Mathieu Souzy

Imen Zaier

Henri Lhuissier

Tanguy Le Borgne

Bloen Metzger

Mixing lamellae in a shear flow

Introduction

Mixing is a key process in many industrial applications and natural phenomena [START_REF] Ottino | The Kinematics of Mixing: Stretching, Chaos, and Transport[END_REF]. Practical examples include glass manufacture, processing of food, micro-fluidic manipulations, and contaminant transport in the atmosphere, oceans and hydrological systems. During the last decades, substantial progresses have been made in the description of mixing in systems as complex as turbulent flows [START_REF] Warhaft | Passive scalars in turbulent flows[END_REF][START_REF] Shraiman | Scalar turbulence[END_REF][START_REF] Falkovich | Particles and fields in fluid turbulence[END_REF], Duplat & Villermaux 2008[START_REF] Kalda | Simple model of intermittent passive scalar turbulence[END_REF], oceanic and atmospheric flows [START_REF] Rhines | How rapidly is a passive scalar mixed within closed streamlines?[END_REF], the earth mantle [START_REF] Allègre | Implications of a two-component marble-cake mantle[END_REF], porous media [START_REF] Dentz | Mixing, spreading and reaction in heterogeneous media: A brief review[END_REF][START_REF] Villermaux | Mixing by porous media[END_REF][START_REF] Borgne | The lamellar description of mixing in porous media[END_REF] and sheared particulate suspensions [START_REF] Souzy | Stretching and mixing in sheared particulate suspensions[END_REF]. In particular, the conceptualization of scalar mixtures as ensembles of 'lamellae' evolving through stretching, diffusion and aggregation [START_REF] Ranz | Applications of a stretch model diffusion, and reaction in laminar and turbulent flows[END_REF], Villermaux & Duplat 2003) has allowed deriving quantitative accurate theoretical predictions for the evolution of the full concentration Probability Density Functions (PDF) for a broad range of flows. Within this framework (valid for lamellae thinner than the smallest scale within the flow), mixing is driven by three processes occurring simultaneously: i) stretching a scalar field by a given flow creating elongated lamellar structures, ii) diffusion and compression that compete to define concentration gradients and iii) lamella coalescence leading ultimately to the final homogeneity of the system (Duplat & Villermaux 2008).

In complex configurations such as those listed above, the evolution of the concentration distribution is sensitive to a number of characteristics of the system: the distribution of stretching rates, the macroscopic dispersion rate, the scalar molecular diffusion coefficient and the rate at which lamellae aggregate. While their influence on the dynamics of mixing is clear from a theoretical point of view, they can rarely be observed independently. Moreover, in spite of the numerous experimental studies on mixing, the question of the spatial resolution required to quantify the evolution of a concentration field has received little attention. This latter point is obviously crucial since under-resolved images, by artificially broadening the concentration distribution of isolated lamellae or conversely, by sharpening the concentration distribution of bundles of adjacents lamellae, lead to an erroneous appreciation of the mixing rate.

Here, we present highly resolved experiments where the basic mechanisms governing mixing are quantified individually: the formation of elongated lamellae by stretching, the establishment of a Batchelor scale by competition between compression and diffusion, the enhancement of diffusive mixing by stretching and the diffusive aggregation of lamellae. We consider for this a lamella formed by photo-bleaching of a fluorescent dye in a laminar shear flow where the concentration distribution can be quantified using a well controlled experimental set-up, built specifically to resolve small length scales. This benchmark experiment was chosen precisely because it is well established theoretically. Unambiguous conclusions can therefore be drawn regarding the spatial resolution required to capture the evolution of the concentration distribution of an isolated lamella, and more generally within any mixing protocols. Moreover, it illustrates with an unprecedented resolution and level of details the basic lamellar theories for mixing at the scale of a single lamella. Last, we investigate the coalescence of two nearby lamellae specifically focusing on its impact on the evolution of the concentration distribution. The theoretical prerequisites are recalled in § 2. After presenting the experimental set-up in § 3, the measurements are reported in § 4. Conclusions are drawn in § 5.

Mixing in a laminar shear flow

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 γ , its length increases as l(t) = l 0 1 + ( γt) 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 / 1 + ( γt) 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 ds A /dt 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 2 A , where D 0 is the molecular diffusion coefficient of the scalar. Balancing these two rates, -1/s A ds A /dt ∼ D 0 /s 2 A and assuming γt 1, naturally defines a time-scale, also called the Batchelor time, t B ∼ P e 1/3 / γ where P e = γs 2 0 /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 significantly decay.

Exact solution

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

C(n, t) C(x, t) s AD σ x x n C( C C n C( C C x x x n n n n x y s A s AD l 0 s 0 θ a) b) c) l 0 γt l(t) z n γ 2 2 2 2
Figure 1. a) Schematic of a lamella of dye of initial length l0 and half width s0 advected in a laminar shear flow. b) Effect of the advection field alone: the strain γt has stretched the lamella and thinned down its transverse dimension to 2sA(t). c) Effect of both advection and molecular diffusion: at the same strain, the half width of the lamella is denoted sAD(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).

∂C ∂t + u.∇C = D 0 ∇ 2 C, (2.1)
where u = ( γy, 0, 0) denotes the advection field, and C the concentration field. This equation can be simplified [START_REF] Ranz | Applications of a stretch model diffusion, and reaction in laminar and turbulent flows[END_REF][START_REF] Rhines | How rapidly is a passive scalar mixed within closed streamlines?[END_REF][START_REF] Ottino | The Kinematics of Mixing: Stretching, Chaos, and Transport[END_REF][START_REF] Meunier | How vortices mix[END_REF] 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 [START_REF] Ranz | Applications of a stretch model diffusion, and reaction in laminar and turbulent flows[END_REF]) which uses warped time units, τ = t 0 D 0 /s 2 A (t )dt , and distances normalized by the lamella transverse dimension, ξ = n/s A (t). Equation (2.1) then reduces to a simple diffusion equation

∂C ∂τ = ∂ 2 C ∂ξ 2 , (2.2)
whose solution for an initial gaussian concentration profile with maximum C 0 is

C(ξ, τ ) = C 0 √ 1 + 4τ e -ξ 2 1+4τ . (2.3)
The maximum concentration of the lamella thus decays with time according to

C max (t) = C 0 √ 1 + 4τ . (2.4)
Since the half width of the lamella in Ranz's units is

σ ξ = (1 + 4τ ) 1/2 (see equation 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

s AD (t) = σ ξ s A (t) = s 0 (1 + 4τ ) 1/2 1 + ( γt) 2 .
(2.5)

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

σ x (t) = s AD (t) cos θ = σ 0 (1 + 4τ ) 1/2 , (2.6)
with σ 0 = s 0 and cos θ = 1/ 1 + ( γt) 2 (see Figure 1).

We now have access to a more accurate estimation of the Batchelor time which by definition is reached when τ = 1 [START_REF] Ranz | Applications of a stretch model diffusion, and reaction in laminar and turbulent flows[END_REF]. Since τ = D 0 (t+ γ2 t 3 /3)/s 2 0 , the Batchelor time, for P e 1, is

t B ≈ (3P e) 1 3
γ .

(2.7)

At this time, the transverse dimension of the lamella, from now on referred to as the Batchelor scale [START_REF] Batchelor | Small-scale variation of convected quantities like temperature in a turbulent fluid. part 1. general discussion and the case of small conductivity[END_REF], is found to be

s AD (t B ) = s 0 √ 5(3P e) -1 3 . (2.8)
Last, the evolution of the concentration distribution P (C, t) of the lamella can be derived. Using the following change of variables P (C)dC = P (x)dx yields P (C) = P (x)|dC/dx| -1 which can easily be expressed from

C(x, t) = C max (t)e - x 2 σ 2 0 (1+4τ ) ,
(2.9)

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

P (C, t) = 1 βC ln(C max /C) , (2.10)
where β is a normalizing prefactor insuring that

Cmax(t) C th P (C, t)dC = 1.
The above set of equations fully describes the evolution of a lamella of dye advected in a laminar shear flow.

Experimental set-up

The experimental set-up is shown in Figure 2. It consists of a shear-cell made of two parallel plexiglass plates sandwiching a fluid cell 85 mm long, 25 mm wide and of height h = 3 mm. The fluid is sheared by setting the two plates into opposite motion with two high-resolution translation stages (not shown on the schematic). The travel-range of the plate enables a maximum strain of 30. The cell is sealed on both ends by two PTFE plates (grey in Figure 2), and on the sides by two transparent side walls (not shown on the schematic) mounted on the bottom moving plate.

The fluid is a Newtonian mixture of triton X-100 (77.4 wt%), zinc chloride (13.4 wt%) and water (9.2 wt%) having a density ρ = 1.19 g.cm -3 and viscosity η = 4.2 Pa.s which ensures laminar flow conditions (over the full range of shear rates investigated Re = ρ γh 2 /η ≤ 10 -3 ). Prior to filling the cell, some fluorescent dye (Rhodamine 6G) is thoroughly mixed to the fluid at a concentration of 2 10 -6 g.mL -1 . The molecular diffusion coefficient of this dye was measured in the cell with the fluid at rest using a technic similar to that reported in [START_REF] Souzy | Stretching and mixing in sheared particulate suspensions[END_REF]. We find D 0 ≈ 2.66 10 -13 m 2 s -1 at a temperature of 22 • C.

The initial lamella is generated using fluorescent recovery after photo-bleaching, [START_REF] Axelrod | Mobility measurement by analysis of fluorescence photobleaching recovery kinetics[END_REF]. A high-power-laser-diode beam (Laser Quantum Gem512-2.5W) is shaped into a thin laser sheet using the combination of lenses set in their 'bleaching' configuration as shown in Figure 2.a. This sheet, oriented in the yz plane (perpendicular to the flow) and used with full intensity, locally changes the conformation of the rhodamine molecules which irreversibly become unable to fluoresce.

Then, the set-up is set into its 'visualization' configuration by rotating the cylindrical lens 90 • : this orients the laser sheet along the xy plane (along the flow). The lamella then appears as a thin dark vertical line, see Figure 2.b. Note that the initial lamella is in fact a transverse sheet, rather than just a line. This avoids diffusion in the z-direction and enables us to study a purely 2D process. This photo-bleaching technic, relative to a direct injection of fluorescent dye, also insures well-controlled initial conditions: the lamella is uniform and its initial concentration profile is gaussian owing to the gaussian nature of the impacting laser beam.

Immediately after bleaching, the fluid is sheared at the desired shear rate and images of the evolution of the lamella are acquired using a camera (Basler Ace2000-50gm, 2048×1080 pixel 2 , 12 bit) coupled to a high-resolution macroscope (Leica Z16 APO×1). The set-up was designed to provide an image resolution (0.8 µm/pixel) small enough to resolve the Batchelor scale (equation 2.8). For instance, with γ = 0.01 s -1 and s 0 = 25µm, we have s

AD (t B ) = s 0 √ 5(3 γs 2 0 /D 0 ) -1 3 ≈ 10µm.
A high-pass filter (590 nm) is positioned between the sample and the camera to eliminate direct light reflexions. To avoid photobleaching during the image acquisition, the intensity of the laser is lowered to 100 mW and the image acquisition is synchronized with a shutter opening only during acquisition times. Note that all experiments are performed at T = 22 ± 0.05 • C by setting the temperature of the water running through the bottom moving plate with a cryo-thermostat.

Experimental results

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 trans- lating 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 well captured by fitting each concentration profile with a gaussian of the form C(x, t) = C max (t)e -x 2 /σ 2 x (t)

(see Figure 3.b).

The resulting maximum concentration C max (t) and width σ x (t) are plotted in Figure 3.c and 3.d versus time for experiments performed at different Péclet number (4.5 ≤ P e ≤ 1190). The Péclet number was varied by repeating the experiment at various shear-rates γ = [6 × 10 -4 -0.3] s -1 . The agreement with equations 2.4 and 2.6 is very good for both C max (t) and σ x (t). Note that in both cases, γ, 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 / γ is the Batchelor time, these data are found to collapse, for all Pe, on the same master curve, see Figure 3.e and f. For t < t B , C max and σ x remain constant. Then when the effect of molecular diffusion becomes significant, i.e for t > t B , C max (respectively σ x ) starts to decrease (respectively increase) following the power law t -3/2 (respectively t 3/2 ), consistently with the long time trends of equations 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 equation 2.5, s AD (t) ∼ s 0 ( γt) -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 √ 5(3P e) -1 3 , see equation 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 extend 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 [START_REF] Ottino | The Kinematics of Mixing: Stretching, Chaos, and Transport[END_REF], 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 equation 2.10 is in good agreement with the measured concentration distributions obtained for all Péclet number and again, without adjustable parameter.

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 large resolution is not achieved. Moreover, when mixing is investigated in complex systems such as turbulent flows [START_REF] Villermaux | Coarse grained scale of turbulent mixtures Physical review letters[END_REF], porous media (Le [START_REF] Borgne | The lamellar description of mixing in porous media[END_REF] or recently in sheared particulate suspensions [START_REF] Souzy | Stretching and mixing in sheared particulate suspensions[END_REF], 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 2×2 pixels into one, 4×4 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 pixels 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 pixels 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.

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 photo-bleaching technic is used to produce two adjacent lamellae and investigate a single coalescence event. 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) [START_REF] Fourier | Théorie Analytique de la Chaleur[END_REF]. 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, lets recall that P (C) ∼ |dC/dx| -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 large concentrations (corresponding to the maximum of concentration of each lamellae), but also one peak at small 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 small 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.

Discussion and conclusions

We have explored experimentally the basic mechanisms of mixing in stirred flows by thoroughly investigating the evolution of an initially segregated scalar (a lamella of flu-orescent dye) stretched within a laminar shear flow. A high resolution set-up, using a photo-bleaching technic to generate and control the shape of the initial lamella, was built in order to resolve length scales at which diffusion plays a significant role. Our measurements of the evolution of the lamella concentration profiles and concentration distributions are, without adjustable parameter, in excellent quantitative agreement, for P e = [4.5 -1190], with the theoretical predictions.

We also investigated the evolution of the lamella's transverse dimension which conspicuously illustrates the advection-diffusion coupling yielding at intermediate time to a t -1 compression of the lamella dominated by the kinematics of the flow, followed, after the Batchelor time, by a t 1/2 broadening dominated by molecular diffusion. The Batchelor scale, which to our knowledge was experimentally observed only once [START_REF] Meunier | Transport and diffusion around a homoclinic Point[END_REF], was here measured systematically for various Péclet number and found to follow the expected behavior, s AD (t B )/s 0 = √ 5(3P e) -1 3 . Most importantly, through a coarsening protocol, we determined the minimal experimental spatial resolution required to resolve the concentration distribution of a single lamella: its half width must be larger than 1 pixel. This requirement is general and constrains the measurement of any mixing protocol. Indeed, for all stretching protocol, lamellae reach their minimum thickness at the Batchelor time while they are still isolated individual entities. Resolving P (C) at this time, which requires to resolve each individual lamella, therefore is the most demanding in term of spatial resolution: the half width of the lamella at the Batchelor time, s AD (t B ), shall at least span 1 pixel. Note that measuring P(C) at longer times can be less stringent. Indeed, beyond the Batchelor time, lamellae diffuse sufficiently that, when their spacing becomes sufficiently small, nearby lamellae start merging together. The concentration gradients then no longer vary over the lamellae transverse dimension but instead, over a larger length-scale which reflects the size of the bundles of merging lamellae. In the context of turbulent mixtures for instance, this 'coarse-grained scale' was proposed to follow η = LSc -2/5 where L denotes the stirring length and Sc the Schmidt number [START_REF] Villermaux | Coarse grained scale of turbulent mixtures Physical review letters[END_REF]). However, the latter length-scale is only relevant after lamellae have merged which occurs after the Batchelor time. Therefore, resolving P(C) at all times requires an experimental spatial resolution satisfying s AD (t B ) > 1 pixel.

Finally, we have investigated the coalescence between two lamellae and its impact on the evolution of the concentration distribution. Lamellae's overlap gives rise to a nontrivial peak at low concentration. This observation is important as it may be relevant to interpret more complex situations.

To conclude, the high resolution experimental technics developed for the present study and the determination of its limitations, open promising perspectives for future studies on mixing.

Figure 2 .

 2 Figure 2. (color version online) a) Schematic of the set-up. b) Typical image of a lamella obtained by photo-bleaching.

Figure 3 .

 3 Figure 3. (color version online) a) Successive images of a lamella undergoing shear (see also supplementary movie 1). b) Corresponding averaged concentration profiles along the flow (x-direction). The black lines are fitting gaussian profiles. c) Normalized maximum concentration Cmax/C0 and d) normalized half width of the concentration profiles σx/σ0 versus time for experiments performed at different Péclet numbers. The black lines correspond to equation 2.4 in c) and equation 2.6 in d). In both cases, γ, s0 and D0 are set and fixed by the experimental conditions. e) and f) Same data plotted versus t/tB.

Figure 4 .

 4 Figure 4. (color version online) a) Evolution of transverse dimension of the lamella sAD(t)/s0 versus time for experiments performed at different Péclet numbers. The black lines correspond to equation 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 sAD(tB)/s0 versus Péclet number; the black line corresponds to equation 2.8. c) Concentration distribution P (C/C0) measured at successive strains, γt, for a lamella sheared at P e = 4.5. The black lines correspond to equation 2.10 with γ. In all cases, γ, s0 and D0 are fixed by the experimental conditions.

Figure 5 .

 5 Figure 5. (color version online) Distributions of concentration P (C/Cmax) obtained from the progressive digital coarsening of an experimental image.

Figure 6 .

 6 Figure 6. (color version online) 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 γt. Predictions for the evolution of d) the concentration profiles and e) the concentration distributions at the same strains. Again, γ, s0 and D0 are fixed by the experimental conditions.

We thank E. Villermaux and P. Meunier for their inspiring comments and suggestions. We would also like to thank Sady Noel for machining the experimental set-up. This work was supported by ANR JCJC SIMI 9, by the Labex MEC ANR-10-LABX-0092, by ANR-11-IDEX-0001-02 and by ERC consolidator grant ReactiveFronts 648377.


M Souzy

H Lhuissier

E Villermaux

B Metzger

Stretching and mixing in sheared particulate suspensions

Stretching and mixing in sheared

particulate suspensions

Introduction

Sheared particulate suspensions represent a quasi-unique system where efficient dispersion spontaneously occurs even under low Reynolds number flow conditions. For instance, the transfer of heat [START_REF] Sohn | Heat Transfer enhancement in laminar slurry pipe flows with power law thermal conductivities[END_REF], Metzger 2013) or mass [START_REF] Wang | Hydrodynamic diffusion and mass transfer across a sheared suspension of neutrally buoyant spheres[END_REF][START_REF] Wang | Augmented Transport of Extracellular Solutes in Concentrated Erythrocyte Suspensions in Couette Flow[END_REF][START_REF] Souzy | Super-diffusion in sheared suspensions[END_REF] across a suspension of non-Brownian particles is significantly enhanced when the suspension is submitted to a macroscopic shear. This would not happen in a pure Newtonian fluid where the laminar streamlines remain perpendicular to the scalar (heat or concentration) gradients. In a sheared suspension, the macroscopic stationary imposed shear results at the particle scale in an unstationary flow: particles constantly collide with one another, change streamlines and thus generate disturbances within the fluid which promote the dispersion of the scalar, prelude to its subsequent mixing. Two mechanisms have been identified to explain the origin of the transfer enhancement. First, the particle translational shear-induced diffusivity, a phenomenon which has been widely investigated over the last decades [START_REF] Eckstein | Self-diffusion of particles in shear flow of a suspension[END_REF][START_REF] Arp | The Kinetics of Flowing Dispersions IX. Doublets of Rigid Spheres (Experimental)[END_REF][START_REF] Cunha | Shear-induced dispersion in a dilute suspension of rough spheres[END_REF][START_REF] Breedveld | Measurement of the full shear-induced self-diffusion tensor of noncolloidal suspensions[END_REF][START_REF] Sierou | Shear-induced self-diffusion in non-colloidal suspensions[END_REF], Metzger 2013). Second, the particle rotation whose impact is particularly important at the boundaries, where particles disrupt the diffusive boundary layer by a 'rolling-coating' effect [START_REF] Souzy | Super-diffusion in sheared suspensions[END_REF]. These studies mainly focused on the rate of transfer across sheared suspensions which is customary characterized by an effective diffusion coefficient much larger than the scalar molecular diffusivity. Another aspect of transport enhancement concerns the mixing properties of the system, namely its ability, starting from a given spatial scalar distribution, to reach homogeneity. Figure 1 shows how a blob of dye with initial size s 0 , diffuses while it is deformed by the complex flow in the interstitial fluid of a suspension. The important question which naturally arises is to understand how this initially segregated system reaches homogeneity, and particularly how long this process takes. By essence, it involves both advection by the flow and molecular diffusion of the scalar. Such a problem has been studied in a wide range of situations involving a single fluid phase such as shear flows [START_REF] Ranz | Applications of a stretch model diffusion, and reaction in laminar and turbulent flows[END_REF], vortice flows (Meunier 2003), turbulent jets [START_REF] Duplat | A nonsequential turbulent mixing process[END_REF], or flows in porous media (Le [START_REF] Borgne | The lamellar description of mixing in porous media[END_REF]. These studies all underline the crucial importance of the rate at which fluid material lines are elongated by the flow [START_REF] Villermaux | On dissipation in stirred mixtures[END_REF]. The knowledge of these 'stretching laws' allows to estimate the mixing time: the time when the scalar concentration fluctuations start to significantly decay [START_REF] Batchelor | Small-scale variation of convected quantities like temperature in a turbulent fluid. part 1. general discussion and the case of small conductivity[END_REF]. For instance, in a simple shear flow with rate γ, the material lines grow as γt. In the limit of large Péclet number P e = γs 2 0 /D, the mixing time for a scalar blob of initial size s 0 is t mix ∼ γ-1 P e 1/3 , where D denotes the molecular diffusivity of the dye. In chaotic flows, where the stretching rate is maintained, the material lines stretch exponentially, as e γt , and t mix ∼ γ-1 ln P e.

In spite of their crucial importance for mixing issues, stretching laws in particulate suspensions have never been studied experimentally, nor have the general question about the mixing time in such a system been. Stretching in particulate suspension has been addressed indirectly using numerical simulations through the measurement of the suspension largest Lyapunov exponent [START_REF] Dasan | Stress fluctuations in sheared Stokesian suspensions[END_REF][START_REF] Drazer | Deterministic and stochastic behaviour of non-Brownian spheres in sheared suspensions[END_REF], Metzger 2012, Metzger 2013). In such a chaotic system, the mean stretching rate of fluid elements can be assimilated to the largest Lyapunov exponent. The reported positive Lyapunov exponents indicate that the stretching laws must be exponential. Stretching has also been explored theoretically with the motivation of understanding the rheology of such system when the suspending fluid is viscoelastic. It was shown that the expected exponential stretching of the polymers should affect the pressure drop in fixed beds of spheres or fibres [START_REF] Shaqfeh | Polymer stretch in dilute fixed beds of fibres or spheres[END_REF] or the viscosity of freely suspended fibres in a simple shear flow [START_REF] Harlen | Simple shear flow of a suspension of fibres in a dilute polymer solution at high Deborah number[END_REF].

In this paper, we specifically address the question of the stretching kinematics by performing experiments on non-Brownian and spherical particulates suspended in a viscous and Newtonian fluid that is steadily and uniformly sheared. In this limit, the flow kinematics is independent of both the shear rate γ and the molecular diffusivity. The sole parameter expected to affect the stretching process is the particulate volume fraction φ. We investigate the stretching laws in particulate suspensions varying the volume fraction over a wide range 20% φ 55%, for which collective effects between particles are present but the suspension still flows easily, since it is still far from jamming. After presenting the experimental set-up in § 2, we first compare the evolution of a blob of dye sheared in a pure fluid (without particles) to that of a blob sheared in a suspension ( § 3). This experiment illustrates the complexity of the advection field induced by the presence of the particles. Then, following the Diffusive Strip Method of Meunier 2010, accurate velocity field measurements of the fluid phase ( § 2.3) are used to determine the stretching laws. Material lines are found to stretch, on average, exponentially with time ( § 4), at a rate which agrees with the largest Lyapunov exponents reported in 3D Stokesian dynamic simulations [START_REF] Drazer | Deterministic and stochastic behaviour of non-Brownian spheres in sheared suspensions[END_REF][START_REF] Dasan | Stress fluctuations in sheared Stokesian suspensions[END_REF]. Beyond the mean, we tackle the complete statistics of stretching, that is to say, the distributions of elongation as a function of strain and particle volume fraction, which are found to converge towards log-normal distributions. In § 5, we present a model, based on a multiplicative stretching process, which explains quantitatively the experimental distributions of the material line elongation and its dependance to γt and φ. Finally, the crucial implication of these findings for scalar mixing are developed and discussed in §6, before we conclude in §7.

Experimental set-up

The experimental set-up is shown in figure 2. It aims at steadily and uniformly shearing a viscous particulate suspension, injecting a small blob of dyed fluid, and observing both the flow and the mixing of the dye. The set-up consists of a transparent cell in which a transparent mylar belt is tightly mounted at the top of the cell on two cylinders and at the bottom on two ball bearings. One cylinder is entrained by a rotating stage (M-061.PD from PI Piezo-Nano Positioning) with high angular resolution (3×10 -5 rad). The motion of the belt generates in its central region a linear shear flow. The suspension is allowed to flow below the cylinders and a constant spacing between the belt and the inner wall of the cell is maintained all around the cell. This specific design, which is an evolution of that used in Metzger 2012, minimizes secondary flows and ensures a velocity profile with constant shear rate within the belt.

Particles and liquid

The particles and the liquid are carefully chosen to allow the visualization of the dye and of the flow inside the suspension, as well as to ensure a purely viscous flow without buoyancy effects. This requires using a transparent media, matching both the density and the refraction index of the particles, and using a fairly viscous liquid.

To fulfill the above requirements, we use mono-disperse spherical particles (PMMA from Engineering Laboratories Inc.) with density ρ = 1.18 kg/m 3 and diameter d = 2 mm, especially chosen for their smooth surface and good transparency. The liquid is a Newtonian mixture of Triton X-100 (77.4 wt %), Zinc Chloride (13.4 wt %) and water (9.2 wt %) with viscosity η = 3 Pa s and having the same density as the particles at room temperature. Its composition is optimized to match both the refractive index and the density of the particles. A small amount of hydrochloric acid (≈ 0.05 wt%) is added to the solution to prevent the formation of zinc hypochlorite precipitate, thereby significantly improving the optical transparency of the solution. Last, to finely tune the index matching between the particles and the liquid, the temperature of the set-up is adjusted with a water bath surrounding the shear cell.

The solid volume fraction φ of the suspension is varied between 20 and 55%. To ensure that inertial effects are negligible, the shear rate γ is set to typically 0.15 s -1 , which corresponds to a particulate Reynolds number ρ γd 2 /η ∼ 10 -4 .

Imaging

The suspension is observed in the flow-gradient plane (xy plane): a slice of suspension is illuminated by a laser sheet across the transparent belt and imaged from the top (see figure 2).

The laser sheet is formed by reflecting a laser beam (2 W, 532 nm) on a standard laserprinter mirror (rotating at ∼ 10000 rpm). This technique was found to produce a light sheet with a better spatial homogeneity than that obtained with classical cylindrical or Powel lense techniques. The sheet is collimated and focused to a thickness of ∼ 60 µm with the help of two perpendicular plano-convex lenses. Last, a high-pass filter (590 nm) eliminates direct light reflexions. The suspension is imaged with a high-resolution camera (Basler Ace2000-50gm, 2048x1080 pixel 2 , 12bit) coupled to a high-quality magnification lens (Sigma APO-Macro-180 mm-F3.5-DG). To avoid that the particles distord the free surface of the suspension through which the visualization is realized, a small plexiglass window is positioned on the free surface, above the region of interest, which locally ensures a flat interface. The window has a small hole allowing the injection of a blob of dyed fluid with a syringe.

Velocity field measurements

The velocity field in the suspending liquid is measured in the plane of the laser sheet (xy), at half-distance between the bottom and the free surface (see figure 3), performing particle image velocimetry (PIV), which yields the two-dimensional velocity field {u, v}, which does not necessarily verify incompressibility. To perform PIV, the liquid is seeded with small passive fluorescent tracers (3.23 µm PMMA B-particles from MF-Rhodamine) at a very low volume fraction (∼ 10 -5 φ). These small and diluted tracers do not affect the flow but allow its visualization and quantification, as shown in figure 3 and Movie 1. The large (2 mm) particles of the suspension do not interact with the laser sheet and appear as black discs. Note that all the particles have the same size; the apparent size differences arise from their different vertical positions relative to the laser sheet plane. The PIV routine is adapted from a Matlab code developed by Meunier 2003. Images are captured every 0.1 s, which corresponds to a strain increment of 0.015. To perform PIV, the images are divided into equally spaced and overlapping sub-images with a typical size of d/20 (32 pixels). The local velocity field is computed by cross-correlating successive sub-images. The presence of a particle in the sub-image is detected with the help of two filters (for the maximum of correlation and for the standard deviation of the sub-images), in which case the corresponding velocity vector is not used (see figure 3b). For each volume fraction, three independent runs over a strain of 20 are performed.

The independence of the measured velocity field on the PIV sub-image size was verified by decreasing the latter to ∼ d/40 (16 pixels). Besides increasing the data noise, no significant effect was found on the measured velocities.

Molecular diffusivity measurements

The molecular diffusion coefficient of the dye (rhodamine 6G) is measured by observing the spreading, in the absence of flow, of a slice of liquid depleted in dye. A small Hele-Shaw cell (100 µm thick) is filled with dye-doped suspending liquid without particles. A thin slice of liquid is initially depleted in dye by bleaching the dye with a high power laser sheet across the cell (see figure 4). The depleted slice appears as a dark line having a gaussian profile which diffuses with the diffusion coefficient of the dye. The spatial variance of the gaussian profile χ 2 is measured over one day, and the diffusivity is determined from D = [χ 2 (t)χ 2 (0)]/2t 1.44 10 -13 m 2 s -1 . This value is consistent with that of 4.14 10 -10 m 2 s -1 found by [START_REF] Culbertson | Diffusion coefficient measurements in microfluidic devices[END_REF] for the diffusivity of the same dye in water, given that water is 3000 times less viscous than the suspending liquid and that, according to the Einstein-Sutherland law, D ∝ 1/η.

General observations

To illustrate the influence of particles on mixing in a shear flow, we first compare the evolution of a blob of dye sheared in a pure fluid (without particles) to that of a blob sheared in the suspension of particles, see Movie 2. A cylindrical blob of dyed fluid is injected, at rest and at t = 0, in the middle of the shear cell. Initially, the blob has a diameter s 0 2 mm, is aligned with the vorticity direction and is centered on the neutral velocity plane. This results in a macroscopically two-dimensional initial configuration, and ensures that the blob does not drift with the flow but only deforms. Figure 5 shows, for a Péclet number P e ≈ 10 6 , how mixing proceeds in the two sheared media, from the initial segregated state up to a strain γt = 20. In the pure liquid, the blob of dye stretches homogeneously. Its length increases linearly with time and the blob transverse dimension thus decreases as 1/t. In the suspension, the situation is markedly different: the fluctuations induced by the particles in the fluid phase strongly impact the evolution of the blob. Several conspicuous features deserve being highlighted i) the dispersion and the unfolded length of the blob are significantly enhanced by the particles, ii) both translational diffusivity (transverse undulations of the blob, see figure 5b) and rotation (blob winding around particles, see figure 5c) of the particles contribute to these enhancements, iii) the blob stretching is highly inhomogeneous: at some locations, its transverse dimension becomes much thinner and at others larger than in the pure fluid case, revealing regions of enhanced stretching and regions of compression, iv) at large strains (figure 5d), the blob has separated into several filaments which means that some regions of the blob have already mixed, while in the pure liquid (without particles) mixing has not occurred yet, v) In some regions, the blob evolves into bundles composed of several nearly overlapping filaments [START_REF] Duplat | Mixing by random stirring in confined mixtures[END_REF]. This suggests an underlying stretching/folding mechanism similar to the well known baker's transform [START_REF] Ottino | The Kinematics of Mixing: Stretching, Chaos, and Transport[END_REF].

The above features are generic to the flow of a viscous suspension at large Péclet number. Since inertial effects are negligible, these features are independent of the rate γ at which the suspension is sheared. Similarly, the value of the Péclet number does not influence the general stretching pattern of the blob, but only prescribes the strain γt at which diffusion starts to becomes effective.

This direct comparison clearly illustrates how the liquid velocity fluctuations generated by the particles dramatically accelerate the blob deformation and dispersion. This acceleration is apparent here from the beginning of the shear, because the blob size s 0 is similar to the particle size d. It is however crucial to realize that the strain at which this acceleration establishes is expected to depend on the ratio s 0 /d. If the initial size of the blob s 0 is larger than d, the blob is essentially not stretched by the particle fluctuation motions. It is thus essentially stretched by the linear macroscopic shear until the blob transverse size has thinned down to d, after a typical strain s 0 /d. From that strain on, the particulate fluctuations are expected to contribute directly to the blob stretching.

In stirred flows, such as the case considered here, mixing results from the coupling between advection and molecular diffusion. In the experiment described above, the blob of dye is stretched by the local velocity field: the blob is stretched along its own longitudinal direction and conversely compressed along its transverse direction. The blob thus evolves towards a topology constituted of sheets, or filaments [START_REF] Ottino | The Kinematics of Mixing: Stretching, Chaos, and Transport[END_REF][START_REF] Buch | Experimental study of the fine-scale structure of conserved scalar mixing in turbulent shear flows[END_REF]. Conversely to the effect of advection, molecular diffusion tends to broaden the filaments. This diffusive broadening will at some point counter-balances the rate of compression of the blob caused by the advection. As we already mentioned, this naturally sets a time-scale called the mixing time, t mix , beyond which the concentration levels drop significantly. The mixing time, a key element to understand the overall mixing process, can be estimated from the sole knowledge of the dye molecular diffusion coefficient and from the history of the transverse dimension of the blob. If one assumes that the flow is two-dimensional (this assumption is discussed in 5.3), incompressibility and mass conservation relate at any time the transverse size of the blob to its length l, through s 0 l 0 = s(t)l(t). The mixing time can therefore be estimated from the characterization of the evolution of l(t). Our goal in the following is thus to determine the so-called 'stretching laws', i.e., the time dependence of l in sheared particulate suspensions.

Experimental stretching laws

Our first attempt to measure the unfolded length of the blob l(t) was naturally to perform direct image analysis on images such as those shown in Figure 5. However, the intrinsic dispersion process rapidly distorts the blob into bundles of very close (sometimes merging) filaments, which renders image analysis ineffective above strains of typically 5.

To overcome these limitations, we adopted a different approach inspired from the Diffusive Strip Method. This method happens to be a very powerful experimental tool allowing the determination of the stretching laws over unlimited strains. The key idea is to use the experimental fluid velocity field to numerically advect passive material lines representing portions of the blob. The lines are initially composed of three passive tracers separated by a distance d/20 which discretize a fluid material line. The lines are randomly located in the two-dimensional velocity field with a random orientation (see figure 6a). Each tracer with coordinate x is advected independently from each other according to the local fluid velocity v(x) (obtained by linear interpolation of the instantaneous PIV velocity field) as x(t + ∆t) = x(t) + v(x)∆t, where ∆t is the time between consecutive measurements of the velocity field. As a material line is advected, it is refined by adding more tracers when its length increases or when its local curvature becomes too large (see Meunier 2010 for a detailed description of the refinement procedure).

Figure 6 shows the evolution of two material lines up to a strain of 15, see also Movie 3. The red line successively stretches and folds very similarly to what is observed in the blob experiments (figure 5). Interestingly, the blue line behaves very differently. Although it sustains the same macroscopic strain as the red one, it experiences a much softer stretching only because it started from a different initial location. These different stretching histories reveal the stochastic nature of the stretching induced within particulate suspensions. The stretching laws therefore have to be sought in a statistical sense by repeating the advection procedure over a large number of independent material lines. However, as the material lines lengthen, they may reach the boundaries of the measured velocity field, which limits the maximum strain that can practically be investigated (typically γt < 10). This problem is easily circumvented by realizing that, as long as the stretching laws are concerned, the object of interest is not the material line as a whole but rather the small segments which compose this line and which all stretch differently from each other. We thus perform a new set of calculations focusing on segments: i) initial segments (composed of two tracers) with length d/20 are positioned and oriented randomly in the flow, ii) each time the length of a segment doubles, it is splited in two individual segments that are subsequently advected independently, iii) if a segment reaches the boundary of the velocity field, it is re-injected near the center of the velocity field, iv) when a segment overlaps with a particle, where the velocity field is undefined (as can happen due to the finite time ∆t), it is frozen until the particle has moved away. Owing to these rules, virtually unlimited strains can be considered, and the stretching history of each segment that have been created over this strain can be determined. We define the elongation of these segments as the ratio

ρ(t) ≡ δl(t)/δl 0 (4.1)
of their current length δl(t) to their initial length δl 0 where δl 0 = (d/20)/2 n , with n the number of times the sub-segment was splited in two. Note that to compute the distributions of elongations we present below, the contribution of each segment is weighted by its initial length. Note also that times for which a segment is frozen are not considered. The distribution of elongations at time t therefore represents the portion of the blob that has reached a given elongation after being advected for a duration t. It was built from the stretching histories of 25000 segments advected over 3 independent experimental velocity fields, each of them recorded for a total strain of 20 (typically 4000 images).

Figure 7 shows the experimental stretching laws obtained for a suspension with φ = 35 %, which is generic to the volume fraction range 20 to 55 % investigated. It presents the mean value ρ and the standard deviation σ ρ ≡ ρ 2ρ 2 of the elongation for strains up to 20. At γt = 20, the segments have on average lengthened by typically 10 3 , which is about one hundred times larger than in the case of a pure liquid. The striking result is that the presence of particles in a shear flow changes the very nature of the stretching laws from linear to exponential. Indeed, the elongation of material line in a simple shear (without particles) follows

ρ lin (θ, t) = 1 + 2 cos θ sin θ γt + sin 2 θ γ2 t 2 , (4.2)
where θ denotes the angle between the line initial orientation and the flow direction. On averaging ρ 2 lin (θ, t) over all possible orientations, we obtain

ρ lin (t) = 1 + γ2 t 2 /2, (4.3)
which is only of order 10 for a strain of 20, and increases linearly with time for large strains. Equation (4.3) is ploted in figure 7 to illustrate the contrast with the elongations actually measured in particulate suspensions: the mean elongation in suspensions is different both in magnitude and in law. Moreover, by contrast with the pure fluid case, the stretching variability of individual material lines is very broad as evidenced by the exponential growth of the standard deviation σ p . These results corroborate the preliminary blob experimental visualizations where, in the suspension, many filaments having very different transverse thickness can be observed while the pure fluid case solely exhibits one uniform thickness, (figure 5d). More precisely, figure 7b shows the distributions of the relative elongations P (ρ/ ρ ) at successive strains. The distribution of elongations broadens rapidly such that at a strain γt = 20, it spans more than eight decades. At that strain, the right tail of the distribution contains segments elongated 10 4 times relative to the average ρ , which corresponds to an absolute elongation of ρ ∼ 10 7 , in stark contrast with the uniform average elongation of 10 obtained in a simple shear. As figure 7b shows, these distributions are found to be well fitted by log-normal distributions (shown as dashed lines). Note that the apparent absence of data on the left hand side of the distributions is fully consistent with lognormal distributions. Indeed, for broad distributions, the statistical weight of the left hand side of the distribution vanishes. Our data thus fully resolve the meaningful part of the distribution.

The advective strip method presented above was repeated with velocity fields measured in suspensions with different volume fractions φ ranging from 20% to 55%. The same trends as those detailed for φ = 35 % are systematically observed. As shown in figure 8, it is moreover found that larger particulate volume fractions increase both the growth rate of the average elongation ρ and that of the standard deviation σ ρ . This indicates that a larger volume fraction results in larger fluid disturbances which, in turn, induce a faster and more random elongation of the fluid material lines. Fitting these curves with exponential growths in strain e κ γt yields for ρ , κ ρ = 0.09 + 0.74 φ, and for σ ρ , κ σρ = 0.12 + 1.03 φ. In the range of volume fraction investigated, the growth rates are found to increase linearly with φ. No measurements could be performed above 55% as the large normal stress built in the suspension starts to deform the belt.

To summarize, by kinematically advecting passive segments using the experimental velocity fields of the fluid phase, we measured the elongation of fluid material lines in sheared particulate suspensions. Two important features characterize these elongations: i) the mean and the standard deviation grow exponentially, ii) the distribution converges to a log-normal. In the following, using two measurable properties of the fluid velocity field, namely the local shear rate distribution and the Lagrangian correlation time, we present a mechanism accounting for these observations.

ρ φ = 20% φ = 25% φ = 30% φ = 35% φ = 40% φ = 45% φ = 50% φ = 55%

Origin of the stretching laws

Principle

We consider the elementary component of a fluid material line: the segment, see figure 9. At that scale, much smaller than the particle size, the local shear γloc is uniform.

Considering the broad distribution of the segment orientations, we assume that the local shear rate has a random orientation with respect to the segment. Therefore, as long as the local shear rate γloc persists, the average elongation of the segment is (see equation 4.3)

ρ = 1 + γ2 loc t 2 /2.
(5.1)

Note that an individual segment can be stretched or compressed depending on wether it is located in a diverging or compressive region of the flow, respectively. However, once averaged over all possible orientations, the segment net elongation is strictly larger than unity. Two questions then naturally emerge: what are the local shear rates? and how long do these shear rates persist? In the following two sections, we address these questions by providing information about the local shear rates and the Lagrangian correlation time of the velocity field.

0 2 4 φ = 20% φ = 25% φ = 30% φ = 35% φ = 40% φ = 45% φ = 50% φ = 55% 0

Local shear rate

We measure the local shear rate from the experimental two-dimensional velocity fields. To this end, we define the local shear rate by the norm of the symmetric part of the strain tensor:

γloc = 2 ∂u ∂x 2 + 2 ∂v ∂y 2 + ∂u ∂y + ∂v ∂x 2 , (5.2)
where {u, v} are the {x, y} components of the velocity field. This definition disregards the rotation part of the strain tensor. For a simple shear, one has γloc = cte = γ. Figure 10a shows a typical local shear rate map, obtained in a suspension with volume fraction φ = 35%. The color-scale represents γloc amplitude normalized by the applied macroscopic shear rate γ, that is to say the amplification of the shear due to the presence of particles. The local shear rate is highly non-uniform and its value can greatly exceed the macroscopic shear rate. Interestingly, large local shear rates occur preferentially in the vicinity of the particles, however there is no apparent correlation between large local shear rates and small inter-particle distances.

φ = 20% φ = 25% φ = 30% φ = 35% φ = 40% φ = 45% φ = 50% φ = 55% fit for φ= 40% 0 V/ γd 0 1 2 -2 -1 φ = 20% 0 1 2 -2 -1 1 2 4 3 5 0 1 2 4 3 5 V V γt γt φ = 50% a) b) c) d) V/ γd
More quantitatively, we report in figure 10b the distribution of normalized local shear rates obtained for various volume fractions. Clearly, the local shear rate exceeds most of the time the imposed macroscopic shear rate, sometimes by one order of magnitude, and this trend accentuates with increasing volume fractions. The mean normalized value γloc / γ is plotted versus φ in the inset of figure 10c. It is found to fit well γloc / γ = A/(φ cφ) δ . Fixing φ c = 0.58 this yields A 0.56 and δ 0.7. Note that the last point, corresponding to φ = 55%, was not included in the fitting procedure since we suspect that it is biased by the deflexion of the belt mentioned above. Note also that PIV using smaller boxes resulted in very similar local shear rate distributions with less than 6% difference on the average. The trends discussed above may also be interpreted in terms of a macroscopic viscosity. In such case, the relevant quantity to investigate is the second moment of the local shear rate distribution γ2 loc [START_REF] Chateau | Homogenization approach to the behavior of suspensions of noncolloidal particles in yield stress fluids[END_REF], Lerner 2012, Dagois-Bogy 2015). Values of this quantity have recently been obtained by Trulson et al. from numerical simulations of dense frictional suspensions [START_REF] Trulsson | Effect of Friction on Dense Suspension Flows of Hard Particles[END_REF]. They report that γ2 loc / γ ∼ (J/µ) -1/3 , where J = γη f /P is the viscous number, with P the confining pressure, and µ the suspension macroscopic friction coefficient. Since η s /η f = σ/ γη f = µ/J, this results in γ2 loc / γ ∼ (η s /η f ) 1/3 . Combining this with η s /η f ∼ (φ cφ) -2 and using φ c = 0.58 [START_REF] Boyer | Unifying suspension and granular rheology[END_REF]) leads to γ2 loc / γ ∼ (φ cφ) -2/3 , which is in fairly good agreement with the measured scaling, as figure 10c shows.

Lagrangian correlation time

The second important quantity of the suspending liquid flow is the persistence time of the velocity fluctuations induced by the particles. Figures 11a andb show the transverse Lagrangian velocity V (perpendicular to the flow) of a passive tracer advected by the fluid at a low and a large volume fraction, respectively. Consistently with the magnitude of the local shear, more concentrated particulate suspensions develop velocity fluctuations with larger amplitudes. However these fluctuations are found to persist on a much shorter time as φ increases. The duration t c for which a segment is coherently stretched by the flow is directly prescribed by this persistence time, which we define from the Lagrangian velocity auto-correlation functions. As shown in figure 11c, these functions decorrelate exponentially with strain. In the range of volume fraction investigated, the dimensionless correlation time γτ , inferred from this exponential decay, decreases linearly with φ as γτ 0.62 -1.08 φ,

(5.3) (see figure 11d). We expect t c to be of the order of τ and thus write (5.4) with α an order one constant. Note that as shown in figure 7, this persistence time ( γ-1 ) is much shorter than the observation period ( 10 γ-1 ).

t c = ατ,

Multiplicative stretching process

With informations about the local shear rates and their persistence time at hand, we now explain the elongations of fluid material lines as a sequence of uncorrelated cycles of stretching. During the first cycle of duration t c , a given segment of a material line is elongated by the local shear rate γloc,1 resulting in a stretching

∆ρ 1 = 1 + ( γloc,1 t c ) 2 /2, (5.5)
where γloc,1 is a local shear rate which probability is prescribed by the distribution P ( γloc / γ), cf. figure 10b. After the duration t c , the local velocity field de-correlates and the local shear rate map is entirely redistributed. The segment then experiences a new local shear rate γloc,2 , which at t = 2t c yields ρ = ∆ρ 1 1 + ( γloc,2 t c ) 2 /2, and so on. The total elongation at time t, after N = t/t c cycles, is the product of all the elementary elongations occurring at each cycle ρ(t) =

N =t/tc i=1 ∆ρ i . The logarithm of this expression can be written as a sum

ln ρ ≡ t/tc i=1 ln ∆ρ i = 1 2 t/tc i=1 ln[1 + ( γloc,i t c ) 2 /2].
(5.6)

Since the elementary stretchings are independent, the distribution of ln ρ is expected, by virtue of the central limit theorem, to be normal. This multiplicative stretching model thus predicts ρ to converge, after a few t/t c cycles, to a log-normal distribution. This prediction is in agreement with the experimental results shown in figure 7b. The distribution of ln ρ, i.e. the normal distribution, writes

P (x = ln ρ) = 1 √ 2πσ e -(x-µ) 2 2σ 2 , (5.7)
with a non-zero mean µ ≡ ln ρ = ln ∆ρ γt c γt, (5.8) and variance

σ 2 ≡ ln 2 ρ -µ 2 = ln 2 ∆ρ -ln ∆ρ 2 γt c γt.
(5.9)

Both the mean and the variance of the distribution of ln ρ increase linearly with time.

They also vary with the particulate volume fraction due to the φ-dependence of γloc and t c . This variation with φ is better appreciated by recasting equations (5.8) and (5.9) into µ = f (φ) γt, (5.10) (5.11) with f (φ) ≡ ln ∆ρ / γt c and g(φ) ≡ ( ln 2 ∆ρln ∆ρ 2 )/ γt c only depending on φ. Note that f (φ) and g(φ) are crucial quantities. Since the time dependency is known, they contain all the information about the asymptotic of the stretching laws in suspensions. The multiplicative stretching model not only explains the origin of the log-normal distributions of elongations measured experimentally, but also the exponential increase of the mean elongation ρ and variance σ 2 ρ shown in figure 7. Indeed, the mean and variance of the (log-normal) distribution of ρ can be deduced from the mean and the variance of the (normal) distribution of ln ρ following ρ = e (f +g/2) γt , (5.12) and σ 2 ρ = (e g γt -1)e (2f +g) γt e 2(f +g) γt , (

σ 2 = g(φ) γt,
the last simplification in σ 2 ρ becoming true after a few t c . Furthermore, the particulate volume fraction dependence of f (φ) and g(φ) can be computed from the persistence time t c and the distribution of local shear rates, using equations (5.10) and (5.11) together with equations (5.8) and (5.9). In the experimental range 20 % < φ < 55 %, this yields f (φ) 0.104 + 0.298 φ,

(5.14) and g(φ) -0.069 + 0.810 φ, (5.15)

with the structure constant α, set once and for all φ, to 0.3 for computing µ, and to 3.9 for computing σ 2 . These rates f and g both increase with φ, in agreement with the experimental trends. Note that this dependence on the volume fraction is non-trivial, since f and g result from the product of γloc and t c , which have opposite trends with φ: the former increases whereas the latter decreases with increasing φ.

The predictions of the multiplicative stretching model are compared to the experimental stretching laws obtained by the Diffusive Strip Method in figure 12. The agreement is good for all volume fractions and all strains, which suggests that the multiplicative stretching model presented above captures the relevant mechanisms at the origin of the stretching laws.

Comments on the stretching process

Stretching of material elements in nature, may they be passive like in the present case, or with internal restoration forces like polymers [START_REF] Shaqfeh | Polymer stretch in dilute fixed beds of fibres or spheres[END_REF][START_REF] Afonso | Nonlinear elastic polymers in random flow[END_REF] may have different origins. The stochastic models to describe them usually present a net drift, and a random noise terms. The relative amplitudes of these two contributions are in our analysis given by f (φ) and g(φ), respectively (see equations (5.10) and (5.11)). The first term sets the growth of ln ρ , while the second sets that of ln 2 ρln ρ 2 . At a 10 0 10 2 10 4 10 0 10 2 10 4 10 1 10 0 10 3 10 2 10 1 10 0 10 2

φ= 20% φ= 25% φ= 30% φ= 35% φ= 40% φ= 45% φ= 50% φ= 55% c) d) σ model ρ σ exp ρ ρ exp ρ model 0.2 0.3 0.4 0.5 φ 0.2 0.4 0.6 0.4 0.8 1.2 0.2 0.3 0.4 0.5 φ 1.6 a) b) f + g 2 2f + 2g
Rates from advective strip method Model: Model:

0.8 Figure 12.
Comparison between the experimental stretching (extracted from figure 8) and the multiplicative stretching model. (a-b) Mean elongation ρ and standard deviation σρ. The values from the advective strip method are plotted versus those predicted by the multiplicative stretching model. (c) Comparison between the exponential rates of the mean elongation κρ obtained from the advective strip method (see figure 8a) and the model prediction f + g/2 (5.12). (d) Comparison between the exponential rates of the variance of the elongation 2κσ ρ (see figure 8b) and the model prediction 2(f + g) (5.13).

microscopic level, the growth of a given material line depends on its orientation with that of the local velocity gradient. The line length l(t) may increase or decrease depending on wether it is aligned with a diverging or compressive region of the flow. For instance, in a flow corresponding to the pure Brownian motion limit [START_REF] Cocke | Turbulent Hydrodynamic Line-stretching: The Random Walk Limit[END_REF], for which ρ/ρ = B(t) with B(t) a zero-mean, Delta correlated noise, i.e. B(t) = 0 and B(t )B(t ) = (1/τ 0 )δ(tt ), these two contributions are balanced and the net line growth d ln ρ(t)/dt = l(t)/l(t) is, on averaging over all directions, identically zero. In that case, representative of t c → 0, the material lines only grow through the contribution of the fluctuations of B(t), which results in d ln ρ /dt = 0, and (ln ρ) 2 ∼ 2t/τ 0 : the logarithm of the elongation diffuses.

In particulate suspensions, the situation is different since if the direction of the stretch indeed changes at random, it has a finite persistence time. In such case, it has been shown that material lines tend to preferentially align in the direction of elongations (see [START_REF] Cocke | Turbulent Hydrodynamic Line Stretching: Consequences of Isotropy[END_REF][START_REF] Orszag | Comments on 'Turbulent Hydrodynamic Line Stretching: Consequences of Isotropy[END_REF][START_REF] Girimaji | Material-element Deformation in Isotropic Turbulence[END_REF][START_REF] Duplat | Persistence of Material Element Deformation in Isotropic Flows and Growth Rate of Lines and Surfaces[END_REF] and also equation 5.5). Thus, over an observation period larger than the (non-zero) correlation time t c , we expect ln ρ in figure 13(b)), on average, the logarithm of the elongations ln ρ increases with time. Material lines in particulate suspensions thus grow from the contribution of both a drift and a noise. The stretching process thus corresponds to a noisy multiplicative sequence of correlated motions, like the random Sine Flow [START_REF] Meunier | The diffusive strip method for scalar mixing in two dimensions[END_REF], or porous media flows (Le Borgne 2015). Porous media and sheared particulate suspensions have similar exponential stretching laws. This is true in 3D systems as in both cases the fluid trajectories are chaotic. Note however that for 2D systems the implications of steadiness change the picture qualitatively. In a 2D porous media, the flow is steady and there are only two degrees of freedom : the flow is thus not chaotic. The elongation of material lines in 2D synthetic porous media have been shown to grow algebraically rather than exponentially (Le [START_REF] Borgne | The lamellar description of mixing in porous media[END_REF]. Conversely in 2D sheared suspensions, the time dependence of the flow allows the system to be chaotic (Metzger 2013). One therefore expect to observe exponential stretching laws in sheared particulate suspensions also in purely 2D configurations.

Further remarks

We would like to point out certain limitations of the present study. First, the present findings and their analysis are restricted to the particulate volume fraction 20 % φ 55 %, for which material lines in the suspending liquid stretch exponentially with strain. This is not necessarily the case outside form this range. In particular, as φ → 0, this exponential trend must cross-over to linear since the elongation of material lines in a simple shear is linear with strain. We however anticipate that the exponential trend could hold down to fairly low volume fractions but only emerge after increasingly large strains, since the velocity correlation time in the dilute limit should follow τ ∼ ( γφ) -1 and diverge at low φ. Further investigations are needed to characterize this dilute regime (φ < 20%). Second, the PIV measurements performed here are two-dimensional and provide the fluid velocity projected in the (xy) plane only. They therefore neglect part of the stretching of the material lines, namely that involving deformations in the vorticity direction (z). However, we believe that they resolve the stretching mechanism and most of its magnitude for the following reasons: i) these measurements resolve the fluid displacements in the gradient direction (y), which is the only direction for which displacements couple with the main shear flow to produce an enhance stretching. The fluctuations in the vorticity direction are thus expected to produce less stretching than those occurring in the gradient direction. ii) Particles in a shear flow rotate mainly about the vorticity axis thereby inducing fluid disturbances mostly in the velocity-gradient plane, which we consider. Here again, the effects of the velocity disturbances induced by the particle rotation should be smaller in the vorticity direction than those occurring in the velocity-gradient plane. iii) More quantitatively, the stretching rates f + g/2 predicted by the present model based on 2D data are in good agreement with the largest Lyapunov exponents obtained from 3D Stokesian simulations, see figure 14. From the above considerations, it is likely that the mechanisms at the origin of the scalar dispersion, stretching and subsequent mixing are well characterized by the present measurements, even though those are limited to the information contained in the xy plane. Third, as already mentioned in section 3, the stretching of material lines is exponential at every scale, but the stretching of a material blob with thickness s 0 is expected to follow that of material lines only if its thickness is smaller than the correlation scale of the fluid motion, which is of order d (in the other case, the blob is first essentially stretched by the macroscopic shear γ until s d). In the following, we will therefore only consider the relevant case s 0 d.

The latter considerations have important consequences on the estimation of the blob thickness s, hence on the mixing time that we will address in the next section. For an arbitrary elongation w/w 0 in the vorticity direction (z), mass conservation gives s 0 l 0 w 0 = s(t)l(t)w(t). However, in light of the above discussion, the flow is assumed to be two dimensional with w/w 0 l/l 0 . Mass conservation thus results in s 0 l 0 = s(t)l(t).

(5.17)

Using direct image analysis, we have checked that this is experimentally verified. A blob with initial surface s 0 l 0 being converted into a strip with length l(t) and thickness s(t) The median (blue line), the most stretched t 3% mix (green line), and the less stretched t 97% mix (red line) dimensionless mixing times can be compared to the dimensionless mixing time ∼ P e 1/3 expected in a pure fluid (dashed line).

indeed obeys, before it starts mixing, to equation (5.17), suggesting that the flow is indeed area preserving.

Implications for mixing

In such area preserving flow, the thickness s(t) of a distorted blob decreases in inverse proportion of its length l(t) according to equation (5.17). As recalled in the introduction, the mixing time for a given blob portion of thickness s is reached when its compression rateṡ/s is balanced by its rate of diffusive broadening D/s 2 . At that time, called the mixing time, the scalar concentration carried by that portion of the blob starts to significantly decay i.e., mix. Since in particulate suspensions ρ = l(t)/l 0 = e κ γt , the mixing time writes t mix γ-1 ln(κP e)/(2κ). We also found that the logarithm of the elongations of an ensemble of such material lines is normally distributed with a mean and a variance growing linearly with time following µ = ln ρ = f (φ) γt and σ 2 = g(φ) γt (see equations (5.10) and (5.11), respectively). These results are illustrated in figure 15a. Since, similarly to the logarithm of the elongations, the stretching rates, κ γ = ln ρ/t are normally distributed, the median mixing time, obtained for the mean stretching rate, i.e. for κ γ = ln ρ /t = µ/t = f (φ) γ, is

t med mix ≈ 1 2f (φ)
γ ln(f (φ)P e). (6.1)

Considering a blob distorted in such a way that it samples all the possible elongations in the global statistics, the above estimate provides the time at which half of the blob has reached its mixing time. The logarithmic dependence of the mixing time on the Péclet number is different from that obtained in a simple shear flow (without particles) for which ρ γt yields t mix γ-1 P e 1/3 . Introducing particles in a viscous fluid therefore becomes more and more efficient at reducing the mixing time as the Péclet number increases. In the present study, the Péclet number is P e ∼ 10 6 . The median mixing time for φ = 35 % is thus t med mix 30/ γ, which has to be compared with t mix 100/ γ in a pure shear flow. Note that varying the volume fraction from 20 % to 55 % increases f (φ) only by a typical factor of 2, which decreases the median mixing time by about the same moderate factor.

In practical situations, mixing half of the scalar may not be the relevant question, precisely because in particulate suspensions, elongations are, as seen in figure 5, broadly distributed. So are mixing times. To address this point, we estimate, for the same conditions as previously, the mixing times for the portions of the blob that undergo the largest and the lowest stretching rates respectively, i.e. the mixing times corresponding to both tails of the distribution (highlighted in figure 15a). The 3 % most strongly stretched portions of the blob are bounded by ln ρ = µ + 2σ. The expressionṡ/s = D/s 2 results in 2f (φ) γt + 4 g(φ) γt = ln[(f (φ) + g(φ)/ γt)P e], which yields the mean field mixing time t 3% mix 14/ γ. On the other end of the distribution, the less stretched portions of the blob, bounded by ln ρ = µ -2σ, reach their mixing time at t 97% mix 64/ γ, later than if it were sheared in a pure fluid. In figure 15b, the median (blue line), the most stretched t 3% mix (green line), and the less stretched t 97% mix (red line) dimensionless mixing times are plotted as a function of the Péclet number. This shows that if the concern is to mix essentially all the scalar, large Péclet numbers ( 10 5 ) are required before mixing in a suspension becomes more efficient than in a pure fluid. Persistent poorly stretched regions are deterring. The relative width σ/µ of the stretching rate distribution decreases in time like t -1/2 but this only mildly decreases the spreading of the mixing times as P e increases, since t mix ∝ ln P e. For instance, at P e = 10 20 , the mixing times remain fairly distributed with t 97% mix /t 3% mix > 2. Finally, the results obtained on the stretching laws must be related to the overall dispersion of the blob. In a random flow, line stretching, and dispersion, are two different things: because the extent of the area occupied by the blob grows more slowly than the area where the scalar constitutive of the blob is dispersed, the blob will at some point unavoidably reconnect and merge by overlapping onto itself [START_REF] Duplat | Mixing by random stirring in confined mixtures[END_REF]. Let us see how: after the mixing time, a scalar blob with length growing like l(t) = l 0 e γt has a transverse concentration profile whose width is confined to the Batchelor scale √ Dt. The area A occupied by the scalar is thus A = √ Dt l 0 e γt , growing exponentially in time. Now, the spatial support of the blob undergoes a dispersion induced by the particle effective dispersion coefficient D eff ∼ γd 2 [START_REF] Eckstein | Self-diffusion of particles in shear flow of a suspension[END_REF]. The total area explored by the blob of dye, within which the blob folds, is typically (see also [START_REF] Taylor | Dispersion of soluble matter in solvent flowing slowly through a tube[END_REF] in a related, but different context), Σ ∼ (l 0 + √ D eff t) × (s 0 + √ D eff t) γt ∼ d 2 ( γt) 2 , growing algebraically in time. Because an exponential will always beat a power law, there will necessary be a time for which the area occupied by the scalar overcomes that visited by the blob (i.e. Σ/A < 1), and from that instant of time, overlaps of the folded scalar filaments will be unavoidable. Such an event is illustrated in figure 16. These overlaps will locally delay the mixing process and therefore affect the whole route of the mixture towards homogenization. This aspect, and more generally all aspects regarding the concentration content of the mixture and its evolution, are left for future research.

Conclusions

Motivated by the need to understand on a firm basis the mixing properties of particulate flows, we have provided a complete characterization of the kinematics of stretchings and consecutive elongations of materials lines in non-Brownian particulate suspensions under a simple macroscopic shear. Our observations rely on high resolution PIV measurements of the interstitial fluid velocity field, and our findings are as follows:

i) Following the Diffusive Strip Method of Meunier 2010, we used the experimentally measured velocity fields to numerically advect passive segments in order to reconstruct the stretching histories of fluid material lines. In agreement with previous theoretical predictions and simulation results, we observe that adding particles in a shear flow changes the very nature of the stretching laws from linear to exponential in strain. The growth rate for the mean elongation are found to closely agree with the largest Lyapunov exponent obtained from 3D numerical simulations [START_REF] Drazer | Deterministic and stochastic behaviour of non-Brownian spheres in sheared suspensions[END_REF][START_REF] Dasan | Stress fluctuations in sheared Stokesian suspensions[END_REF]. Besides the mean, our analysis also provides the full statistics of the material lines elongation: the variances of the elongations also grow exponentially in strain and the distributions of elongations converge toward log-normals. This statistics of elongation was characterized for a large range of volume fractions 20% φ 55%.

ii) Using the same velocity fields, we determined the distribution of the local shear rates intensities and their persistence time. From these, we have shown how the fluid material lines undergo a multiplicative stretching process consisting in a noisy multiplicative sequence of correlated motions. We also discussed the important role of the finite correlation time of the velocity field. The model quantitatively predicts the evolution of the mean and the variance of the elongations of the fluid material lines as well as their evolution towards a log-normal distribution.

iii) We have discussed the importance of this characterization of the flow kinematics to understand how mixing proceeds in sheared particulate suspensions. The exponential stretching results in a mixing time increasing logarithmically with the Péclet number. Moreover, the broad distribution of stretching rates implies a broad distribution of mixing times. The stochastic nature of the stretching process thus allows stretching rates that are smaller than in a pure shear flow. However, our analysis shows that the occurrence of such events becomes negligible at large Péclet number ( 10 5 ) as mixing occurs at larger deformations.

The present study opens the way for a complete description of the mixing process occurring in sheared particulate suspension. In particular, it allows the prediction of the evolution of the concentration distribution P (C, t) [START_REF] Duplat | A nonsequential turbulent mixing process[END_REF]. A quantitative verification of these predictions requires a specific experimental device that resolves the Batchelor scale s(t mix ) which corresponds to the transverse dimension of the filaments at the time when diffusion significantly modifies the concentration levels. Such challenging measurements will be addressed in future studies.

We would like to thank P. Meunier for letting us use his DSM code, S. Dagois Bohy, S. Gallier, E. DeGiuli, M. Wyart, O. Pouliquen for having thoughful discussions and P. Cervetti, S. Noel and S. Martinez for helping us build the experimental set-up. This work was supported by ANR JCJC SIMI 9 and by the Labex MEC ANR-10-LABX-0092 et ANR-11-IDEX-0001-02.

Figure 1 .

 1 Figure 1. Some dye, initially confined to a small blob in a flowing particulate suspension (a), mixes with the rest of the suspension (b) by diffusing while the blob is stretched in the complex micro-flow generated by the particles.

Figure 2 .

 2 Figure 2. Schematics of the set-up.

Figure 3

 3 Figure 3. a) Flow streamlines in the bulk of a suspension. b) Slice of a sheared suspension illuminated by a laser sheet. The small fluorescent tracers seeding the suspending fluid appear as bright whereas the particle intersections with the laser sheet appear as dark, see also Movie 1. c) Magnified view of the suspending liquid velocity field obtained from the PIV (the velocity is not computed in the particles).

Figure 4 .

 4 Figure 4. a) Schematics of the set-up used to measure the molecular diffusivity D of the dye (rhodamine 6G), in the suspending liquid (TritonX+ZnCl2+H2O). b) Diffusive thickening of the bleached line at t = 0, 4800, 21600 and 64800 s (the image width is 5 mm). c) Concentration profiles at successive times (0 s< t <64800 s). d) Increase of the spatial variance of the concentration χ 2 (t) -χ 2 (0) versus time. Its fit to 2Dt yields D = 1.44 ± 0.2 × 10 -13 m 2 s -1 .

Figure 5 .

 5 Figure 5. Comparison of the stretching processes of a blob of dye sheared at high Péclet (∼ 10 6 ) and low Reynolds numbers (∼ 10 -4 ), in a pure fluid (top), and in a particulate suspension with volume fraction φ = 35% (bottom). The dye appears as dark, and the beads appear as bright, see also Movie 2.

Figure 6 .

 6 Figure 6. Example of stretching for two material lines numerically advected using the experimental fluid velocity field, see also Movie 3.

Figure 7 .

 7 Figure 7. Stretching laws measured for a suspension with volume fraction φ = 35%. a) Mean value ρ and standard deviation σρ = ρ 2 -ρ 2 of the distribution of elongations versus macroscopic strain in a semilogarithmic representation. The dashed line corresponds to the mean elongation in a pure fluid ρ lin (t) = 1 + γ2 t 2 /2. b) Distribution of the normalized elongations ρ/ ρ at different strains. The dashed curves are log-normal distributions built from the mean value ρ and standard deviation σρ of the experimental elongation distributions.

Figure 8 .Figure 9 .

 89 Figure 8. Mean elongation ρ (a) and standard deviation σρ (b) versus strain for increasing volume fractions ranging from 20 to 55%. Insets: growth rate κ of the exponential fit e κ γt to the main curves, as a function of φ. The lines show κρ = 0.09 + 0.74 φ (a), and κσ ρ = 0.12 + 1.03 φ (b).

Figure 10

 10 Figure 10. a) Typical local shear rate map for a suspension with volume fraction φ = 35%. b) Experimental distributions of normalized local shear rate P ( γloc / γ) for different volume fractions (the solid line is not a fit but the experimental data, sparse markers are used for sake of clarity). c) γ2 loc / γ versus φc -φ. The best fit γ2 loc / γ ∼ (φc -φ) -β , with φc = 0.58, yields β = 0.601 (see text). Inset: mean normalized local shear rate γloc / γ versus φ. The line is the best fit by A/(φc -φ) δ .

Figure 11 .

 11 Figure 11. (a-b) Lagrangian velocity transverse to the flow of a tracer passively advected by the suspending liquid V , as a function of the strain γt. a) φ = 20% and b) φ = 50%. c) Average Lagrangian velocity auto-correlation function V V obtained for different volume fractions versus strain. The velocity auto-correlation functions fit well e -γt/ γτ where τ denotes the correlation time. d) Correlation strain γτ versus φ and corresponding linear fit γτ = 0.62 -1.08 φ.

Figure 13

 13 Figure 13. a) Mean logarithm of the material line elongations, ln ρ versus γt for a suspension of volume fraction φ = 35%. b) PDF of the logarithm of the material line elongations P (ln ρ) at successive times.

Figure 14 .

 14 Figure14. Comparison between the stretching rates obtained in the present study and the largest Lyapunov exponent obtained from 3D Stokesian dynamics simulations[START_REF] Dasan | Stress fluctuations in sheared Stokesian suspensions[END_REF][START_REF] Drazer | Deterministic and stochastic behaviour of non-Brownian spheres in sheared suspensions[END_REF]).

Figure 15

 15 Figure15. a) Evolution of the distribution P (ln ρ) of the logarithm of the material line elongations in a particulate suspension. The mean of the distribution µ ∼ t and its standard deviation σ ∼ √ t. b) Dimensionless mixing times γtmix in a suspension (φ = 35%) as a function of P e. The median (blue line), the most stretched t 3% mix (green line), and the less stretched t 97% mix (red line) dimensionless mixing times can be compared to the dimensionless mixing time ∼ P e 1/3 expected in a pure fluid (dashed line).

Figure 16 .

 16 Figure 16. Picture illustrating the complexity of folding of the stretched blob of dye and the potential interaction (merging) of nearby filaments.