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### Abstract

We introduce a simple phenomenological modification to the hydrodynamic equations for dense flows of identical, frictionless, inelastic disks and show that the resulting theory describes the area fraction dependence of quantities that are measured in numerical simulations of steady, homogeneous shearing flows and steady, fully developed flows down inclines. The modification involves the incorporation of a length scale other than the particle diameter in the expression for the rate of collisional dissipation. The idea is that enduring contacts between grains forced by the shearing reduce the collisional rate of dissipation while continuing to transmit momentum and force. The length and orientation of the chains of particles in contact are determined by a simple algebraic equation. When the resulting expression for the rate of dissipation is incorporated into the theory,numerical solutions of the boundary-value problem for steady, fully developed flow of circular disks down a bumpy incline exhibit a core with a uniform area fraction that decreases with increasing angles of inclination. When the height at which an inclined flow stops is assumed to be proportional to this chain length, a scaling between the average velocity, flow height, and stopping height similar to that seen in experiments and numerical simulations is obtained from the balance of fluctuation energy.

The author is grateful to J. Carlson, G. Lois, N. Mitarai, and H. Nakanishi for sharing data from their numerical simulations and for conversations related to their interpretation, to H. Shen for discussions on structure formation in shearing flows, and to J. McElwaine for suggesting the alternative way to determine the length scale in the rate of dissipation. This research was supported by NASA Microgravity Fluid Physics Program Grants No. NAG3-2353 to Cornell University and No. NAG3-2717 to Clarkson University and by NSF Grant No. PHY99-07949 to the Kavli Institute of Theoretical Physics, University of California, Santa Barbara, where the work was initiated.

I. INTRODUCTION

II. BACKGROUND

III. THEORY

A. Constitutive relations

B. Chain length

C. Predictions

IV. NUMERICAL SOLUTIONS

V. HOMOGENEOUS ENERGY BALANCE

VI. STOPPING HEIGHT

VII. SCALING

VIII. CONCLUSIONS

### Key Topics

- Kinetic theory
- 12.0
- Constitutive relations
- 10.0
- Elasticity theory
- 9.0
- Number theory
- 9.0
- Boundary value problems
- 7.0

## Figures

Dimensionless chain length or cluster size vs area fraction for three values of the coefficient of restitution .

Dimensionless chain length or cluster size vs area fraction for three values of the coefficient of restitution .

Values of the scaled temperature vs measured in numerical simulations of simple shear by Lois *et al.* (open symbols) and the predicted behavior (line).

Values of the scaled temperature vs measured in numerical simulations of simple shear by Lois *et al.* (open symbols) and the predicted behavior (line).

Values of the scaled shear stress vs measured in numerical simulations of simple shear (open symbols as in Fig. 2) and the predicted behavior: (line).

Values of the scaled shear stress vs measured in numerical simulations of simple shear (open symbols as in Fig. 2) and the predicted behavior: (line).

Values of the scaled pressure vs measured in numerical simulations of simple shear (open symbols as in Fig. 2) and the predicted behavior: (line).

Values of the scaled pressure vs measured in numerical simulations of simple shear (open symbols as in Fig. 2) and the predicted behavior: (line).

Values of the stress ratio vs measured in numerical simulations of simple shear (open symbols as in Fig. 2), numerical simulations of inclined flow by Mitarai and Nakanishi (stars), and the predicted behavior for : (line).

Values of the stress ratio vs measured in numerical simulations of simple shear (open symbols as in Fig. 2), numerical simulations of inclined flow by Mitarai and Nakanishi (stars), and the predicted behavior for : (line).

Values of the dimensionless rate of dissipation vs measured in numerical simulations of inclined flow (stars as in Fig. 5) and the predicted behavior: (line).

Values of the dimensionless rate of dissipation vs measured in numerical simulations of inclined flow (stars as in Fig. 5) and the predicted behavior: (line).

Profiles of dimensionless fluctuation velocity vs height in particle diameters for , , , and , unless otherwise indicated.

Profiles of dimensionless fluctuation velocity vs height in particle diameters for , , , and , unless otherwise indicated.

Profiles of dimensionless mean velocity vs height in particle diameters. The line designations are the same as in Fig. 7.

Profiles of dimensionless mean velocity vs height in particle diameters. The line designations are the same as in Fig. 7.

Profiles of area fraction vs height in particle diameters. The line designations are the same as in Fig. 7.

Profiles of area fraction vs height in particle diameters. The line designations are the same as in Fig. 7.

Profiles of dimensionless fluctuation velocity vs height in particle diameters that result from the numerical solution of the full theory (full lines) and the algebraic approximation in the dense theory (dotted lines).

Profiles of dimensionless fluctuation velocity vs height in particle diameters that result from the numerical solution of the full theory (full lines) and the algebraic approximation in the dense theory (dotted lines).

in particle diameters vs the tangent of the angle of inclination for and calculated from the dependence of on in simple shearing (full line) and from the numerical solution of the boundary-value problem for the full theory with and (open circles).

in particle diameters vs the tangent of the angle of inclination for and calculated from the dependence of on in simple shearing (full line) and from the numerical solution of the boundary-value problem for the full theory with and (open circles).

Prochnow’s simulation data for the inclined flow of circular disks at (solid lines), 22° (dotted lines), and 17° (dashed lines), with , 1.7, and 5.4, respectively.

Prochnow’s simulation data for the inclined flow of circular disks at (solid lines), 22° (dotted lines), and 17° (dashed lines), with , 1.7, and 5.4, respectively.

Two scalings of Pouliquen’s experimental data for the inclined flow of spheres at (solid lines), 25° (dotted lines), and 22° (dashed lines), with , 2.1, and 4.2, respectively.

Two scalings of Pouliquen’s experimental data for the inclined flow of spheres at (solid lines), 25° (dotted lines), and 22° (dashed lines), with , 2.1, and 4.2, respectively.

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