Subdiffusive and superdiffusive transport in plane steady viscous flows

Dispersion of particles in chaotic, turbulent or random flows has been studied for a long time. It is known that the action of advection on large spatial and temporal sales typically can be described as an (anisotropic) normal diffusion process. However, as found by Kraichnan in 1970, that is not the case for steady two-dimensional flows of incompressible viscous fluids. We show that the deterministic transport of particles through lattices of solid bodies or arrays of steady vortices can be anomalous. Motion along regular patterns of streamlines is typically aperiodic. Repeated slow passages near stagnation points and/or solid surfaces serve for eventual decorrelation. Singularities of passage times near the obstacles, determined by the boundary conditions, affect the character of transport anomalies. Flows past regular arrays of vorticies are subdiffusive; the temporal evolution of MSD displays some nontrivial similarity properties. Tracers advected through lattices of solid obstacles can feature superdiffusion. The particle transport in spatially irregular flows is also considered. The analytical predictions match the results of numerical simulations.