  We study the dissipation enhancement by cellular flows. Previous work by
Iyer, Xu, and Zlato\v{s} produces a family of cellular flows that can enhance
dissipation by an arbitrarily large amount. We improve this result by providing
quantitative bounds on the dissipation enhancement in terms of the flow
amplitude, cell size and diffusivity. Explicitly we show that the mixing time
is bounded by the exit time from one cell when the flow amplitude is large
enough, and by the reciprocal of the effective diffusivity when the flow
amplitude is small. This agrees with the optimal heuristics. We also prove a
general result relating the dissipation time of incompressible flows to the
mixing time. The main idea behind the proof is to study the dynamics
probabilistically and construct a successful coupling.
