Topological optimisation of rodstirring 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2010
 arXiv:
 arXiv:1004.0639
 Bibcode:
 2010arXiv1004.0639F
 Keywords:

 Nonlinear Sciences  Chaotic Dynamics;
 Condensed Matter  Soft Condensed Matter;
 Mathematics  Dynamical Systems;
 Mathematics  Geometric Topology;
 Physics  Fluid Dynamics
 EPrint:
 22 pages, 53 figures. PDFLaTeX with RevTex4 macros.