FSU Topology Week and FSU-UF Meeting 2013 (and FSU colloquium)

Title: Two-dimensional topological fluid mechanics

Speaker: Phil Boyland (University of Florida)

Abstract: In a two-dimensional fluid flow the trajectories of a collection of points may become suf- ficiently tangled that their existence has consequences for the surrounding fluid’s evolution. The simplest case to analyze is when the entangled points are actually physical stirrers mov- ing in a periodic protocol. The topological structure of the system is then governed by the Thurston-Nielsen theory which implies that the essential topological length of arcs (material lines) grows either exponentially or linearly. When this growth is exponential the norms of gradients of any transported scalar function also grow exponentially. This has consequences for material mixing as well as for Euler flows. Using the Helmholtz-Kelvin Theorem we show that there are periodic stirring protocols for which generic initial vorticity yields a solution to Euler’s equations which is not periodic and further, the L∞ and L1-norms of the gradient of its vorticity grow exponentially in time. A second application investigates which stirring protocols maximize the topological entropy efficiency of mixing. This application uses an action on homology in an Abelian cover as well as a nonlinear generalization of the joint spectral radius to actions on free groups. The talk will not assume any prior background in fluid mechanics orThurston-Nielsen theory.