Fluid Stirring on a Sphere -- a Topological Approach

The stirring and mixing of a fluid with moving rods is vital in many physical applications in order to achieve homogeneity within the mixture. These rods act as an obstacle that stretches and folds together fluid elements. Over time, the permutation of these rods comprise a mathematical braid whose properties dictate a minimum topological entropy, a number to describe the total disorder or chaos of a system. A braid whose topological entropy is greater than one exhibits chaotic behavior which guarantees an optimal mixing pattern. These rod stirring braids have been previously studied on both the disk as well as the two dimensional torus. The trajectory of fluid mixing on a sphere poses an intriguing starting inquiry to overall mixing on spherical surfaces like the ocean, stars, etc. We use a recipe established by Yvon Verberne to create pseudo-Anosov maps on a punctured sphere using Dehn twist in order to construct similar maps on a 4-times punctured sphere. Since a quotient of the torus under a hyperelliptic involution of torus is the 2-sphere with 4 marked points, we are able to use various methods in order to estimate the topological entropy of a stirring protocol on a 4-times punctured sphere