# Flow regimes and repose angle in a rotating drum filled with highly concave particles

Weiyi Wang, Jonathan Barés, Mathieu Renouf, Emilien Azéma

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Weiyi Wang, Jonathan Barés, Mathieu Renouf, Emilien Azéma. Flow regimes and repose angle in a rotating drum filled with highly concave particles. Physical Review E, 2025, 112 (1), pp.015417. 10.1103/lygy-vjlj. hal-05172308

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# Flow regimes and repose angle in a rotating drum filled with highly concave particles

Weiyi Wang,<sup>1,*</sup> Jonathan Barés,<sup>1,†</sup> Mathieu Renouf,<sup>1,‡</sup> and Emilien Azéma<sup>1,2,3,§</sup>

<sup>1</sup>LMGC, Université de Montpellier, CNRS, Montpellier, France

<sup>2</sup>Department of Civil, Geological, and Mining Engineering, Polytechnique Montréal, Montréal, Canada

<sup>3</sup>Institut Universitaire de France (IUF), Paris, France

<figure><img src="image_2.png" /></figure>

(Received 7 February 2025; accepted 10 June 2025; published 18 July 2025)

We present a series of experiments investigating the flow regimes and repose angles of highly concave particle packings in a rotating drum. By varying grain geometry from spherical to highly nonconvex shapes, adjusting frictional properties and the particle number of branches, we examine how these parameters and the drum speed influence the flow behavior. Our study identifies two distinct flow regimes: the rolling regime, where granular matter exhibits solid-like behavior near the walls and flows like a liquid near the free surface, and the slumping regime, characterized by cyclic avalanches and solid body rotations. Using quantitative criteria such as the repose angle difference and the area ratio of particle packings, we construct phase diagrams delineating the crossover between these regimes. Our findings highlight the significant effects of particle concavity, friction, and rotation speed on the flow dynamics of granular materials, providing new insights into the mechanical behaviors of *metagranular matter*.

DOI: 10.1103/lygy-vjlj

## I. INTRODUCTION

Granular matter has predominantly been studied through the lens of spherical particles. However, grain shape is a crucial factor in understanding the mechanical behavior of aggregates [1–6]. For example, particle angularity has been shown to induce high shear stress and dilatancy [1,7–9], while particle elongation has been shown to induce nematic ordering, leading to anisotropic flow and bulk properties [10,11]. Some studies have also explored more exotic shapes, such as highly nonconvex particles, revealing the unconventional properties of packings made of Z-shaped [12,13], U-shaped [13–16], and star-shaped or polypod particles [6,17–21]. These studies focus particularly on jamming and stability properties, attempting to explain how particle entanglement can lead to very stable free-standing and even hanging structures.

This new *metagranular matter* paves the way for more intriguing mechanical behaviors and applications [22–24]. Nevertheless, in most cases, such materials have been studied in static or quasistatic conditions, primarily probing their solid-like behavior. With the exception of a split-bottom Couette shear experiment [25], little is known about how these highly specific granular systems behave under dynamic loading or how they flow.

In the experimental work presented here, we aim to narrow this gap by investigating the flow behavior of concave particles. Sets of particles, ranging progressively from spherical to

highly concave and from slippery to highly frictional, flow in one of the most paradigmatic flowing systems, the rotating drum. In this specific loading geometry, provided that the rotation is not too fast and that the entire packing does not slip on the drum edges, it has been shown that the flow can exhibit two distinct regimes [26–28]: (i) In the *rolling regime*, the granular matter displays a solid-like behavior near the walls and flows like a liquid near the free surface. (ii) In the *slumping regime*, the granular packing cyclically exhibits sequences of avalanches and solid body rotations.

In this paper, we investigate the flow regime of polypod aggregates. Very few studies have focused on the flow of such concave particles, even though they can be present in nature and industry [22,24]. While our previous work [20] focused on the flow of these particles in the well-behaved rolling regime and for a certain type of particles, here we explore the effects of particle concavity, ranging from very low to very high, their frictional properties, as well as the drum rotation speed.

In the first part, we present our experimental setups and the polypod particles. In the second part, we present results for two specific cases to illustrate the main flow regimes and associated experimental observations, particularly the evolution of the free surface. In the third part, we construct phase diagrams delineating the crossover between slumping and rolling regimes as a function of grain shape, rotation speed, and coefficient of friction. Finally, these results are discussed in the fourth part.

## II. EXPERIMENTAL SETUP AND MATERIAL

The experimental setup consists of a rotating drum, as depicted in Fig. 1(a), which was previously described in [20]. This apparatus is a custom-made device comprising a tube with an inner diameter of 28 cm and variable depth ranging

<footnote>*Contact author: weiyi.wang@umontpellier.fr</footnote>

<footnote>†Contact author: jonathan.bares@umontpellier.fr</footnote>

<footnote>‡Contact author: mathieu.renouf@umontpellier.fr</footnote>

<footnote>§Contact author: emilien.azema@umontpellier.fr</footnote>
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<figure><img src="image_7.png" /><figcaption>FIG. 1. (a) Picture of the experimental setup: A represents the camera, B denotes the drum, C indicates the stepper motors, and D represents the lighting system. (b) HDPE particles and (c) EPDM particles exhibit varying shapes, ranging from spherical ($\eta = 0$) to highly concave ($\eta = 0.87$). As the concavity, $\eta$, increases, the branch diameter gradually decreases from 12 to 1.5 mm. (d) PA12 particles exhibit the same concavity ($\eta = 0.87$) with an increasing number of branches, ranging from $n_b = 4$ to $n_b = 20$. (e) PP particles showcase a range of shapes, from short branches ($\eta = 0.71$) to long branches ($\eta = 0.94$).</figcaption></figure>

from 5 to 10 cm. The tube, closed by two glass discs, is mounted on two slick pinions driven by two synchronized stepper motors capable of variable speed control. The system operates within a rotational speed range of 2 to 5 rpm, while

being recorded at 10 frames per second (fps) using a 14 megapixel camera equipped with a 50 mm lens.

The drum is filled with monodisperse particles of various shapes and materials, as illustrated in Figs. 1(b)–1(e). All particles depicted in Figs. 1(b)–1(d) are inscribed within a sphere of diameter $d = 12$ mm. In Figs. 1(b) and 1(c), they consist of six spherocylinders (also called branches in this paper) extending toward the faces of a regular cube. The radius of the branches, denoted as $r_0$, gradually increases from 0.75 to 6 mm (sphere case). These particles are fabricated from high-density polyethylene (HDPE) (b) and ethylene propylene diene monomer (EPDM) (c). The particles depicted in Fig. 1(e) are of the same nature, composed of polypropylene (PP), with a branch section radius of $r_0 = 3$ mm, while their diameter, $d$, varies from 13 to 63 mm. All these particles are produced via injection molding using a custom-made mold [20], generating clusters of particles. In contrast, the particles presented in Fig. 1(d) are 3D printed using the slice laser sintering (SLS) process with nylon PA12 material. Their diameter and branch section are constant ($d = 12$ mm and $r_0 = 1.5$ mm), but the number of branches, denoted as $n_b$, varies from 3 to 20, allowing the branches to emerge from the five platonic solids ($n_b \in [4, 6, 8, 12, 20]$). The choice of materials for the particles ensures they can be considered non-deformable (with Young's moduli around 1 GPa) and exhibit Coulomb friction coefficients, $\mu$, ranging from 0.2 (HDPE) to 0.8 (PA12) (approximately 0.5 for EPDM, and 0.6 for PP).

To quantify the particle geometry, a concavity parameter, denoted as $\eta$, is defined as $\eta = (d - 2r_0)/d$. For HDPE and EPDM particles [Figs. 1(b) and 1(c)], this parameter varies such that $\eta \in [0, 0.33, 0.5, 0.58, 0.67, 0.71, 0.75, 0.79, 0.83, 0.875]$. For PA12 particles [Fig. 1(d)], it remains constant at 0.875, while for PP particles [Fig. 1(e)], it varies such that $\eta \in [0.77, 0.87, 0.91, 0.93, 0.94, 0.95]$. This geometrical parameter is derived from the one introduced in [29]. It is evidently a coarse, low-order shape parameter that cannot distinguish all shapes. This is why we have also previously introduced $n_b$.

The drum features smooth, transparent glass on both axial sides, while the inner radial side is regularly crenellated to prevent sliding. For HDPE, EPDM, and PA12 particles [see Figs. 1(b)–1(d)], a constant volume of 900 ml of particles is loaded into the drum with a depth of 5 cm. For PP particles [see Fig. 1(e)], the drum is half-filled with a depth of 10 cm, since the largest particles are 63 mm wide. Illumination is provided by LED lights, and the camera is positioned perpendicular to the front glass to capture images of the system. The rotation speed of the drum is controlled by adjusting the speed of two synchronized stepper motors positioned below it [see Fig. 1(a)]. We tested six angular velocities for the drum speed: $\Omega \in [1.93, 2.14, 2.91, 3.20, 4.01, 4.83]$ rpm. For each experiment, the drum is initially rotated for 2 minutes, followed by a 5-minute recording period. The mean behavior for each set of grain shapes and $\Omega$ is obtained by averaging over 3000 frames.

For each frame, the image is thresholded to discriminate the bulk of particles from the rest of the picture. Subsequently, the top of the packing is isolated and described by a polygonal line. This line represents the *free surface*. If this line is regular enough, a slope is fitted at the center of the drum to extract
<figure><img src="image_8.png" /></figure>

FIG. 2. Free surface and repose angle for spherical ($\eta = 0$) HDPE particles ($\mu \approx 0.2$) with a rotation speed of $\Omega = 4.01$ rpm. (a) Spatial density distribution of the free surface with a frame image as background. Density ranges from blue (low) to red (high). (b) Evolution of the inclination angle, $\theta$, as a function of time. Red (resp. green) circles indicate destabilization (resp. stabilization) angles. (c) Diagram illustrating the packing splitting. The average free surface is represented by a solid black line, with the inclined angle denoted as $\theta$. The gray dashed line perpendicular to the angle, passing through the center of the drum, divides the space occupied by grains into two parts, represented by dark and clear blue zones. (d) Probability density function of the angle of stabilization, $\theta_{\text{stab}}$, and destabilization, $\theta_{\text{destab}}$, represented by the green and red lines, respectively. Vertical lines show the 95% error bars.

its inclination angle, $\theta$, which corresponds to the *repose angle* when the packing is at rest.

### III. FREE SURFACE AND REPOSE ANGLE

When conducting experiments, and as anticipated from the literature [26–28], two distinct flow regimes are observed depending on the drum speed, $\Omega$, and the exact nature of the particles ($\eta$ and $\mu$). In the simplest case, as previously explored in [20], the system exhibits a solid-like behavior near the walls and deep within the bulk, where particle motion mirrors the solid motion of the drum. Conversely, in the upper part, a liquid-like behavior emerges, characterized by layers of grains exhibiting varying degrees of inertia. This corresponds to the rolling regime. In a more intricate scenario, the particle bed undergoes a cyclical process of avalanche and solid body rotation. Specifically, when the bed surface reaches the maximum angle of inclination, a slump occurs during which particles from the upper part of the bed rapidly slide downwards. This corresponds to the slumping regime.

In this paragraph, we present the evolution of the main experimental observables for two significant experiments. Specifically, for HDPE particles ($\mu \approx 0.2$), we demonstrate results obtained with spheres ($\eta = 0$) with a drum rotating at 4.01 rpm (see Fig. 2), and with highly concave particles ($\eta = 0.83$) with a drum rotating at 3.20 rpm (see Fig. 3).

For spheres turning rapidly, as depicted in Fig. 2(b), concerning the variation of the free surface angle, $\theta(t)$, the rolling

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FIG. 3. Free surface and repose angle for highly concave ($\eta = 0.83$) HDPE particles ($\mu \approx 0.2$) with a rotation speed of $\Omega = 3.20$ rpm. (a) Spatial density distribution of the free surface with a frame image as background. Density ranges from blue (low) to red (high). (b) Evolution of the inclination angle, $\theta$, as a function of time. Red (resp. green) circles indicate destabilization (resp. stabilization) angles. (c) Cartoon illustrating the packing splitting. The average free surface is represented by a solid black line, with the inclined angle denoted as $\theta$. The gray dashed line perpendicular to the angle, passing through the center of the drum, divides the space occupied by grains into two parts, represented by dark and clear blue zones. (d) Probability density function of the angle of stabilization, $\theta_{\text{stab}}$, and destabilization, $\theta_{\text{destab}}$, represented by the green and red lines, respectively. Vertical lines show the 95% error bars.

regime indicates that $\theta$ gently oscillates around a fixed value with narrow amplitude. Subsequently, by extracting the upper (resp. lower) peaks of this signal, we measure the destabilization (resp. stabilization) angle of repose, $\theta_{\text{destab}}$ (resp. $\theta_{\text{stab}}$). When plotting the probability density functions (PDF) of these quantities, they exhibit a Gaussian-like distribution, as illustrated in Fig. 2(d). Additionally, in this regime, the average position of the free surface is characteristic. As demonstrated in Fig. 2(a), it forms a homogeneous narrow straight cloud oriented at an angle equal to the average repose angle. Furthermore, in Fig. 2(c), a line that cuts the average free surface perpendicularly in its middle divides the particle packing into two distinct regions: dark blue at the top and clear blue at the bottom. We call $\mathcal{R}$ the ratio between both areas. Notably, in the specific case of the rolling regime, this areas are very similar, resulting in the ratio $\mathcal{R}$ being close to 1.

For highly concave particles turning slowly, in Fig. 3, the same quantities as those shown for the rolling regime are plotted for the slumping case. First, as illustrated in Fig. 3(b), the free surface angle evolves with a much larger amplitude. The PDFs of the stabilization-destabilization repose angles are also distinctive, it is bimodal, since it corresponds to a distribution with two modes. They form two separate Gaussian-like distributions, with average values clearly distinguished, as depicted in Fig. 3(d). Additionally, the average free surface position differs significantly from the rolling regime case. As shown in Fig. 3(a), it forms a wider cloud with a wavy
backbone. When dividing the packing into two distinct areas as before, it is evident that they are not equal in terms of surface area, resulting in the $R$ ratio being significantly lower than 1.

Key elements in the definition of the flow regime are the free surface and the inclination angle. From the results presented in this paragraph, two quantitative criteria emerge to discriminate between the rolling and slumping flowing regimes. First, we define the repose angle difference, $\Delta\theta$, as the difference between the average stabilisation and destabilisation repose angles: $\Delta\theta = \langle\theta_{\text{destab}}\rangle - \langle\theta_{\text{stab}}\rangle$. When the value of this criterion is low, stabilization and destabilization angles are close, indicating that the system is more likely in the rolling regime. Conversely, when it is large, stabilization and destabilization angles are separated by a significant avalanche phase, suggesting that the system is more likely in the slumping regime. A second criterion also emerges: the top and lower area packing ratio, $R$. It is defined as the ratio between the areas separated by a line cutting the average free surface perpendicularly in its middle. If this ratio is close to 1, both areas are similar, indicating that the top and bottom packing are balanced, and the system is more likely to be gently rolling. Conversely, if this ratio is significantly lower than 1, the average free surface is no longer straight, suggesting that the system is more likely to be in the slumping regime.

#### IV. PHASE DIAGRAM OF FLOW STATES

As discussed in the previous section, two flow regimes are qualitatively observed when particles are rolled into our drum. In parallel, two quantitative observables are constructed, taking very different values for systems in both regimes. In Figs. 4(a) and 4(b), we plot the evolution of these observables, $\Delta\theta$ and $R$, respectively, for EPDM particles, varying the drum rotation speed, $\Omega$, and the particle concavity, $\eta$. In both figures, an isoline is plotted for a value in the middle of the full scale. For the first criterion, this is typically $\Delta\theta = 8^{\circ}$, and, for the second, $R = 0.92$. These values are not critical thresholds marking a transition, as the system does not undergo a true transition. Instead, they represent central values of the crossover. First, we note that both observables give very similar results, as the phase diagrams resemble each other, and their isolines almost superimpose. Additionally, we observe a rapid variation of $\Delta\theta$ or $R$ around this isoline, from the minimum value of the observable to its maximum value. This crossover implies that both regimes are significantly distinct: (i) When $\Delta\theta$ is low, around 4, or $R$ is high, around 1, the system is in the rolling regime. (ii) Conversely, when $\Delta\theta$ is high, around 12, or $R$ is low, around 0.84, the system is in the slumping regime. Thus, from Figs. 4(a) and 4(b), we deduce that slumping intermittently occurs when the concavity is high and the rotation speed is low whereas the flow is continuous otherwise in our observation domain.

In Figs. 4(c) and 4(d), we explore the same phase diagram for slippery particles, namely HDPE. The conclusions are similar to those for the top of the figure. However, we observe that, when decreasing the interparticle friction coefficient, the slumping regime region shrinks significantly to higher concavity levels and slower rotation speeds. This makes it less likely to observe the intermittent flow regime when particles

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FIG. 4. Flow phase diagram for high and low friction. Evolution of the flow criteria as a function of the particle concavity, $\eta$, and the drum rotation speed, $\Omega$. The left column represents the $\Delta\theta$ criterion, while the right column represents the $R$ criterion. The top row is for frictional EPDM particles, while the bottom row is for lower friction HDPE particles. For each phase diagram, a dashed isoline is plotted at $\Delta\theta = 8^{\circ}$ and $R = 0.92$, near the regime crossover. The grey points indicate the locations where measurements were taken, while the rest of the map is obtained through linear interpolation from these points.

are less frictional. This tendency is once again observed from both criteria.

In Fig. 5, phase diagrams analogous to those in Fig. 4 are presented for PA12 particles. As observed previously, increased friction enhances the likelihood of the system entering the slumping regime. Here, PA12 particles are highly frictional, and, consistently, the systems are shown to slump intermittently regardless of the number of branches, $n_b$, or the rotation speed, $\Omega$. Indeed, in Fig. 5(a), $\Delta\theta$ remains above the identified crossover value of $8^{\circ}$. Similarly, in Fig. 5(b), $R$ consistently stays below 0.92.

Although the system remains within the same regime, discrepancies are observed when varying the rotation speed and the number of particle branches. Depending on these control parameters, one regime is more or less prominent. It appears from the phase diagrams in Fig. 5(a) that regime bimodality is more pronounced for packings made of particles with fewer branches and lower rotation speeds. Additionally, in Fig. 5(b), we observe that the imbalance between the lower and upper parts of the packing is greater when the number of branches is fewer, irrespective of the rotation speed, $\Omega$. However, when $\Omega$ is high, this imbalance is also significant for particles with a high number of branches.

In Fig. 6, phase diagrams analogous to the previous ones are presented for PP particles. Once again, despite displaying different quantities, these phase diagrams convey the same information. They are also consistent with Fig. 5, but allow for a more detailed exploration of the effect of concavity, for higher values. Consistently, we observe that the system is more likely to be in the rolling regime when particle
<figure><img src="image_11.png" /><figcaption>FIG. 5. Flow phase diagram varying the number of branches for highly frictional particles. Evolution of the flow criteria as a function of the number of particle branches, $n_b$. (a) depicts the $\Delta\theta$ criterion, while (b) illustrates the $\mathcal{R}$ criterion for PA12 particles. The grey points indicate the locations where measurements were taken, while the rest of the map is obtained through linear interpolation from these points.</figcaption></figure>

concavity is lower and rotation speed is higher. However, we also note that, for high concavity, $\eta$, the effect of the drum rotation speed becomes less significant, as the phase diagrams are vertically invariant on their right-hand side. As $\eta$ increases, we observe that regime bimodality also increases, indicating that higher concavity leads to greater intermittency in the flow regime. No saturation is observed within the range of concavity explored, even though it is quite high.

Based on the foregoing discussion and classification, we have delineated the flow regime of granular materials under various conditions. Notably, for these different experiments, the angle in the rolling regime exhibits relative stability, with fluctuations confined to a narrow range. Thus, it is possible to consider this as a stable angle of repose and to measure its average value, providing insight into the angle evolution for a given experiment [see Fig. 2(b)]. In the slumping regime, the average angle of repose can be measured from the spatial density distribution of the free surface [see Fig. 3(a)].

In Fig. 7, we present variations of the time-averaged angle for all the different particles, for both the maximum and minimum rotation speeds. We observe first that, for lower interparticle friction coefficients, the repose angle is significantly lower regardless of the particle shape. Additionally, at low concavity levels, the rotation speed affects the average angle of repose, but this effect diminishes as concavity increases.

<figure><img src="image_12.png" /><figcaption>FIG. 6. Flow phase diagram for highly concave particles. Evolution of the flow criteria as a function of particle concavity, $\eta$. (a) depicts the $\Delta\theta$ criterion, while (b) illustrates the $\mathcal{R}$ criterion for PP particles. For each phase diagram, a dashed isoline is plotted at $\Delta\theta = 8$ and $\mathcal{R} = 0.92$, near the regime crossover. The grey points indicate the locations where measurements were taken, while the rest of the map is obtained through linear interpolation from these points.</figcaption></figure>

Specifically, the angle of repose is higher for higher rotation speeds, but only for particle concavity values lower than ~0.7. It also becomes clear that the angle of repose increases with particle concavity. Below a certain level, this increase appears linear, but for higher concavity values, such as those observed in PP particles, the increase is much steeper. Even if not presented here, we note that the average angle of repose does not significantly vary with the number of branches. We believe this is mainly due to the fact that for PA12 particles' behavior is dominated by frictional processes.

## V. CONCLUDING DISCUSSION

The experiments presented in this study illustrate the rolling and slumping flow regimes of metagranular matter in a rotating drum. The granular systems are composed of star-shaped particles with varying concavity, $\eta$, number of branches, $n_b$, and friction coefficient, $\mu$. This study focuses on the free surface, particularly on the inclination angle when different particles are tested at varying rotation speeds, $\Omega$. Two physical observables are identified to discriminate the flow regime: (i) $\Delta\theta$, the difference between the average stabilization/destabilization angles, and (ii) $\mathcal{R}$, the ratio between the average number of particles at the top and bottom of the drum flow [cf. Fig. 2(c)]. In the rolling regime, $\Delta\theta$ is small and $\mathcal{R}$ is close to 1, whereas in the slumping regime, $\Delta\theta$ increases
<figure><img src="image_13.png" /><figcaption>FIG. 7. Angle of Repose: Variation of the time-average free surface angle $\langle\theta\rangle$ as a function of concavity $\eta$ for different materials of different friction coefficients: EPDM and PA12 (a), and HDPE and PP (b). Data at maximum drum speed $\Omega = 4.83$ rpm are shown as a dark plot, while data at minimum drum speed $\Omega = 1.93$ rpm are shown as a lighter plot. The solid markers indicate that the flow is in the rolling regime, while the empty markers indicate the slumping regime. Vertical bars show the 95% confidence interval of the average angle.</figcaption></figure>

while $\mathcal{R}$ decreases. Both observables show similar sensitivity in distinguishing the flow regimes and reveal a crossover between the two regimes when the control parameters $(\eta, n_b, \mu$, and $\Omega$) are adjusted. This provides a quantitative method to evaluate the flow regime by measuring the evolution of the free surface over time. Also, the phase diagrams provide a way to predict granular flow as a function of the grain properties and forcing parameters.

More precisely, within the observational domain covered by this experimental study, it is found that a flowing system is more likely to be in the rolling regime when (i) particle concavity is low, (ii) drum rotation speed is high, and (iii) interparticle friction is low. Discrepancies have also been observed in the slumping regime when varying the number of particle branches: packing oscillation is higher when the number of branches is either low or high (not intermediate). This observation aligns with our previous numerical studies on the stability of such metagranular matter [21,30], which demonstrated that these particles exhibit *geometric cohesion*. First introduced by Franklin in 2012 [14], geometric cohesion describes systems composed of grains with peculiar shapes that interlock and/or mutually entangle, causing the bulk system to behave as a solid even in the absence

of adhesive forces between grains. Our recent studies [21] have shown that geometric cohesion for polypods is a non-monotonic function of the number of branches and concavity level. This is due to the fact that, in 2D, a low number of branches does not allow sufficient rotation interlocking, while a high number of branches, given constant concavity, tends toward spherical shapes, reducing interlocking capabilities.

Extending the concept of geometric cohesion to systems where external forcing continuously destabilizes the granular system, such as in the drum case, we propose the concept of *geometrical consistency*. Indeed, geometrical cohesion describes systems that jam due to the shape of their particles. In our case, however, the system does not jam in the steady state, as it remains in a flowing regime. Nevertheless, particle shape still plays a crucial role: their geometry induces a certain *viscosity* in the granular packing, preventing it from flowing as fluidly as a packing of spheres. To some extent, this *viscosity* can become so high that the system intermittently jams, as observed in the bimodal regime. Still, we refrain from referring to it as viscosity, since the term has a precise physical definition associated with a measurable quantity that does not describe what we observe here. Instead, we propose the term *consistency* to more accurately describe this effect. Hence, geometrical consistency characterizes systems composed of grains with exotic shapes that entangle, preventing them from easily moving relative to each other. This causes the bulk system to resist flowing, exhibiting properties such as thickness, viscosity, or even stickiness. It has a direct effect on the angle of repose of the packing or on the $\Delta\theta$ and $\mathcal{R}$ parameters. This study illustrates that, for a given interparticle friction coefficient, the ability of particle packing to rearrange in response to external loading and flow varies with particle concavity and the number of branches. Because geometrical consistency influences various aspects of the flow, it is, so far, not possible to associate it with a single measurable quantity.

It is observed that it is more difficult for a granular system to flow when the concavity is high or when the number of particle branches is either very large or very small. These results differ somewhat from what has been observed in two-dimensional (2D) systems [21] with a large number of branches. In this latter case, the explored branch densities were significantly higher, leading to particles that behaved almost like spheres. The results also differ for low $n_b$ values, as the additional degree of freedom in 3D allows particle centers to come closer—down to approximately $2r_0$—which enables stronger interlocking compared to the 2D case.

Furthermore, when geometrically induced consistency is high, flow bimodality is also high. This geometrically induced consistency implies a variation in the repose angle, which increases with particle concavity, friction, and the number of branches, provided the number is not too high. This is consistent with observations of geometric cohesion [6,21]. Additionally, lower consistency increases the likelihood of the system rolling, whereas higher consistency favors slumping. This is likely because rolling requires particles to rearrange
freely and quickly relative to the drum speed, a process hindered by friction or particle interlocking.

Results obtained with PP particles are consistent with observations for other particles, regardless of changes in the drum width aspect ratio, suggesting limited influence of this geometrical parameter.

The findings reported here represent a significant step forward in understanding the flow of metagranular systems. Although this results are consistent with a recently developed theoretical model describing the flow of these particles in the rolling regime [20], we remain far from establishing a unified theoretical framework that fully incorporates particle geometry, frictional properties, and flow behavior. Developing such a comprehensive model should be considered a long-term objective for future research. We still introduce the concept of granular *geometric consistency*, extending *geometric cohesion* to flowing systems. Beyond the scientific implications, this work provides mechanical rules for designing granular flow, offering heuristic solutions to customize flow behavior. From this point of view, future studies should also explore more

complex flowing geometries, other particle shape and also consider the particle softness [31,32].

## ACKNOWLEDGMENTS

The authors acknowledge financial support from ANR MICROPROGRAM (Grant No. ANR-20-CE92-0009). We would also like to express our gratitude to Gille Camp, Gilles Genevois, and Quentin Chapelier for their technical assistance in setting up the experimental device. Additionally, we extend our thanks to Arnaud Regazzi, Benjamin Gallard, and Sylvain Buonomo for their assistance in particle preparation. We are grateful to Karen Daniels and Yuchen Zhao for generously sharing the PP particles with us.

The authors declare that they have no conflict of interest.

## DATA AVAILABILITY

The data that support the findings of this article are available upon request.

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