Jean-Luc Thiffeault: A topological theory of stirring
Abstract:
In a fluid, stirring is usually necessary to overcome the slow diffusion of most substances. This is important in a wide range of applications, from industry to geophysics. Here I focus on a prototypical application, the stirring of a two-dimensional viscous fluid with rods. Mathematically, stirring rods can be viewed as 'punctures' in a two-dimensional surface. They present topological obstructions to material lines in the fluid. The theory developed by Nielsen and Thurston to classify the possible periodic motions in such a system can be used to decide which stirring methods are best. Global aspects of the concentration field of a substance in the stirring device can then be determined via 'train tracks', which are skeletons of an important structure called the unstable foliation. Since there are only a limited number of possible train tracks, all stirring protocols can be classified, as well as their properties. This can be extended to completely general situations where rods are replaced by periodic orbits. I will present experimental and numerical examples of all these concepts. Finally, I will introduce a stirring device designed and optimized using these topological principles.