Topological optimisation of rod-stirring devices

TL;DR

This paper investigates the topological optimization of rod-stirring devices to maximize material stretching, identifying optimal growth rates linked to the golden and silver ratios, and proposing designs called silver mixers.

Contribution

It introduces a topological approach to optimize rod-stirring devices, revealing optimal growth rates and constructing devices called silver mixers for practical implementation.

Findings

Optimal growth rate is the logarithm of the golden ratio.

For realistic costs, the optimal growth rate is the logarithm of the silver ratio.

Constructed devices called silver mixers realize these optimal growth rates.

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.