Numerical Analysis Seminars: Exponential growth in two-dimensional topological fluid mechanics
26 October 2012 14:05 in CM105
In a two-dimensional fluid flow the trajectories of a collection of points may become sufficiently tangled
that their existence has consequences for the surrounding fluids evolution. The simplest case to analyze is when the entangled points are actually physical stirrers moving in a periodic protocol. The topological kinematics of the system is then governed by the Thurston-Nielsen (TN) theory which implies that the essential topological length of material lines grows either exponentially or linearly. When this growth is exponential the norms of gradients of any passively transported scalar also grow exponentially. Using the Helmholtz-Kelvin Theorem we then 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 $L^\infty$ and $L^1$-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.
Contact David.Bourne@durham.ac.uk for more information