Iterative bounds on effective transport for advection diffusion in periodic flow fields

Over three decades ago a Stieltjes integral representation for the effective diffusivity of a tracer in a steady fluid velocity field was developed, involving the spectral measure of a compact self-adjoint operator and the P\'eclet number of the flow. Rigorous bounds on the homogenized diffusivity could then be obtained from knowledge of the moments of the spectral measure. A recent extension to space-time periodic flows involves an unbounded self-adjoint operator. Though Pad\'e approximants provide upper and lower bounds in terms of the moments, the lack of a general method for calculating them has significantly limited the utility of this approach. Here we develop an iterative method that enables an arbitrary number of moments, hence bounds, to be calculated analytically in closed form for spatially and space-time periodic flows. The known behavior of the effective diffusivity for a 2D steady cellular flow is accurately captured by high order upper and lower bounds. The bounds extend to 3D steady and time periodic flow fields away from the advection dominated regime where an open issue remains concerning the divergence of the bounds.