Dynamic Angle of Repose Measurements

• Dynamic angle of repose is a measure of the steady slope of granular materials in motion, reflecting the effects of particle cohesion, friction, and shape.
• Measurement protocols use rotating drums with image-based linear fits and center-of-mass methods to extract precise dynamic angles under various flow regimes.
• Scaling analyses identify transitions between rolling and avalanching regimes, emphasizing the influence of rotation speed, particle properties, and cohesive forces.

Dynamic angle of repose measurements quantify the inclination of a flowing free surface in granular or powder assemblies under sustained mechanical excitation—most commonly, in rotating-drum geometries. Unlike the static angle of repose, which reflects the limit of pile stability at rest, the dynamic (or flowing) angle captures the steady-state slope during continuous or intermittent flow. This metric is foundational for assessing granular flow regimes, calibrating particle-scale models, characterizing bulk rheology under weak confining stresses, and understanding the interplay of particle shape, cohesion, friction, and kinematics in both industrial and geological scenarios.

1. Experimental and Computational Geometries

Dynamic angle of repose measurements are almost universally performed in horizontally mounted rotating drums. Standard implementations include:

• Macroscopic Drums: Transparent glass or polycarbonate drums with diameters typically in the 10–30 cm range, widths accommodating either quasi-2D (∼1–2 particle layers) or fully 3D (bulk) flow. Drums are rotated at controlled angular velocity $\Omega$ covering quasi-static to rapid-flow regimes (Wang et al., 2024, Pourandi et al., 9 Dec 2025).
• Micro-scale Drums: Miniaturized PDMS drums (e.g., $D=100\,\mu$m) loaded with Brownian-size grains to probe the crossover between thermal and gravitational effects (Bérut et al., 2017).
• DEM and Numerical Setups: Simulations replicate cylindrical or quasi-2D wedge geometries with explicit solid boundaries, particle insertion protocols (pouring, settling), and wall-removal or continuous rotation to initiate flow (Dong et al., 2023, Shoji et al., 2019).
• Protocols: Experiments routinely employ stepwise or continuous variation of $\Omega$, usually recording both ramp-up (spin-up) and ramp-down (spin-down) to assess hysteresis or history effects (Irie et al., 2021).

Optical access (backlighting, high-speed imaging) is critical for precise surface detection. In the case of cohesive powders or wet materials, careful sample conditioning (precise wetting, mixing) precedes flow initialization (Pourandi et al., 9 Dec 2025).

2. Measurement, Definition, and Extraction of the Dynamic Angle

The dynamic angle of repose, denoted generically as $\theta_{\mathrm{dyn}}(t)$ or $\theta(t)$, is quantified from the geometric contour of the free surface during steady-state flow:

• Direct Linear Fit Methods: Pixelated free-surface profiles are extracted from binarized images or particle centers, and a linear or segmented fit (typically over the central portion to avoid wall and toe effects) yields $\theta(t)$ per frame (Wang et al., 2024, Shi et al., 2020).
• Center-of-Mass Algorithms: For symmetry or bulk averaging, $\theta_i$ is computed as the angle between the vertical axis and the line joining the drum center to the center of mass (CM) of the powder phase in frame $i$ (Pourandi et al., 9 Dec 2025).
• Time-Averaging: After at least several drum revolutions (post-transient), the mean dynamic angle $\overline{\theta}_{\mathrm{dyn}}$ is computed as:

$$\overline{\theta}_{\mathrm{dyn}} = \frac{1}{N} \sum_{i=1}^N \theta_i$$

with $N$ frames spanning steady-state (Pourandi et al., 9 Dec 2025).

• Dynamic Regimes: For intermittent (slumping or avalanching) flow, both the maximum (“destabilization” or $\theta_\max$) and minimum (“stabilization” or $\theta_\min$) surface angles are tracked. Their difference $\Delta\theta=\langle\theta_\max\rangle-\langle\theta_\min\rangle$ provides a regime diagnostic (Wang et al., 2024).

A summary of extraction protocols:

Method/Parameter	Description	Key References
Surface linear fit	Fit over central bulk	(Wang et al., 2024, Shi et al., 2020)
CM-based angle	Line drum center ↔ CM of powder	(Pourandi et al., 9 Dec 2025)
Peak/trough detection	Identifies $\theta_\max$, $\theta_\min$ during avalanches	(Wang et al., 2024)

3. Scaling and Regime Classification

Dynamic angle of repose measurements delineate between distinct flow regimes and provide a framework for scaling against particle and system properties:

• Rolling vs. Slumping/Avalanching: Rolling regime is characterized by continuous thin-layer flow with small oscillations, yielding $\Delta\theta \sim 4^\circ$, area ratio $R \sim 1$. The slumping regime manifests as intermittent avalanching with larger $\Delta\theta \sim 12^\circ$, $R \sim 0.84$ (Wang et al., 2024).
• Criticality Criteria: Empirically determined thresholds, such as $\Delta\theta_c \approx 8^\circ$, $R_c \approx 0.92$, or the transition Froude or Weber numbers in cohesive flows ($We_c \sim 400$) separate these regimes.
• Control Parameters: Systematic variation of:
  • Rotation speed $\Omega$ / Froude number $Fr$: Higher $\Omega$ generally reduces $\Delta\theta$, promoting rolling; lower $\Omega$ (or $Fr$) promotes intermittent slumping (Wang et al., 2024, Dong et al., 2023).
  • Particle properties: Shape concavity $\eta$, friction coefficient $\mu$, and wettability/capillarity (surface tension $\gamma$, liquid content) directly modulate $\langle\theta\rangle$ and flow regime boundaries.
  • Cohesion: Capillary liquid-bridge forces (Bond number $Bo$, Weber number $We$) systematically steepen $\theta_{\mathrm{dyn}}$ and delay transition to cascading (Dong et al., 2023, Pourandi et al., 9 Dec 2025).

Comprehensive phase diagrams in $(\Omega,\,\eta)$, $(\Omega,\,\mu)$, or $(\gamma,\,\omega)$ space allow a posteriori prediction of the operative regime for given granular configurations (Wang et al., 2024, Dong et al., 2023).

4. Influence of Particle Shape, Cohesion, and History

Dynamic angle of repose is highly sensitive to microstructure, interparticle forces, and loading protocol:

• Shape and Friction: For meta-granular systems, increasing concavity parameter $\eta$ or friction $\mu$ raises both the mean $\overline{\theta}$ and $\Delta\theta$. For $\eta\lesssim0.7$, $\langle\theta\rangle$ increases moderately with $\eta$ and $\Omega$; for strongly non-convex shapes ($\eta\gtrsim0.7$), $\langle\theta\rangle$ grows superlinearly, decoupling from $\Omega$ (Wang et al., 2024).
• Cohesive/Wet Granular Media: Liquid-induced capillarity, parameterized by $V_\text{L}/V_\text{S}$ (liquid:solid volume), $Bo$, or $We$, systematically increases $\overline{\theta}$. Thresholds for bridge formation ($V_{\min}$) and bridge saturation/agglomeration ($V_{\max}$) are practically extracted via flow curve calibration in DEM (Pourandi et al., 9 Dec 2025, Dong et al., 2023). Large agglomerates or clusters correlate with increased dynamic AoR and suppress well-defined regime transitions.
• Loading History and Hysteresis: Under cyclic loading (spin-up/spin-down), the effective bulk friction $\mu$ (and thus the dynamic angle) exhibits hysteresis: $\mu$ can increase by up to 33% under moderate $\Gamma=r_0\omega^2/g$ during spin-up, but does not fully recover upon reversal (Irie et al., 2021). This reflects metastable microstructural reorganization and compactification.
• Powder Cohesion Regimes: In limestone powders, dynamic angle $\phi_{\mathrm{dyn}}(\Omega)$ can either increase (+) with $\Omega$ (non-cohesive) or decrease (−) due to agglomerate breakup (strongly cohesive). The extrapolated zero-rate dynamic angle falls below the static heap angle for cohesive systems (Shi et al., 2020).

5. Mathematical and Empirical Descriptions

Multiple empirical and theoretical relationships capture the dependence of dynamic angle of repose on system variables:

• Empirical Scaling for Meta-Granular Matter:

$$\langle\theta\rangle(\eta, \Omega, \mu) \simeq \theta_0(\mu) + k_1(\mu)\eta + k_2(\mu)\Omega\quad\text{for}\ \eta\lesssim 0.7$$

with $\langle\theta\rangle$ increasing more steeply with $\eta$ for $\eta\gtrsim 0.7$ (Wang et al., 2024).

• Combined Cohesion-Inertia Number:

$C_E = \sqrt{Bo} + \lambda Fr$

with $\theta_{\mathrm{dyn}} \approx A C_E$, encapsulating competition between capillary cohesion and inertial forcing ($A \simeq 1$, $\lambda \simeq 12$ fitted) (Dong et al., 2023).

• Fr and We Parametrizations: At fixed $\gamma$, $\theta_{\mathrm{dyn}}$ increases linearly with $Fr$; at fixed $\Omega$, it increases linearly with $\sqrt{Bo}$ or, equivalently, as $1/We$ (Dong et al., 2023).
• Quasi-2D Force Balance: In rotating cell experiments,

$$\tan\theta(r) = \frac{\Gamma(r)-\mu}{1+\Gamma(r)\mu}$$

with extraction of $\mu(\Gamma)$ by fitting measured surface profiles (Irie et al., 2021).

• Ring-Shear and Stress-Dependence:

$$\phi(\Omega) = \phi_{\Omega 0} + \phi_{\Omega 1} \Omega,\qquad \phi(\sigma) = \phi_1 - \Delta\phi \ln(\sigma/\sigma_1)$$

linking dynamic and quasi-static frictional angles under drum or ring-shear conditions (Shi et al., 2020).

No universal closed-form “master law” is reported, but all significant dependencies are empirical or semi-empirical, and must be calibrated for each particle system.

6. Applications, Limitations, and Extensions

Dynamic angle of repose measurements provide a stringent, reproducible benchmark for granular flow models and industrial powder-handling scenarios:

• Model Calibration: DEM contact parameters, especially in the context of cohesion (liquid-bridge strength, cluster size), are routinely adjusted to best replicate experimentally measured $\overline{\theta}$ over a range of moisture contents or particle shapes (Pourandi et al., 9 Dec 2025).
• Regime Prediction: Phase diagrams (e.g., $(\Omega,\,\eta)$, $(\gamma,\,\Omega)$) guide the design and operation of drums and silos to avoid or exploit regime transitions.
• Planetary and Geological Processes: 2D DEM indicates that minor liquid volumes ($V/V_\text{particle}\sim 10^{-7}$), as would occur from humidity cycles, significantly raise the Martian sand repose angle and potentially trigger slope lineae (Shoji et al., 2019).
• Rheological Generalization: For Brownian or deeply agitated grains, a Kramers‐escape framework connects $\theta_{\mathrm{dyn}}$ with energy barrier-hopping rates, giving logarithmic relaxation below athermal thresholds (Bérut et al., 2017).
• Limitations and Open Problems: Many DEM and experimental studies measure only static or steady-flow angles, with less focus on detailed time-resolved, cyclic, or kinetic dynamic angles. Full 3D microstructure, in situ cluster evolution, and complex humidity cycles remain underexplored in most protocols. Explicit dynamic angle protocols in DEM, especially under time-varying cohesive or frictional parameters, are largely absent from current work.

7. Representative Systematics Across Studies

Study / Regime	Dynamic Measurement Protocol	Key Observations
(Wang et al., 2024)	Rotating drum, video tracking	Bimodal $\theta_{\max}$/$\theta_{\min}$, rolling vs. slumping, phase diagrams with $\eta$, $\mu$, $\Omega$
(Pourandi et al., 9 Dec 2025)	Rotating drum, CM algorithm	DEM calibration vs. capillary parameters, cluster-induced intermittency
(Dong et al., 2023)	Drum & DEM, $\max\theta(x)$	$C_E$ scaling, cohesion-delay of transition, cluster statistics
(Shi et al., 2020)	Rotating drum, interface slope	Linear $\phi(\Omega)$, effect of cohesion in raising static-dynamic gap
(Irie et al., 2021)	Quasi-2D spin-up/spin-down	Hysteresis in $\mu(\Gamma)$, compaction effects
(Bérut et al., 2017)	Microdrum, barrier model	Creeping log relaxation below $\theta_c$ (athermal), Pe number scaling
(Shoji et al., 2019)	DEM, 2D wedge geometry	Capillary-bridge-induced increase in static angle, no direct dynamic measurements

The versatility and technical sophistication of dynamic angle of repose measurements make them indispensable for dissecting granular-flow mechanisms, charting regime boundaries, and feeding high-fidelity particle simulations. However, nuances in definition, extraction, and physical interpretation necessitate rigorous methodological transparency and system-specific calibration.