Geodesic Theory of Transport Barriers

I describe a unified approach to locating key material transport barriers in unsteady flows induced by two-dimensional, non-autonomous dynamical systems. Seeking transport barriers as minimally stretching material lines, one obtains that such barriers must be shadowed by minimal geodesics under the metric induced by the Cauchy-Green strain tensor field associated with the flow map. As a result, snapshots of transport barriers can be explicitly computed as trajectories of ordinary differential equations. Using this approach, hyperbolic barriers (generalized stable and unstable manifolds), elliptic barriers (generalized KAM curves) and parabolic barriers (generalized shear jets) can be found with high precision in temporally aperiodic flows defined over a finite time interval. I illustrate these results on unsteady flows arising in mechanics and fluid dynamics.