Maximal Effective Diffusivity for Time-Periodic Incompressible Fluid Flows

In this paper we establish conditions for the maximal, Pe, behavior of the effective diffusivity in time-periodic incompressible velocity fields for both the Pe oc and Pe 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 Poincar6 map of the velocity field. The connection between the Pe2 asymptotic behavior of the effective diffusivity and 2 asymptotic dispersion of a nondiffusive tracer is established. Several examples are analyzed: we relate the existence of accelerator modes in a flow with Pe2 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 Pe in time-dependent flows is established. Key words, effective diffustivity, homogenization, ergodic theory AMS subject classifications. 76R50, 60H25, 35R60, 58F