kind of general information about mixing could be gained from a topological analysis of Lagrangian trajectories, or patterns of dye filaments.
See [1] to learn more about topological mixing.
Back to chaotic mixing
Recent publications
- J.-L. Thiffeault, M. D. Finn, E. Gouillart, and T. Hall, Topology of Chaotic Mixing Patterns, submitted to Chaos.
- M. D. Finn, J.-L. Thiffeault, and E. Gouillart, Topological Chaos in Spatially Periodic Mixers, Physica D 221 (1), 92-100, September 2006.
- E. Gouillart, J.-L. Thiffeault, and M. D. Finn, Topological Mixing with Ghost Rods , Physical Review E 73, 036311, March 2006.
Collaborations
- Jean-Luc Thiffeault, University of Madison, WI USA.
- Matt Finn, University of Adelaide, Australia.