The mixing efficiency of open flows

Mixing in open incompressible flows is studied in a model problem with inhomogeneous passive scalar injection on an inlet boundary. As a measure of the efficiency of stirring, the bulk scalar concentration variance is bounded, and the bound is shown to be sharp at low Péclet number. Although no specific flow saturating the bound at high Péclet number is produced here, the estimate is conjectured to be approached for flows possessing sufficiently sustained chaotic regions.