The effective diffusivity of cellular flows

In the previous chapter we dealt with unidirectional flows. Now we turn to the slightly more complicated case of incompressible two-dimensional cellular flows. The velocity field can be obtained from a streamfunction ψ(x, y) according to our usual convention u = (u, v) = (−ψy, ψx). The domain is a periodic array of square cells, each with side , so that the streamfunction has the periodicity ψ(x + m , y + n ) = ψ(x, y) where m and n are integers. We use the notation