Maximal Effective Diffusivity for Time-Periodic Incompressible Fluid Flows

In this paper we establish conditions for the maximal, Pe², behavior of the effective diffusivity in time-periodic incompressible velocity fields for both the Pe → ∞ 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 Poincaré map of the velocity field. The connection between the Pe² asymptotic behavior of the effective diffusivity and t² 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 Pe² in time-dependent flows is established.