Nonuniform mixing

Fluid mixing usually involves the interplay between advection and diffusion, which together cause any initial distribution of passive scalar to homogenize and ultimately reach a uniform state. However, this scenario only holds when the velocity field is nondivergent and has no normal component to the boundary. If either condition is unmet, such as for active particles in a bounded region, floating particles, or for filters, then the ultimate state after a long time is not uniform and may be time dependent. We show that in those cases of nonuniform mixing it is preferable to characterize the degree of mixing in terms of an $f$-divergence, which is a generalization of relative entropy, or to use the $L^1$ norm. Unlike concentration variance ($L^2$ norm), the $f$-divergence and $L^1$ norm always decay monotonically, even for nonuniform mixing, which facilitates measuring the rate of mixing. We show by an example that flows that mix well for the nonuniform case can be drastically different from efficient uniformly mixing flows.

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Figure 1

In the presence of a suction flow, passive particles such as viruses accumulate near a filter. The filter is permeable to fluid but not to particles.

Figure 2

The Green's function Eq. (5.5) with $x_0=1/4$, plotted at different times for $D=L=1$ and (a) $U=-4$; (b) $U=4$. The ultimate state is the invariant density Eq. (5.4).

Figure 3

Density $p(x,t)$ for the periodic flow of Sec. 5b, for two different initial conditions. The two initial conditions converge to the same periodic pattern $\phi (x,t)$ after about three periods. Here the period $\tau =0.05$ and drift $U=20$, for a domain of width $L=1$ and with diffusivity $D=1$. The vertical lines indicate period boundaries, and the dashed lines are half-periods when the flow switches from right to left.

Figure 4

Variance or $L^2$ norm (solid line), squared $L^1$ norm (dotted line), and Jensen–Shannon divergence (dashed line) between the two solutions in Fig. 3. The variance is nonmonotonic, whereas the $L^1$ norm and $H_{\mathrm{JS}}$ decreases monotonically.

Figure 5

Density $p(x,t)$ for the random flow of Sec. 5c, for two different initial conditions. The two initial conditions rapidly converge to the same random pattern $\phi (x,t)$. The vertical lines indicate period boundaries $t_k$, and the dashed lines are the random times $t_k+{\alpha }_k\tau$ when the flow switches from right to left. Parameter values are as in Fig. 3.

Figure 6

Variance or $L^2$ norm (solid line), squared $L^1$ norm (dotted line), and Jensen–Shannon divergence (dashed line) between the two solutions in Fig. 5. The variance is nonmonotonic, whereas the $L^1$ norm and $H_{\mathrm{JS}}$ decreases monotonically.