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Analysis of granular rheology in a quasi-two-dimensional slow flow by means of discrete element method based simulations

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Pith's one-line read This paper establishes that slow dense granular flow with two velocity components behaves as a slightly compressible power-law fluid, with viscosity, volume fraction, and fluctuation velocity collapsing onto single curves of the inertial…

desk verdict Careful DEM data and a useful coordinate transform, but the continuum model is validated only by a tuned amplitude fit, so the no-yield-stress claim outruns the evidence. read the letter →

arxiv 1908.05080 v2 pith:QRHAA4IM submitted 2019-08-14 cond-mat.soft

Ashish Bhateja , Devang V. Khakhar This is my paper · ORCID
classification cond-mat.soft PACS 45.70.Mg83.80.Fg
keywords granularrheologydiscreteelementmethodinertialnumberpower-lawfluidquasi-two-dimensionalflowbininclinedsurfacenormalstressdifferences
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper uses discrete element simulations of steady gravity-driven flow in a rectangular bin, a flow with two nonzero velocity components, to test whether a simple local rheology describes slow dense granular flow. The central claim is that the material behaves as a slightly compressible power-law fluid: the shear-rate-scaled viscosity follows $1/\hat{\eta}=m I^n$, the solid fraction follows $\varphi=\varphi_m-r I^s$, and data from three outlet sizes collapse onto single curves of the inertial number $I$. A coordinate transformation casts any two-dimensional isochoric flow into a canonical form that makes the local viscosity and pressure components directly measurable, and the resulting continuum model reproduces the full velocity profile of an inclined-surface flow. The authors conclude that a power-law relation can be used over the whole range of $I$ studied, without invoking a yield stress.

What carries the argument

The load-bearing object is the pointwise coordinate transformation of Sec. III.B, which maps any two-dimensional isochoric velocity gradient tensor into the canonical form of Eq. (16) by solving the eigenvalue problem of the rotated isochoric gradient. For a generalized Newtonian fluid with stress $\sigma=P I-2\eta D$, this transformation makes the shear stress and the individual pressure components directly computable: $\eta=k_1\cdot\sigma\cdot k_2/(k_1\cdot D\cdot k_2)$, $P_{x'}=k_1\cdot\sigma\cdot k_1$, and similarly for the other components. The resulting scalings with the inertial number $I=\dot{\gamma}d/\sqrt{P/\rho}$—power laws for $1/\hat{\eta}$, $\bar{u}^2$, and $\varphi$—are the empirical content that closes the continuum model.

What would settle it

Measure the effective friction coefficient $\mu=|\tau_{xy}|/P$ in a quasi-two-dimensional bin flow at inertial numbers below 0.003: the power-law claim predicts $\mu\to 0$ as $I\to 0$, while a yield-stress material would level off at a finite $\mu_s$; a companion check is to repeat the bin simulation at thickness $L=16d$ and see whether the extracted viscosity and pressure relations shift outside the stated error, which would expose the isochoric two-dimensional assumption.

Watch

Extended reading notes

Core claim

The local rheology of slow dense granular flow in the bin is a slightly compressible power-law fluid, not a yield-stress fluid. The paper derives this from DEM data by transforming the local velocity gradient to the canonical form $v_{x'}=\dot{\gamma} y'$, $v_{y'}=\psi \dot{\gamma} x'$, which is possible for every two-dimensional isochoric flow, and then reading off viscosity and pressure from the stress tensor. The viscosity scaled by shear rate obeys $1/\hat{\eta}=m I^n$ with $m=1.713$, $n=1.872$ for bin flow, and the solid fraction obeys $\varphi=\varphi_m-r I^s$ with separate fits below and above $I=0.03$. The same power-law form describes flow on a rough inclined surface with a slightly different prefactor and a lower viscosity caused by particle layering. Because the power law extends to $I\approx 0.003$, the paper argues that a yield stress is not needed for the range of slow flows studied.

Load-bearing premise

The entire extraction of viscosity and pressure assumes the flow is exactly two-dimensional and volume-preserving at the point of measurement, even though the bin has finite thickness (8 particle diameters) and measured dilation reaches about 10% of the shear rate.

Editorial extensions

If this is right

  • Continuum simulations of slow dense granular flows in complex geometries can use Eq. (33) and Eq. (37) directly, eliminating the need for a yield-stress term in the constitutive relation.
  • The collapse of bin-flow data for three outlet widths onto a single curve means the local rheology is fully characterized by the inertial number and a few material parameters, at least in the slow regime studied.
  • The explicit correlation for fluctuation velocity ($\bar{u}^2=g I^h$) closes the momentum balance without solving a separate fluctuation-energy equation.
  • The same constitutive relations, with parameters refit for the layered surface flow, reproduce the inclined-surface velocity profile through Eq. (34).
  • In-plane normal stresses are nearly equal and the out-of-plane component is smaller, with each pressure component proportional to the corresponding fluctuation-velocity component, so pressure anisotropy carries no independent information beyond the fluctuation anisotropy.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the power law holds all the way to $I\to 0$, the apparent yield stress seen in many granular measurements may be a finite-shear-rate or finite-size artifact; this could be tested with extremely slow creep experiments or DEM at $I<10^{-3}$.
  • The layering-induced offset between inclined-surface and bin-flow rheology suggests that rheological parameters calibrated in unidirectional, layered shear flows may mis-predict non-layered complex flows; a bin-flow-calibrated model should be tested against hopper or silo discharge.
  • The transformation's assumption of isochoric two-dimensional flow limits the scheme's accuracy wherever out-of-plane motion or dilation contributes; extending the extraction to weakly compressible three-dimensional flows would require a third invariant or a volumetric correction term.
  • The equality of in-plane pressure components and the anisotropy of pressure with fluctuation components suggests a possible closure modeling the full pressure tensor from the granular temperature tensor rather than a single scalar pressure.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. Using soft-particle DEM, the paper studies steady gravity-driven discharge from a rectangular bin (Do = 6d, 7d, 8d) and compares it with bumpy inclined-surface flows. After establishing that the flow is nearly isochoric, that the stress tensor is symmetric, and that the principal axes of stress and rate of strain are nearly aligned, the authors introduce a pointwise coordinate transformation that reduces any two-dimensional isochoric velocity gradient to the canonical form of Eq. (16), enabling direct extraction of an effective viscosity and pressure components. They report power-law fits for the inverse scaled viscosity (1/eta = a I^b), squared scaled fluctuation velocity (u^2 = g I^h), shear-rate-scaled viscosity (1/eta-hat = m I^n), and solid fraction (phi = phi_m - r I^s), with good collapse across outlet sizes for I above about 0.003-0.01. These correlations are assembled into a continuum model (Eqs. 12, 13, 33, 37, 38). The model is tested against inclined-surface velocity profiles via Eq. (34), and agreement is obtained after tuning the exponent n from 1.922 to 1.905. The paper concludes that a power-law viscosity relation such as Eq. (33) can be used over the entire range of I in place of a yield-stress description.

Significance. The paper contains substantial careful work: coarse-grained averaging with explicit standard-error filtering, checks of independence of the bin thickness L, a clean argument that every two-dimensional isochoric flow can be locally transformed to the canonical form, detailed data collapse for three outlet widths, and direct comparison with the Jop et al. mu(I) model. If the proposed constitutive model were independently validated, it would be a useful empirical closure for slow dense granular flows, especially because the fluctuation-velocity scaling obviates the need for a separate energy equation. The main limitation is that the single continuum validation is in-sample and hand-tuned, and the no-yield-stress conclusion extrapolates beyond the fitted inertial-number range. The central claim is therefore defensible only after re-scoping or after an out-of-sample test.

major comments (3)
  1. [Section III.C, Fig. 12 and Eq. (34)] The velocity-profile comparison is not an independent validation. The parameters m = 2.024 and n = 1.922 are obtained by fitting 1/eta-hat versus I for the inclined-surface flow (Fig. 11), and the profile in Eq. (34) is then evaluated using that same power law and compared with the same flow class. When the initial prediction differs substantially, the exponent is changed to n = 1.905 'to obtain a good match' (Sec. III.C). With a parameter adjusted to the target curve, Fig. 12 demonstrates consistency rather than predictive skill. Please provide an out-of-sample test (for example, a different geometry, a different flow rate, or a profile computed with the directly fitted m and n before any tuning), or explicitly and prominently frame Fig. 12 as a consistency check rather than as validation.
  2. [Section III.C, Fig. 11, and Conclusions] The conclusion that Eq. (33) 'may be used over the entire range of I instead of considering a yield stress' is an extrapolation. The fit of Eq. (33) is restricted to I > 0.003 (Sec. III.C), while Fig. 7 shows a region of steep rise and 'complex yielding behaviour' for I < 0.02, and Fig. 9(b) shows low-I deviations from the power-law form. The data as presented do not exclude a finite yield stress as I approaches zero. Please restrict the claim to the fitted range I > 0.003, or add low-I data and a quantitative comparison between the power-law and yield-stress forms (for example, residual analysis or an information criterion applied over the full I range).
  3. [Section III.C, Eqs. (35)-(36)] The discussion around Eqs. (35)-(36) implies that the dash-dotted curve in Fig. 7 provides independent evidence for a no-yield-stress mu(I). However, Eq. (36) is an algebraic restatement of Eq. (33) combined with the definition 1/eta-hat = m I^n and eta = mu P / gamma-dot; it is not a new prediction. The text should make clear that the mu(I) power law is derived from, not tested against, the 1/eta-hat data, and should not present the agreement of the dash-dotted line with part of the mu-I data as independent support.
minor comments (5)
  1. [Appendix A, Eq. (A4)] The last factor in Eq. (A4) is written as (H - y)^{1/2} but should evidently be (H - y)^{3/2}; the main-text Eq. (34) has the correct form.
  2. [Section III.C, after Eq. (37)] The sentence 'with the bin flow data fitted in two inertial number ranges, i.e., I <= 0.03 and I > 0.03; phi_m.' is incomplete and should be rewritten.
  3. [Section IV, Conclusions] In the Conclusions, 'the inertial member' should read 'the inertial number'.
  4. [Fig. 12 caption and surrounding text] The caption and text should state explicitly that n = 1.905 is the tuned value rather than the value obtained from the direct fit in Fig. 11, so that readers do not mistake the comparison for an independent prediction.
  5. [Section III.C, Fig. 7 discussion] The statement that 'The values obtained for I0 are lower as compared to those reported earlier' lacks a specific citation or numerical comparison; please name the earlier values or the reference to which the comparison is made.

Circularity Check

1 steps flagged · score 6.0 of 10

Inclined-surface profile "validation" is a hand-tuned refit of the same ISF data, so the no-yield-stress power law is not independently confirmed.

  1. fitted input called prediction [Sec. III.C, Fig. 12, Eq. (34)]
    "For m = 2.024 and n = 1.922, the prediction of Eq. (34) differs substantially from the simulation data. However, a good match is obtained between them by tuning the exponent n to 1.905 as shown in Fig. 12. It is reasonable to consider n = 1.905 as there is a small difference between two curves for the inclined surface flow when 1/ηˆ is plotted by considering n = 1.905 (solid and dash-dotted lines in Fig. 11)."

    The inclined-surface velocity profile is presented as a test of Eq. (33), but m and n are fitted to the same inclined-surface-flow data whose velocity is then compared with Eq. (34). The paper explicitly changes n from the fitted value 1.922 to 1.905 to obtain agreement with that target profile, and φ used in Eq. (34) also comes from fitting the same flow. The Fig. 12 match is therefore a refit with a parameter adjusted against the very quantity being 'predicted,' rather than an independent validation of the power-law constitutive model.

full rationale

The paper's core content is an honest empirical characterization: the power-law relations (31), (32), (33) and (37) are fitted to DEM data with stated fit ranges, and the continuum model is described as based on these empirical correlations. Fitting a correlation to data is not circular. The coordinate transformation of Sec. III.B is derived in the paper via the eigenvalue argument of Eqs. (19)-(23), and the citation to Bhateja and Khakhar (45) is a pointer to prior usage, not the load-bearing justification; no self-citation chain forces the result. Eq. (36) is an algebraic consequence of the definitions η̂ = η/(ρ γ̇ d²) = μ/I² and Eq. (33), not an independent prediction, so it is not circular. However, the inclined-surface velocity profile is claimed as a test of Eq. (33), and this validation step is compromised: m = 2.024 and n are fitted to the same inclined-surface-flow data (Fig. 11), and n is then changed from 1.922 to 1.905 explicitly to match that flow's velocity profile (Fig. 12). The profile agreement therefore reduces to parameter adjustment and cannot independently confirm the constitutive model. Separately, the conclusion that Eq. (33) may replace a yield stress over the entire range of I extrapolates beyond the fitted range I > 0.003 and sits uneasily with the reported complex yielding behaviour for I < 0.02 and the low-I deviations in Fig. 9(b); this is a correctness/extrapolation concern rather than circularity. On balance, the circularity is partial: one central validation step is a fitted-input-called-prediction, while the empirical correlations themselves are openly presented as fits rather than disguised derivations.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a set of empirical power-law fits whose parameters are obtained from the same simulations used for validation. The DEM contact model and the generalized-Newtonian-fluid assumption are domain assumptions inherited from prior work. No new entities are introduced.

free parameters (6)
  • Jop model parameters µs, µm, I0 (Eq. 30, comparison only) = Bin: 0.344, 0.57, 0.149; ISF: 0.328, 0.458, 0.053
    Fitted to the µ-I data in Fig. 7; used only to compare with the power-law model, not in the final continuum model.
  • Power-law prefactor a and exponent b for 1/η vs I (Eq. 31) = Bin: a=1.25, b=1.674; ISF: a=1.468, b=1.694
    Fitted to the inverse scaled viscosity data, I>0.01 (Fig. 9).
  • Power-law prefactor g and exponent h for u^2 vs I (Eq. 32) = Bin: g=0.529, h=1.599; ISF: g=0.519, h=1.538
    Fitted to fluctuation velocity data, I>0.01 (Fig. 10).
  • Power-law prefactor m and exponent n for 1/η̂ vs I (Eq. 33) = Bin: m=1.713, n=1.872; ISF: m=2.024, n=1.922
    Fitted to shear-rate-scaled viscosity data, I>0.003 (Fig. 11).
  • Tuned exponent n for velocity profile prediction = n=1.905
    Adjusted from the fitted ISF value 1.922 to match the simulated velocity profile in Fig. 12; this is parameter tuning in the validation step.
  • Solid-fraction fit parameters φm, r, s (Eq. 37) = ISF: 0.64, 0.077, 0.163; Bin I≤0.03: 0.644, 0.256, 0.481; Bin I>0.03: 0.662, 0.112, 0.169
    Fitted piecewise to the φ-I data in Fig. 13.
assumptions (6)
  • domain assumption DEM contact model (linear spring-dashpot with Coulomb friction) is a faithful microscopic model for the granular material.
    All rheological measurements are derived from this contact law (Sec. II, Eqs. 1-3); the results inherit any inaccuracies of the force model.
  • domain assumption The granular material at continuum level is a generalized Newtonian fluid with σ = PI - 2ηD.
    Assumed in Sec. III.B (Eq. 14) and justified only by approximate colinearity of principal axes (Fig. 4) and stress symmetry (Fig. 5).
  • domain assumption The analysis-region flow is incompressible (∇·v=0).
    Used to write Eqs. (12)-(13); measured dilation is below 10% of shear rate in most of the domain (Fig. 3).
  • ad hoc to paper The empirical scaling u^2 = g I^h closes the continuum model without a fluctuation-energy equation.
    Introduced in Sec. III.C (Eq. 32) as a modeling choice; it is fitted to the same simulations and has no independent support.
  • domain assumption The quasi-2D bin flow can be treated as locally two-dimensional isochoric for the coordinate transformation.
    The transformation in Sec. III.B is proven for 2D isochoric flows; the bin thickness is 8d and out-of-plane gradients are assumed negligible.
  • domain assumption Kinetic-theory scalings η = ρudF(φ) and P = ρu^2f(φ) from prior literature motivate the variable normalization.
    Used to define scaled variables in Sec. III.C based on refs. 53, 54, 87; no derivation is given in this paper.

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Pith. "Pith review of Analysis of granular rheology in a quasi-two-dimensional slow flow by means of discrete element method based simulations." pith.science (2026). https://pith.science/paper/QRHAA4IM

@misc{pith2026190805080,
  author       = {Pith},
  title        = {Pith review of: Analysis of granular rheology in a quasi-two-dimensional slow flow by means of discrete element method based simulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QRHAA4IM}},
  note         = {Machine review of arXiv:1908.05080}
}
read the original abstract

The steady flow of spherical particles in a rectangular bin is studied using the Discrete Element Method (DEM) for different flow rates of the particles from the bin, in the slow flow regime. The flow has two non-zero velocity components and is more complex than the widely studied unidirectional shear flows. The objective of the study is to characterize, in detail, the local rheology of the flowing material. The flow is shown to be nearly constant density, with a symmetric stress tensor and the principal directions of the stress and rate of strain tensors nearly colinear. The local rheology is analyzed using a coordinate transformation which enables direct computation of the viscosity and components of the pressure assuming the granular material to be a generalized Newtonian fluid. The scaled viscosity, fluctuation velocity and volume fraction are shown to follow power law relations with the inertial number, a scaled shear rate, and data for different flow rates collapse to a single curve in each case. Results for flow of the particles on an inclined surface, presented for comparison, are similar to those for the bin flow, but with a lower viscosity and a higher solid fraction due to layering of the particles. The in plane normal stresses are nearly equal and slightly larger than the third component. All three normal stresses correlate well with the corresponding fluctuation velocity components. Based on the empirical correlations obtained, a continuum model is presented for computation of granular flows.

Figures

Figures reproduced from arXiv: 1908.05080 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Gravity-induced flow of particles in a rectangular bin. Green particles shown at top [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Streamlines and spatial distribution of the magnitude of mean velocity [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Spatial distribution of scaled dilation rate [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) The distribution function [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Variation of stress component [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. A schematic representation of the streamlines of different kinds of flow. [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Variation of the effective friction coefficient [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Number density ( [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Variation of the inverse of the scaled viscosity 1 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Variation of the square of scaled fluctuation velocity [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Variation of the inverse of the scaled viscosity 1 [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Variation of [PITH_FULL_IMAGE:figures/full_fig_p021_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Variation of solid fraction [PITH_FULL_IMAGE:figures/full_fig_p022_13.png]
Figure 16
Figure 16. Figure 16: The data for all three components collapse to a single curve, indicating that the [PITH_FULL_IMAGE:figures/full_fig_p023_16.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Variation of the scaled normal stress difference (a) [PITH_FULL_IMAGE:figures/full_fig_p024_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. (a) Inverse of the scaled pressure components 1 [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Variation of the inverse of the scaled pressure components 1 [PITH_FULL_IMAGE:figures/full_fig_p026_16.png]

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Reviewed August 14, 2026 · model on record in the stance chip above.