^{1}and James T. Jenkins

^{2}

### Abstract

We study dry flows of two types of spheres down an inclined, rigid, bumpy bed in the absence of sidewalls. The flow is assumed to be steady and uniform in all but the direction normal to the free surface, collisions between particles are dissipative, and the sizes and masses of the particles are not too different. We restrict our analysis to dense flows and use an extension of kinetic theory to predict the concentration of the mixture and the profile of mixture velocity. A kinetic theory for a binary mixture of nearly elastic spheres that do not differ by much in their size or mass is employed to predict profiles of the concentration fraction of one type of sphere. We also determine the ratio of the radii and of the masses of the two species for which there is no segregation. We compare the predictions of the theory to the results of numerical simulations.

The authors are grateful to the Department of Civil, Environmental and Mechanical Engineering at the University of Trento and the School of Civil and Environmental Engineering at Cornell University for their financial support and hospitality.

I. INTRODUCTION

II. THE BALANCE EQUATIONS

A. Segregation

B. Mixture fluctuation velocity, concentration, and average velocity

III. MIXTURE RESULTS

A. Velocity, concentration, and concentration fraction

B. Special case: No segregation

IV. CONCLUSION

### Key Topics

- Kinetic theory
- 20.0
- Elasticity
- 8.0
- Velocity measurement
- 7.0
- Differential equations
- 5.0
- Elasticity theory
- 5.0

## Figures

Profiles of average velocity and concentration for identical spheres with c M = 0.58, α = 0.50, θ = π /3, and e eff = 0.65 at four angles of inclination. Distance and velocity are made dimensionless by particle diameter d and (gd)^{1/2}, respectively. The lines are the predictions for the uniform, dense region of the flow, and the symbols are the values measured in the numerical simulation.

Profiles of average velocity and concentration for identical spheres with c M = 0.58, α = 0.50, θ = π /3, and e eff = 0.65 at four angles of inclination. Distance and velocity are made dimensionless by particle diameter d and (gd)^{1/2}, respectively. The lines are the predictions for the uniform, dense region of the flow, and the symbols are the values measured in the numerical simulation.

The relation between the mixture volume fraction, roughly constant in the dense limit, and the inclination angle of the flow. The symbols are the depth-averaged concentration measurements of Tripathi and Khakhar ^{13} and the solid line is the prediction for the uniform concentration in the dense region of the flow when c M = 0.58, α = 0.50, θ = π/3, and e eff = 0.65.

The relation between the mixture volume fraction, roughly constant in the dense limit, and the inclination angle of the flow. The symbols are the depth-averaged concentration measurements of Tripathi and Khakhar ^{13} and the solid line is the prediction for the uniform concentration in the dense region of the flow when c M = 0.58, α = 0.50, θ = π/3, and e eff = 0.65.

Profiles of the predicted mixture velocity u, local concentration fraction c A /c of the more massive spheres and mixture concentration c when e eff = 0.65, r A /r B = 1, m A /m B = 3, α = 0.50, and f A = 0.50 (equivalent to V A /V = 0.5) for four angles of inclination. Distance from the base and velocity are made dimensionless by particle diameter 2r B and (2gr B )^{1/2}, respectively. The lines are the predictions for the uniform, dense region of the flow, and the symbols are the values measured in the numerical simulation.

Profiles of the predicted mixture velocity u, local concentration fraction c A /c of the more massive spheres and mixture concentration c when e eff = 0.65, r A /r B = 1, m A /m B = 3, α = 0.50, and f A = 0.50 (equivalent to V A /V = 0.5) for four angles of inclination. Distance from the base and velocity are made dimensionless by particle diameter 2r B and (2gr B )^{1/2}, respectively. The lines are the predictions for the uniform, dense region of the flow, and the symbols are the values measured in the numerical simulation.

Profiles of the predicted mixture velocity u, local concentration fraction c A /c of the larger spheres and mixture concentration c in the uniform, dense region of the flow, when e eff = 0.65, r A /r B = 1.5, m A /m B = (1.5)^{3}, α = 0.50, and f A = 0.23 (equivalent to V A /V = 0.5) at four angles of inclination. Distance from the base and velocity are made dimensionless by particle diameter 2r B and (2gr B )^{1/2}, respectively. The lines are the predictions for the uniform, dense region of the flow, and the symbols are the values measured in the numerical simulation.

Profiles of the predicted mixture velocity u, local concentration fraction c A /c of the larger spheres and mixture concentration c in the uniform, dense region of the flow, when e eff = 0.65, r A /r B = 1.5, m A /m B = (1.5)^{3}, α = 0.50, and f A = 0.23 (equivalent to V A /V = 0.5) at four angles of inclination. Distance from the base and velocity are made dimensionless by particle diameter 2r B and (2gr B )^{1/2}, respectively. The lines are the predictions for the uniform, dense region of the flow, and the symbols are the values measured in the numerical simulation.

Profiles of the predicted mixture velocity u, local concentration fraction c A /c of the larger spheres and mixture concentration c in the uniform, dense region of the flow, when e eff = 0.65, r A /r B = 1.5, m A /m B = (1.5)^{3}, α = 0.50, and ϕ = 25° at five values of the total volume fraction of the larger spheres. Distance from the base and velocity are made dimensionless by particle diameter 2r B and (2gr B )^{1/2}, respectively. The lines are the predictions for the uniform, dense region of the flow, and the symbols are the values measured in the numerical simulation.

Profiles of the predicted mixture velocity u, local concentration fraction c A /c of the larger spheres and mixture concentration c in the uniform, dense region of the flow, when e eff = 0.65, r A /r B = 1.5, m A /m B = (1.5)^{3}, α = 0.50, and ϕ = 25° at five values of the total volume fraction of the larger spheres. Distance from the base and velocity are made dimensionless by particle diameter 2r B and (2gr B )^{1/2}, respectively. The lines are the predictions for the uniform, dense region of the flow, and the symbols are the values measured in the numerical simulation.

Profiles of the predicted mixture velocity u, local concentration fraction c A /c of the larger spheres and mixture concentration c in the uniform, dense region of the flow, when e eff = 0.65, V A /V = 0.5, α = 0.50, and ϕ = 25° at four values of the radii ratio. Distance from the base and velocity are made dimensionless by particle diameter 2r B and (2gr B )^{1/2}, respectively. The lines are the predictions for the uniform, dense region of the flow, and the symbols are the values measured in the numerical simulation.

Profiles of the predicted mixture velocity u, local concentration fraction c A /c of the larger spheres and mixture concentration c in the uniform, dense region of the flow, when e eff = 0.65, V A /V = 0.5, α = 0.50, and ϕ = 25° at four values of the radii ratio. Distance from the base and velocity are made dimensionless by particle diameter 2r B and (2gr B )^{1/2}, respectively. The lines are the predictions for the uniform, dense region of the flow, and the symbols are the values measured in the numerical simulation.

Profiles of the predicted mixture velocity u, local concentration fraction c A /c of the larger spheres and mixture concentration c in the uniform, dense region of the flow, when r A /r B = 1.5, e eff = 0.65, α = 0.50, ϕ = 25°, and V A /V = 0.5 (f A = 0.23) at two values of the density ratio. Distance from the base and velocity are made dimensionless by particle diameter 2r B and (2gr B )^{1/2}, respectively. The lines are the predictions for the uniform, dense region of the flow, and the symbols are the values measured in the numerical simulation.

Profiles of the predicted mixture velocity u, local concentration fraction c A /c of the larger spheres and mixture concentration c in the uniform, dense region of the flow, when r A /r B = 1.5, e eff = 0.65, α = 0.50, ϕ = 25°, and V A /V = 0.5 (f A = 0.23) at two values of the density ratio. Distance from the base and velocity are made dimensionless by particle diameter 2r B and (2gr B )^{1/2}, respectively. The lines are the predictions for the uniform, dense region of the flow, and the symbols are the values measured in the numerical simulation.

Curves of the ratio δm/δr versus concentration c for various values of the effective coefficient of restitution e eff along which segregation is predicted to be absent.

Curves of the ratio δm/δr versus concentration c for various values of the effective coefficient of restitution e eff along which segregation is predicted to be absent.

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