Mixing as an Aggregation Process

Experiments show how a stirred scalar mixture relaxes towards uniformity through an aggregation process. The elementary bricks are stretched sheets whose rates of diffusive smoothing and coalescence build up the overall mixture concentration distribution. The cases studied, in particular, include mixtures in two and three dimensions, with different stirring protocols and Reynolds numbers which all lead to a unique family of concentration distributions stable by self-convolution, the signature of the aggregation mechanism from which they originate.

Figure 1

Mixing of a dye discharging from a jet of diameter $d=8\mathrm{m}\mathrm{m}$ in a square ($L\times L$ with $L=3\mathrm{c}\mathrm{m}$) duct. From 1 to 3, successive instantaneous planar cuts of the scalar field at increasing downstream locations in the duct showing the progressive uniformization of the dye concentration.

Figure 2

Left: Downstream evolution of the concentration distribution $P(C)$ as the dye progresses along the duct as shown in Fig. 1. The concentration distribution of the evolving mixture gets narrower around the average concentration $⟨C⟩=0.3$. Solid line: experimental distributions; dashed line: distributions given by Eq. (1). Right: Fitting parameter $n$ of the distributions (1) as a function of the downstream distance $(x-d)/L$. ∘, $\mathrm{R}\mathrm{e}={10}^4$; ●, $\mathrm{R}\mathrm{e}=5\times {10}^3$.

Figure 3

Top: The stirring protocol of a drop of ink deposed at the surface of pure glycerol using a small rod. The sequence displays the initial state, half, and a completed stirring cycle. Bottom: The mixture’s state after $2\frac{1}{2}$ completed stirring cycles.

Figure 4

Left: Contour length of the drop as it deforms through the mixing cycles for three different concentration thresholds. The length increases linearly independently of the concentration thresholds chosen to define the contour up to the mixing cycle $p_s\approx 2.5$. For $p>p_s$, the contour length depends on the concentration threshold and, at fixed threshold, decreases as $p$ increases. Right: Corresponding average transverse sheet, or striation thickness $s$ before the mixing cycle; the line has a slope $-1$.

Figure 5

Left: Evolution of the dye concentration distribution $P(C)$ after 3, 4, 5, 6 and 13 cycles according to the protocol shown on Fig. 3. The concentration distribution of the evolving mixture gets narrower around the average concentration $⟨C⟩$. solid line: experimental distributions, dashed line: distributions given by Eq. (1). Right: Fitting parameter $n$ and average concentration of the distributions (1) as a function of the number of cycles.