2.1 General picture

We consider a lamella of dye of length $l_{0}$ and half-width $s_{0}$ initially positioned perpendicular to the flow, as illustrated in figure 1(a). As the lamella is advected by a laminar shear flow with a shear rate $\dot{\unicode[STIX]{x1D6FE}}$ , its length increases as $l(t)=l_{0}\sqrt{1+(\dot{\unicode[STIX]{x1D6FE}}t)^{2}}$ . Thus, considering the sole effect of the advection field (for now neglecting that of molecular diffusion), the half-width of the lamella $s_{A}(t)$ decreases following $s_{A}(t)=s_{0}/\sqrt{1+(\dot{\unicode[STIX]{x1D6FE}}t)^{2}}$ (see figure 1 b) since in this two-dimensional flow, mass conservation prescribes $s_{0}l_{0}=s_{A}(t)l(t)$ . The effect of the shear flow can thus be quantified by a compression rate $(-1/s_{A})(\text{d}s_{A}/\text{d}t)$ which describes how fast the lamella transverse dimension thins down owing to its stretching by the advection field. Conversely, molecular diffusion tends to broaden the lamella with a diffusive broadening rate $D_{0}/s_{A}^{2}$ , where $D_{0}$ is the molecular diffusion coefficient of the scalar. Balancing these two rates, $(-1/s_{A})(\text{d}s_{A}/\text{d}t)\sim D_{0}/s_{A}^{2}$ and assuming $\dot{\unicode[STIX]{x1D6FE}}t\gg 1$ , naturally defines a time scale, also called the Batchelor time, $t_{B}\sim Pe^{1/3}/\dot{\unicode[STIX]{x1D6FE}}$ , where $Pe=\dot{\unicode[STIX]{x1D6FE}}s_{0}^{2}/D_{0}$ denotes the Péclet number. This time corresponds to the onset of the homogenization of the concentration levels within the system, beyond which the concentration levels within the lamella start to decay significantly.

Figure 1.

(a) Schematic of a lamella of dye of initial length $l_{0}$ and half-width $s_{0}$ advected in a laminar shear flow. (b) Effect of the advection field alone: the strain $\dot{\unicode[STIX]{x1D6FE}}t$ has stretched the lamella and thinned down its transverse dimension to $2s_{A}(t)$ . (c) Effect of both advection and molecular diffusion: at the same strain, the half-width of the lamella is denoted $s_{AD}(t)$ . Inset: schematic of the Gaussian concentration field of the lamella with its concentration profiles along the flow $C(x,t)$ and transverse to the lamella $C(n,t)$ .

2.2 Exact solution

The complete description of the evolution of the lamella of dye can be found by directly solving the full advection–diffusion equation

where $\boldsymbol{u}=(\dot{\unicode[STIX]{x1D6FE}}y,0,0)$ denotes the advection field, and $C$ the concentration field.

This equation can be simplified (Ranz Reference Ranz1979; Rhines & Young Reference Rhines and Young1983; Ottino Reference Ottino1989; Meunier & Villermaux Reference Meunier and Villermaux2003, Reference Meunier and Villermaux2010) if written in a moving frame $(n,z)$ aligned with the directions of maximal compression and stretching of the lamellae (see figure 1 c), and by using Ranz’s transform (Ranz Reference Ranz1979), which uses warped time units, $\unicode[STIX]{x1D70F}=\int _{0}^{t}D_{0}/s_{A}^{2}(t^{\prime })\,\text{d}t^{\prime }$ , and distances normalized by the lamella transverse dimension, $\unicode[STIX]{x1D709}=n/s_{A}(t)$ . Equation (2.1) then reduces to a simple diffusion equation

whose solution for an initial Gaussian concentration profile with maximum $C_{0}$ is

The maximum concentration of the lamella thus decays with time according to

Since the half-width of the lamella in Ranz’s units is $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D709}}=(1+4\unicode[STIX]{x1D70F})^{1/2}$ (see (2.3)), the transverse dimension of the lamella $s_{AD}(t)$ , accounting both for the effect of advection and diffusion, and expressed in standard unit of length, is

The latter expression gives access to the half-width of the lamella along the flow ( $x$ -direction)

with $\unicode[STIX]{x1D70E}_{0}=s_{0}$ and $\cos \unicode[STIX]{x1D703}=1/\sqrt{1+(\dot{\unicode[STIX]{x1D6FE}}t)^{2}}$ (see figure 1).

We now have access to a more accurate estimation of the Batchelor time, which by definition is reached when $\unicode[STIX]{x1D70F}=1$ (Ranz Reference Ranz1979). Since $\unicode[STIX]{x1D70F}=D_{0}(t+\dot{\unicode[STIX]{x1D6FE}}^{2}t^{3}/3)/s_{0}^{2}$ , the Batchelor time, for $Pe\gg 1$ , is

At this time, the transverse dimension of the lamella, from now on referred to as the Batchelor scale (Batchelor Reference Batchelor1959), is found to be

Last, the evolution of the concentration distribution $P(C,t)$ of the lamella can be derived. Using the following change of variables $P(C)\,\text{d}C=P(x)\,\text{d}x$ yields $P(C)=P(x)|\text{d}C/\text{d}x|^{-1}$ , which can easily be expressed from

and the uniformity of $P(x)$ . Considering the range of concentration $C_{th}\leqslant C\leqslant C_{max}(t)$ , above any arbitrary threshold concentration $C_{th}$ (larger than the experimental background noise), one obtains

where $\unicode[STIX]{x1D6FD}$ is a normalizing prefactor ensuring that $\int _{C_{th}}^{C_{max}(t)}P(C,t)\,\text{d}C=1$ .

The above set of equations fully describes the evolution of a lamella of dye advected in a laminar shear flow.

Figure 2.

(a) Schematic of the set-up. (b) Typical image of a lamella obtained by photobleaching.

4.1 A single lamella

Figure 3(a) shows successive pictures of a lamella undergoing a laminar shear (see also supplementary movie 1). Initially vertical and highly contrasted, the lamella progressively tilts under the effect of the shear flow while blurring under the effect of molecular diffusion. Accurate measurement of the lamella’s concentration profile along the flow ( $x$ -direction) are obtained by averaging over all horizontal lines of pixels after translating these lines to make their maximum concentration coincide. The resulting average concentration profile of the lamella is shown in figure 3(b) for successive strains: the maximum concentration decays while the width increases. These trends are captured well by fitting each concentration profile with a Gaussian of the form $C(x,t)=C_{max}(t)\text{e}^{-x^{2}/\unicode[STIX]{x1D70E}_{x}^{2}(t)}$ (see figure 3 b).

Figure 4.

(a) Evolution of transverse dimension of the lamella $s_{AD}(t)/s_{0}$ versus time for experiments performed at different Péclet numbers. The black lines correspond to (2.5). The dotted line corresponds to the solution in the absence of shear, i.e. in the pure diffusion limit ( $Pe=0$ ). (b) Corresponding Batchelor scale $s_{AD}(t_{B})/s_{0}$ versus Péclet number; the black line corresponds to (2.8). (c) Concentration distribution $P(C/C_{0})$ measured at successive strains, $\dot{\unicode[STIX]{x1D6FE}}t$ , for a lamella sheared at $Pe=4.5$ . The black lines correspond to (2.10). In all cases, $\dot{\unicode[STIX]{x1D6FE}}$ , $s_{0}$ and $D_{0}$ are fixed by the experimental conditions.

The resulting maximum concentration $C_{max}(t)$ and width $\unicode[STIX]{x1D70E}_{x}(t)$ are plotted in figures 3(c) and 3(d) versus time for experiments performed at different Péclet number ( $4.5\leqslant Pe\leqslant 1190$ ). The Péclet number was varied by repeating the experiment at various shear rates $\dot{\unicode[STIX]{x1D6FE}}=[6\times 10^{-4}-0.3]~\text{s}^{-1}$ . The agreement with (2.4) and (2.6) is very good for both $C_{max}(t)$ and $\unicode[STIX]{x1D70E}_{x}(t)$ . Note that in both cases, $\dot{\unicode[STIX]{x1D6FE}}$ , $s_{0}$ and $D_{0}$ are fixed by the experimental conditions; there is thus no adjustable parameter. When plotted as a function of the dimensionless time $t/t_{B}$ , where $t_{B}=(3Pe)^{1/3}/\dot{\unicode[STIX]{x1D6FE}}$ is the Batchelor time, these data are found to collapse, for all $Pe$ , on the same master curve (see figure 3 e,f). For $t<t_{B}$ , $C_{max}$ and $\unicode[STIX]{x1D70E}_{x}$ remain constant. Then when the effect of molecular diffusion becomes significant, i.e for $t>t_{B}$ , $C_{max}$ (respectively $\unicode[STIX]{x1D70E}_{x}$ ) starts to decrease (respectively increase) following the power law $t^{-3/2}$ (respectively $t^{3/2}$ ), consistently with the long-time trends of (2.4) and (2.6). These measurements clearly illustrate how mixing is accelerated by imposing an external macroscopic shear: larger applied shear rates (larger Péclet numbers) result in earlier mixing times.

We have so far probed the lamella along the direction of the flow. However, further insight into the mixing process, specifically on the advection–diffusion coupling presented above, are provided by probing the lamella width along its transverse direction (along $n$ , see figure 1). Figure 4(a) shows the evolution of $s_{AD}(t)$ measured experimentally. At an intermediate time, the thickness of the lamella is found to decrease like $t^{-1}$ . After reaching a minimum, it increases like $t^{1/2}$ . These trends precisely illustrate the expected interplay between advection and diffusion. The lamella width initially decreases as imposed by the kinematics of the flow following the intermediate time trend (for $t<t_{B}$ ) of (2.5), $s_{AD}(t)\sim s_{0}(\dot{\unicode[STIX]{x1D6FE}}t)^{-1}$ . However, this compression of the lamella progressively steepens its concentration gradients which, beyond the Batchelor time, eventually makes the broadening effect due to molecular diffusion become dominant. The transverse dimension of the lamella then re-increases diffusively like $t^{1/2}$ . At the Batchelor time $t_{B}$ , the lamella typically reaches its minimum thickness, which is equal to the Batchelor scale $s_{AD}(t_{B})$ (within 3 %). As shown in figure 4(b), reporting this direct measurement of the Batchelor scale obtained for various Péclet numbers matches the expected prediction $s_{AD}(t_{B})=s_{0}\sqrt{5}(3Pe)^{-1/3}$ (see (2.8)).

To fully describe the mixing process, we also measure the evolution of the lamella’s concentration distribution $P(C,t)$ . The distribution $P(C)$ is obtained from the histogram of intensities collected from all the pixels constituting the lamella which are above a threshold $C_{th}=0.1C_{0}$ . This discards the background image noise, enabling us to focus on the peak of interest, that of high concentration. Changing the value of the threshold above the background noise changes the extent of the spatial domain over which $P(C)$ is computed, but it does not affect its shape. We obtain concentration distributions which have a characteristic U-shape (Ottino Reference Ottino1989) (see figure 4 c). The distribution’s maximum probability, initially located at $C=C_{0}$ , progressively drifts to lower values as the lamella is stretched and diffuses. The prediction provided by (2.10) is in good agreement with the measured concentration distributions obtained for all Péclet numbers and, again, without adjustable parameter.

Figure 5.

Distributions of concentration $P(C/C_{max})$ obtained from the progressive digital coarsening of an experimental image.

The distributions shown above are well resolved since the Batchelor scale is always larger than 10 pixels. However, in most studies on mixing, such a high resolution is not achieved. Moreover, when mixing is investigated in complex systems such as turbulent flows (Duplat & Villermaux Reference Villermaux and Duplat2006), porous media (Le Borgne et al. Reference Le Borgne, Dentz and Villermaux2015), or recently in sheared particulate suspensions (Souzy et al. Reference Souzy, Lhuissier, Villermaux and Metzger2017), the initial lamella of dye does not evolve towards a single lamella with uniform thickness but, instead, towards a large number of lamellae having widely distributed thicknesses. In such situations, it is important to know which part of the distribution of lamellae can be resolved experimentally.

To address this point, we systematically investigate how the set-up spatial resolution can bias the measured concentration distribution of a single lamella. This can be achieved by a simple coarsening protocol: the reference concentration distribution (blue dotted line in figure 5) is obtained from a highly resolved image where the lamella half-width spans $8$ pixels. When coarsening the pixels of this reference image, (merging $2\times 2$ pixels into one, $4\times 4$ pixels into one and so on) we find that the concentration distribution remains unchanged for images having $4$ and $2$ pixels across the lamella half-width (green dotted lines in figure 5). Conversely, larger coarsening, i.e. images having $1$ pixel or less across the lamella half-width, yield erroneous results: the concentration distribution departs from the reference one (red dotted lines in figure 5). The limit of $1$ pixel per lamella half-width is surprisingly small and comes as encouraging news for experimentalists. This limit holds as long as the concentration profile is smooth (e.g Gaussian) and there is a finite tilt between the lamella and the lines of pixels. In such case, the progressive drift of the lamella position relative to that of the pixels scans all the possible concentration levels, thereby providing results consistent with the fully resolved PDF.

4.2 Two lamellae

As mentioned in the introduction, mixing in complex systems generally involves coalescence between dispersed lamellae. This latter step is crucial for the system to reach its final homogeneity. Here, the photobleaching technique is used to produce two adjacent lamellae and investigate a single coalescence event.

Figure 6.

(a) Coalescence of two nearby lamellae advected in a laminar shear flow (see also supplementary movie 2). Evolution of the corresponding experimental (b) concentration profiles and (c) concentration distributions obtained at successive strains $\dot{\unicode[STIX]{x1D6FE}}t$ . Predictions for the evolution of (d) the concentration profiles and (e) the concentration distributions at the same strains. Again, $\dot{\unicode[STIX]{x1D6FE}}$ , $s_{0}$ and $D_{0}$ are fixed by the experimental conditions.

Figure 6(a) shows images of two parallel lamellae captured at successive strains (see also supplementary movie 2). The two lamellae are initially distinct from each other and, as they are stretched by the flow and diffuse, they progressively merge to eventually form one single entity. The evolution of their concentration profiles measured along the flow $x$ -direction are shown in figure 6(b). The two maxima of concentration, corresponding to each lamella, decay (blue arrows) while the minimum located in between increases (green arrow). This goes on until the two lamellae merge into one single lamella whose maximum subsequently decreases (yellow arrow). This evolution can easily be predicted owing to the linearity of the diffusion equation: the concentration profile of a set of two lamellae, with individual profiles $C_{1}(x,t)$ and $C_{2}(x,t)$ , is simply obtained from the summation $C(x,t)=C_{1}(x,t)+C_{2}(x,t)$ (Fourier Reference Fourier1822). Using, for each lamella, equation (2.9) with the initial experimental maximum intensity and width, we thereby obtain the concentration profiles shown in figure 6(d).

To anticipate the impact of this coalescence on the evolution of the concentration distribution, let us recall that $P(C)\sim |\text{d}C/\text{d}x|^{-1}$ . Thus, each value of $C(x,t)$ where the concentration profile presents an horizontal slope gives rise to a peak in the concentration distribution. This can be clearly observed on the experimental concentration distributions shown in figure 6(c). As long as the two lamellae are distinct, one not only observes two peaks at high concentration (corresponding to the maximum of concentration of each lamellae), but also one peak at low concentration (corresponding to the minimum concentration located in between the two lamellae). Then, as the lamellae are stretched and diffuse, the two peaks corresponding to the two concentration maxima move to the left (blue arrow in figure 6 c). Conversely, as the concentration in between the two lamellae increases, the low concentration peak moves to the right (green arrow in figure 6 c). Coalescence occurs when the three peaks collide; the distribution eventually recovers its characteristic U-shape as the lamellae have coalesced into a single entity. The same phenomenology can be observed on the concentration distributions obtained from numerically computing the slope of the predicted concentration profiles (see figure 6 e).