We study the dissipation enhancement by cellular flows. Previous work by Iyer, Xu, and Zlatoš 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 C(ε^2/κ + |ln δ|^2/(ε^2A)). Here κ is the diffusivity, ε is the cell size, A/ε is the flow amplitude, and δ = (κ/A)^(1/2) is the thickness of the boundary layer. The above 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.

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