Topological optimisation of rod-stirring devices

Abstract

There are many industrial situations where rods are used to stir a fluid, or where rods repeatedly stretch a material such as bread dough or taffy. The goal in these applications is to stretch either material lines (in a fluid) or the material itself (for dough or taffy) as rapidly as possible. The growth rate of material lines is conveniently given by the topological entropy of the rod motion. We discuss the problem of optimising such rod devices from a topological viewpoint. We express rod motions in terms of generators of the braid group, and assign a cost based on the minimum number of generators needed to write the braid. We show that for one cost function -- the topological entropy per generator -- the optimal growth rate is the logarithm of the golden ratio. For a more realistic cost function,involving the topological entropy per operation where rods are allowed to move together, the optimal growth rate is the logarithm of the silver ratio, $1+\sqrt{2}$. We show how to construct devices that realise this optimal growth, which we call silver mixers.

22 pages, 53 figures. PDFLaTeX with RevTex4 macros.

References in corpus (4)

• Topology, Braids, and Mixing in Fluids
• Topology of Chaotic Mixing Patterns
• Topological Entropy of Braids on the Torus
• Topological Chaos in Spatially Periodic Mixers

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• Lyapunov exponents for the random product of two shears
• Trajectory entanglement in dense granular materials
• Braiding a flock: winding statistics of interacting flying spins
• Using braids to quantify interface growth and coherence in a rotor-oscillator flow
• Estimating topological entropy from the motion of stirring rods
• The entropy efficiency of point-push mapping classes on the punctured disk
• Reynolds number effects on mixing due to topological chaos
• Topological entropy and secondary folding