A bound on mixing efficiency for the advection-diffusion equation

An upper bound on the mixing efficiency is derived for a passive scalar under the influence of advection and diffusion with a body source. For a given stirring velocity field, the mixing efficiency is measured in terms of an equivalent diffusivity, which is the molecular diffusivity that would be required to achieve the same level of fluctuations in the scalar concentration in the absence of stirring, for the same source distribution. The bound on the equivalent diffusivity depends only on the functional 'shape' of both the source and the advecting field. Direct numerical simulations performed for a simple advecting flow to test the bounds are reported.