Quantifying the dissipation enhancement of cellular flows

Abstract

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 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.

Figures (2)

• Figure 1: Stream lines of the cellular flow defined in equation (\ref{['e:v']}). The flow is only non-zero in the shaded region.
• Figure 2: Sample trajectories illustrating the coupling in steps 2 and 3. Here $X_0 = (0.75, 0.25)$, $\tilde{X}_0 = (0.25, 0.25)$, and the trajectory of $X$ is shown in blue. Until $\tilde{X}$ hits a vertical cell boundary the trajectory of $\tilde{X}$ (shown in green) is simply a shift of the trajectory of $X$. After this time, the trajectory of $\tilde{X}$ (shown in red) is a mirror image of the trajectory of $X$ until they hit the same vertical line ($x = 0.5$ in this case).

Theorems & Definitions (39)

• Theorem 2.1
• Proposition 2.2
• Remark 2.3
• proof : Proof of Theorem \ref{['t:main']} when $A \geqslant \kappa / \varepsilon^4$
• Lemma 3.1: Coupling of projections
• Lemma 3.2: Vertical boundary hitting time
• Lemma 3.3: Vertical coupling
• Lemma 3.4: Horizontal boundary hitting time
• Lemma 3.5: Horizontal coupling
• proof : Proof of Theorem \ref{['t:main']} when $A \leqslant \frac{\kappa}{\varepsilon^4}$