Walls Inhibit Chaotic Mixing

We report on experiments of chaotic mixing in a closed vessel, in which a highly viscous fluid is stirred by a moving rod. We analyze quantitatively how the concentration field of a low-diffusivity dye relaxes towards homogeneity, and we observe a slow algebraic decay of the inhomogeneity, at odds with the exponential decay predicted by most previous studies. Visual observations reveal the dominant role of the vessel wall, which strongly influences the concentration field in the entire domain and causes the anomalous scaling. A simplified 1D model supports our experimental results. Quantitative analysis of the concentration pattern leads to scalings for the distributions and the variance of the concentration field consistent with experimental and numerical results.

Figure 1

(a) Chaotic mixing experiment in a closed vessel: a rod moves periodically on a figure-eight path [see Fig. 2a] and transforms an initial spot of dye into a complicated filamentary pattern. Inset: Poincaré section obtained numerically for the corresponding Stokes flow. Note that the chaotic region spans the entire domain. (b) Evolution of the variance of the concentration field in a fixed central region. ■: experiment , ○: numerical simulation (see Fig. 3 for a description). Solid line fits: contribution of the white pixels ${\sigma }_W^2\simeq (2\log t+\log w_B)/(\log \Gamma \times t^2)$ derived below. (c) Experimental concentration PDFs in the bulk at time $t=13$, 17, 31 periods. Both sides of the peak can be fitted by power laws $(C_{\max }-C{)}^{-2}$ (red and blue plots). Inset: left (“light-gray”) tail of the peak, $P(C)$ against $|C_{\max }(t)-C|$.

Figure 2

(a) Transport mechanisms: (i) the rod stretches and folds the mixed region and (ii) a white strip of unmixed fluid is injected between the two folded parts. (b) Stretched strips of dye (black) are smeared out by diffusion on a scale $w_B$ (gray). The concentration $C$ of a pixel $x$ is then given by adding the concentrations coming from strips inside a box of size $w_B$ around $x$.

Figure 3

Numerical concentration PDFs ($t=15$, 18, 25) obtained by letting concentration profiles evolve as $C(x,t+1)=C(f^{-1}(x),t)$, $f(x)\mathrm{\text{:}}{f}_1(x)=x-a{x}^2+(\gamma -1+a)x^3$; $f_2(x)=1-a{x}^2+(\gamma -1+a)x^3$ with $a=0.9$ and $\gamma =0.55$. Upper (lower) inset: light (dark) gray tail, $P(C)$ against $|C_{\max }-C|$. We observe the same power-law decay $(C_{\max }-C{)}^{-2}$ on both sides of the maximum as in the experiment [Fig. 1c].