Characterizing and understanding the behavior of flowing suspensions is an important challenge in materials science and engineering, with significant implications across various industrial sectors [1, 2]. Suspensions exhibit diverse flow behaviors that profoundly influence their properties and applications. Understanding and controlling flow phenomena in suspensions is crucial for optimizing processes and enhancing product performance in industries ranging from pharmaceuticals to cosmetics, from food processing to petroleum extraction [2]. The flow behavior of suspensions is intricately linked to their rheological properties, including shear-thinning and thickening. Shear-thinning, characterized by a decrease in viscosity under increasing shear rates, is commonly observed in many suspensions and plays a vital role in processes such as pumping, painting, inkjet printing and mixing. Conversely, shear thickening, where viscosity increases with shear rate, is important in applications such as body armor and damping systems [3].

Shear thickening typically occurs at relatively large Péclet number, P e = γ ̇ t s (where γ ̇ is the shear rate and t s is the system intrinsic time scale), posing a challenge for accurately measuring the structure and velocities of particles. To tackle this obstacle, our prior work introduced a novel technique aimed at capturing both local structure and particle velocities [1]. In this approach, instead of scanning the focused laser in two dimensions to produce a planar image in which particles are located, the laser is scanned only along the vorticity direction, allowing for very high time resolution. The flow of particles past the scan line enables the measurement of local structure and approximate flow speed. While this method successfully captures the spatio-temporal dynamics of dense suspensions in the thickening regime, there is considerable uncertainty in the measured speed values due to the coupling between the estimated flow speed and the spatial structure of the suspension (discussed below). To address this limitation, here we introduce an alternative approach employing laser scanning in the flow direction, allowing for precise measurement of average particle speed that is independent of the degree or type of order in the suspension (or any other aspects of the suspension structure or particle shape). We compare results obtained by line scanning in each direction in various flow fields and find that both approaches capture the flow profile (below the shear-thickening regime) and speed fluctuations (seen in the shear-thickening regime). However, the line scanning in the vorticity direction, using the parameters from our previous measurements, overestimates the values of speed by approximately 15%–25%. As expected, the exact values of the discrepancy depend on the structure of the suspension under flow. Moreover, we show that line scanning in the flow direction can accurately capture backflows in highly dense suspensions of silica particles in the shear thickening regime, a phenomenon not previously reported. The two approaches in combination enable accurate and robust measurements of speed and order fluctuations in rapidly flowing dense suspensions.

Dense suspensions at shear rates below shear-thickening: We first compare the values of speeds in the flow direction ( v x ) measured independently from line scanning in the flow ( v x flow ) and vorticity ( v x vorticity ) directions. Here, we report the results for a dense suspension of 1.5  μ m silica spheres at a volume fraction ϕ = 0.54 % at different heights from the bottom boundary for a constant shear rate of 5 s − 1 (corresponding to σ ≈ 8 P a ). This shear rate is well below the shear-thickening regime (Supplementary Figure S1). The resulting flow profiles are presented in Figure 4, with error bars given by the standard deviation of each time series (averaged over 100 frames). Flow profiles obtained by line scanning in both directions yield qualitatively similar results, but the speed measured by scanning in the vorticity direction v x vorticity is consistently higher than v x flow , suggesting that the structural correlations in the suspension are not isotropic. We have previously shown that the suspension is disordered at heights above 20 μ m , so the likely explanation for the anisotropy is a larger average particle separation in the flow direction relative to the vorticity direction. Assuming that increase in separation is 1/0.85, the two speed measurements match for heights above 20  μ m . The discrepancy between v x vorticity and v x flow is larger closer to the bottom boundary, consistent with the observation that the particles display significant hexagonal order near the wall, which produce notable asymmetries in the structure in the flow-vorticity plane [1]. Note that the ordering in the flow-vorticity plane is also associated with layering in the gradient direction, which produces a lower viscosity [11–16], as is evident in the increased shear rate (slope of the flow profile) measured near the bottom boundary.

Dense suspensions in the shear-thickening regime: Next, we compare the average velocity values measured by line scanning in the flow and vorticity directions in the shear-thickening regime where a more complex flow field with faster particle speeds is anticipated [1, 17]. Here, we show the results at the bottom layer for the same silica suspension ( ϕ = 54 %), but with a constant applied stress of 200Pa which is well-inside shear-thickening regime (Supplementary Figure S1). Both v x vorticity and v x flow show intermittent spikes (see Figure 5). These velocity fluctuations have been linked to fluctuations in localized stresses at the boundary [1, 17] which leads to fluctuations in the bulk viscosity/shear rate [17–20]. Although both v x vorticity and v x flow show the same trend, their exact value differs; away from spikes where the structure is ordered (Figure 5D), the difference is approximately 50% which is larger than the one observed for a low shear rate of 5 s − 1 (Figure 4). However, the difference reduces to about 10%–20% during the spikes where the local order is disrupted by flow (Figure 5D),

In Figures 5C, D, we show the corresponding kymographs for line scanning in the flow and vorticity directions, respectively, during one spike (highlighted regions in Figures 5A, B). In the case of line scanning in the flow direction (Figure 5C), variations in speed are discernible by changes in the slope of dark lines in the kymograph, representing of particles over time along the direction of flow. In Figure 5C, for t < i , we observe clearly defined dark lines with relatively similar slopes, indicating uniform motion of particles along the flow direction. For i < t < i i , the slope of the dark lines exhibits a sharp increase, indicating that particles are moving faster. For i i < t < i i i , the slope of the dark lines shows a gradual decrease, and finally for t > i i i , one can observe a clear reduction in slope. We also note that the spacing of dark lines can offer insights into the relative distance between particles along the flow direction. A preliminary examination of Figure 5C suggests a noticeable alteration in interparticle spacing during velocity spikes, which may suggest a compression wave as suggested by [17, 21]. However, a detailed quantitative analysis of these variations is beyond the scope of the present study.

For line scanning in the vorticity direction, the change in speed is manifested by the change in the relative size of dark ovals (particles) in the flow direction. A longer oval corresponds to a slower flow, and vice versa. Additionally, as previously mentioned, line scanning in the vorticity direction provides information on the local structure of the suspension in the vorticity-flow plane. As shown in the kymograph of Figure 5D, for t < i , particles appear to have arranged in a hexagonal ordered lattice (see Figure 3B and Ref. [1]). For i < t < i i , the size of ovals reduces significantly, indicating an increase in particle speed. Moreover, the structure seems to be largely disordered and heterogeneous along the vorticity axis. For i i < t < i i i , the structure remains disordered but more homogeneous with smaller oval sizes. For t > i i i , the structure appears still amorphous, but the size of ovals shows a clear increase, indicating a slower flow. Thus by looking at lin scanning measurements along flow and vorticity directions under the same conditions, we can acquire both precise and accurate measurements of the fast, dramatic speed fluctuations (from the flow direction scans) and unambiguous observations of the order/disorder fluctuations associated with the speed fluctuations (from the vorticity direction scans).

Very dense suspensions in the shear-thickening regime: Next, we investigate the flow in a very dense silica suspension ( ϕ = 0.58 %) in the shear-thickening regime. Here, we present data only for line scanning in the flow direction. Figure 6A shows the averaged particle speeds at various heights from the bottom boundary. Similar to lower volume fractions, the averaged particle speeds exhibit intermittent spikes during the application of constant stress ( 200 P a ) . However, the shape of these spikes is more irregular, indicating an even more complex flow which includes the presence of velocities with negative components in the flow direction ( v x values below the blue dashed lines) at all investigated heights from the bottom boundary. These backflows are manifested by dark lines with negative slopes as seen in regions i and i i for the kymograph during one of the speed spikes (Figure 6B). Note that line scanning in the vorticity direction cannot unambiguously identify backflow, since it operates based on the size of ovals, and there is no way to determine if particles are moving in the positive or negative direction.

An additional feature evident in the kymograph is that for times before the sharp increase in speed (times before and within region i ), rather than extended dark lines indicating particles traveling along the scan line, we instead observe ovals, similar to the kymograph patterns captured by line scanning in the vorticity direction. This pattern indicates times when the particle flow field includes significant components orthogonal to the scan direction, so that particles exit the scan line relatively quickly in one of the transverse directions. This observation of non-affine flow is consistent with our previous study on dense cornstarch [17] and calcium carbonate [22] suspensions, where we identified flows in the vorticity direction during the passage of a high-stress front. Thus line scanning in the flow direction reveals a complex flow field in the shear-thickening regime for highly dense suspensions of silica particles, including the presence of backflows and orthogonal flow, highlighting the intricate nature of the flow behavior in such systems.

In conclusion, we have introduced a novel approach for precisely measuring particle speeds in fast-flowing suspensions. This approach involves scanning the laser focal point back and forth along the flow direction, effectively tracking the movement of individual particles and enabling the precise determination of their velocities over time. The rapid line scanning process facilitates the measurement of speeds far exceeding those achievable with conventional techniques such as particle tracking or PIV. We have validated our approach under various flow conditions and compared it with measurements from line scanning in the vorticity direction. Although vorticity direction scanning provides information on the local structure in the flow-vorticity plane, it approximates the values of speed since it requires assumptions about the relative interparticle spacings in the vorticity and flow directions. We first compared values of speed obtained by line scanning in the flow and vorticity directions well below the shear-thickening regime where the suspension is expected to experience a simpler flow field. Our direct comparison showed similar flow profiles obtained by both methods. However, line scanning in the vorticity direction was found to overestimate flow speeds. The exact values of the discrepancy between the two methods depends on the structure of the suspension under flow. Near the bottom boundary of the suspension, where there is layering and ordering of particles, this discrepancy was approximately 25%, whereas farther from the bottom plate where the structure becomes largely disordered the difference was smaller (about 15%). In the thickening regime, both methods showed speed fluctuations during the course of shear with quantitative differences ranging from 50% (away from fluctuations) to about 10%–20% (in the middle of the fluctuations). These fluctuations are linked to a loss of order near the bottom boundary. Linescan measurements in the flow direction detected a more complex flow field for very dense suspension of silica particles in the shear-thickening regime, which includes backflow and presence of orthogonal flows, highlighting the intricate nature of the flow behavior in such systems. Our findings underscore the importance of precise speed measurements in understanding suspension dynamics, particularly in contexts where suspensions experience strong nonaffine flows.

Experiments were performed on the suspensions of silica particles with the size of 1.5  μ m (from Angstorm, Inc.) dispersed in an index-matched mixture of water/glycerol (20/80 v/v). To visualize the particles, we added 20 mM fluorescein sodium salt into the suspension. This enabled the untagged spheres to contrast as dark spots against the fluorescent solvent background. Rheological experiments were conducted on an Anton Paar MCR 301 stress-controlled rheometer which is mounted on an inverted Leica SP5 confocal microscope [23]. A parallel plate geometry with a diameter of 5.2 mm and a gap of 170  μ m was used. The linescan measurements along both flow and vorticity directions were taken at 300  μ m from the edge of the plate on a length of 145  μ m using 63 × objective. A typical frame consists of a 2D array, with horizontal dimension being the number of pixels per scan (1,024) along either flow or vorticity directions, and the vertical dimension being the number of lines scanned per frame (1,025 lines). In the shear-thickening regime due to fast flows, the number of scans was reduced to 256.