Maximal Effective Diffusivity for Time-Periodic Incompressible Fluid Flows

In this paper we establish conditions for the maximal, $Pe^2 $, behavior of the effective diffusivity in time-periodic incompressible velocity fields for both the $Pe \to \infty $ and $Pe \to 0$ limits. Using ergodic theory, these conditions can be interpreted in terms of the Lagrangian time averages of the velocity. We reinterpret the maximal effective diffusivity conditions in terms of a Poincaré map of the velocity field. The connection between the $Pe^2 $ asymptotic behavior of the effective diffusivity and $t^2 $ asymptotic dispersion of a nondiffusive tracer is established. Several examples are analyzed: we relate the existence of accelerator modes in a flow with $Pe^2 $ effective diffusivity and show how maximal effective diffusivity can appear as a result of a time-dependent perturbation of a steady cellular velocity field. Also, three-dimensional, symmetric, time-dependent duct velocity fields are analyzed, and the mechanism for an effective diffusivity with Peclet number dependence other than $P...