Closing the $μ(I,Θ)$ rheology of dense granular flows: a validated fluctuation-energy balance for granular temperature and the universality of the exponent $p$

Kim and Kamrin (2020) showed that rescaling the stress ratio $μ$ by a power of the dimensionless granular temperature $Θ$ collapses discrete-element-simulation data across many dense granular flow geometries onto a single master curve, $μΘ^p = f(I)$ , with $p ≈ 1/6$ in 3D and $p ≈ 1/8$ in 2D — but the framework is not closed: the temperature field must be imported from particle simulations because no validated fluctuation-energy balance exists for the dense, near-jammed, contact-dominated regime. Key later progress — a 2026 preprint deriving $μ(I,Θ)$ from a contact-inclusive granular-temperature equation and recovering a generalized nonlocal-fluidity equation perturbatively, a 2025 vibration-controlled closure, and 2026 continuum implementations — has advanced every sub-part, yet no geometry-independent, near-jammed-accurate, predictive closure with boundary conditions has been validated, and the universality of $p$ remains unsettled (current evidence favors material-dependence, with large uncertainties). The problem remains open in its original generality: derive and validate a fluctuation energy balance for $Θ$ , with boundary conditions, that closes the rheology; quantify its relation to nonlocal fluidity models; and determine the status of $p$ .

Background

Dense granular flows — gravity-driven landslides, chute flows, and industrial handling of grains and powders — are commonly described by the phenomenological $μ(I)$ rheology, in which the stress ratio (macroscopic friction coefficient) $μ = τ/P$ , the shear stress $τ$ normalized by the confining pressure $P$ , is a single-valued function of the inertial number $I = dγ̇/√(P/ρ_s)$ , the ratio of the microscopic rearrangement time to the macroscopic deformation time for particles of diameter $d$ and material density $ρ_s$ under shear rate $γ̇$ (da Cruz et al. 2005; Jop et al. 2006). This 'local' description fails for heterogeneous flows: regions where $μ$ is below the yield threshold still creep, and shear-band widths depend on system size. To capture such nonlocal effects, the Nonlocal Granular Fluidity (NGF) model (Kamrin and Koval 2012) introduces a scalar 'fluidity' field $g = γ̇/μ$ that obeys a phenomenological reaction–diffusion equation, $ξ^2 ∇^2 g = g − g_{loc}$ , with a cooperativity length $ξ$ and a local fluidity $g_{loc}$ set by the homogeneous rheology.

Grain-scale velocity fluctuations, quantified by the granular temperature $T = δv^2/D$ ( $D$ is the spatial dimension and $δv$ the velocity fluctuation magnitude), act as an agitation variable. Classical kinetic theory of granular gases supplies a transport ('fluctuation energy') balance for $T$ , but its assumptions — momentum and energy transported by instantaneous binary collisions — break down in dense, enduring-contact, near-jammed flow. From discrete element method (DEM; particle-resolved dynamics simulations) data across many flow geometries, Kim and Kamrin (2020) found that rescaling the stress ratio by a power of the dimensionless temperature $Θ = ρ_s T/P$ collapses data that $μ(I)$ cannot: $μ Θ^p = f(I)$ , with $p ≈ 1/6$ for 3D sphere systems and $p ≈ 1/8$ for 2D disk systems; particle surface friction shifts the master curve $f(I)$ but not the exponent. In their own predictive chute-flow test, however, the authors imported the $Θ$ field from the particle simulations rather than computing it, precisely because no validated fluctuation-energy balance exists that remains accurate in the dense regime; their conclusion states that including such a balance law 'would provide Θ and complete the rheological model'. The paper also notes that the presumed diffusive spreading of $Θ$ suggests, without establishing, a connection to the NGF fluidity diffusion equation, and it leaves the status of $p$ open in general terms — describing a 'potentially universal scaling exponent $p$ , which, once identified, allows model fitting solely from simple shear data' — while itself varying only spatial dimension and surface friction; the concrete sweep over particle shapes, polydispersity, restitution coefficients, and boundary conditions reflects later treatments of the universality question rather than the original paper.

Problem Statement

Complete the $μ(I,Θ)$ constitutive framework for dense granular flows into a closed, predictive continuum model. Here $μ = τ/P$ is the stress ratio (shear stress $τ$ over confining pressure $P$ ), $I$ the inertial number (the ratio of microscopic to macroscopic deformation time scales), $T$ the granular temperature ( $T = δv^2/D$ , the mean squared velocity fluctuation per dimension, with $D$ the spatial dimension), and $Θ = ρ_s T/P$ its dimensionless form, with $ρ_s$ the particle material density. Specifically: (i) formulate and validate a fluctuation energy balance — a governing transport equation for $T$ , equivalently for $Θ$ — together with closure relations for its flux and dissipation and appropriate boundary conditions (including external agitation sources such as vibrating walls), that remains accurate in the dense, contact-dominated, near-jammed regime where classical kinetic theory fails, so that $Θ$ is predicted by the model rather than imported from experiment or particle simulation; (ii) determine whether, and quantify to what extent, the diffusion/reaction dynamics of $Θ$ within this balance are connected to the nonlocal granular fluidity (NGF) reaction–diffusion equation of Kamrin and Koval; and (iii) determine whether the scaling exponent $p$ in $μΘ^p = f(I)$ is universal or material-dependent, across space dimensions, particle shapes, polydispersity, surface friction, restitution coefficients, and boundary conditions. The three sub-goals (i)-(iii) are jointly posed; the properties listed after 'across' in (iii) define the scope over which the universality of $p$ is to be determined. These deliverables jointly serve one objective: turning the empirical power-law collapse into a self-contained continuum model whose flow-field predictions can be tested against configurations not used for calibration, for use in geophysical and industrial granular-flow prediction.

References

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