Analytical solutions for dense, inclined, granular flow over a rigid, bumpy base

We first phrase a boundary-value problem for a dense, steady, fully-developed, gravitational flow of identical inelastic spheres over in inclined bumpy base in the absence of sidewalls. We then obtain approximate analytical solutions for the profiles of the solid volume fraction, the strength of the velocity fluctuations, and the mean velocity of the flow. We compare these with those obtained in numerical solutions of the exact equations.


Introduction
Flows of granular materials down inclines occur often in Nature and in industry. Consequently, it is important to be able to describe and predict them. Because such flow take place under the influence of gravity, they typically involve solid volume fractions greater than 0.30. In this sense, they are dense. Classical dense flow kinetic theory (e.g., [1]) has the capability of describing such flows (e.g,, [2]), provided that the volume fraction does not exceed about 0.49. Above this, the predictions for the properties of inclined flow fail, and alternatives must be employed.
One such alternative is a nonlocal extension [3] of the µ-I rheology [4,5] that has been applied to dense, inclined flows [6]. Another is an extension of the kinetic theory that takes into account, in a phenomenological way, the existence of velocity correlations [7,8,9]. In fact, the two approaches have much in common [10,11].
Here, we employ the latter to phrase a boundaryvalue problem for a dense, steady, fully-developed, gravitational flow of identical inelastic spheres over an inclined bumpy base in the absence of sidewalls. We then obtain approximate analytical solutions for the profiles of the solid volume fraction, the strength of the velocity fluctuations, and the mean velocity of the flow.

Model
We consider a steady, dense, fully-developed flow of identical, frictional, inelastic spheres of mass m, material density diameter d, with coefficient of restitution, e, and sliding friction, µ. The flow is driven by gravity over a rigid, bumpy bed, inclined at an angle of f, in the absence of sidewalls. The bed is composed of spheres, identical to those of the flow, randomly affixed * Corresponding author: jim.jenkins@cornell.edu to a plane with an average distance of s between their edges. The x-axis is in the direction of flow, the y-axis is normal to the flow, directed upward from the base, and g is the gravitational acceleration. Primes denote derivatives with respect to y. The solid volume fraction is n, the flow velocity is u, and the measure of the strength of the velocity fluctuations, T, is one third their meansquared value. We employ extended kinetic theory to phrase a boundary-value problem for the variations of these fields across the flow and obtain approximate, analytical solutions for them that we compare to numerical solutions of the exact equations.

Momentum balance
The component of the momentum balance along the flow governs the variation of the shear stress S: (1) While the component of the momentum balance normal to the flow governs the variation of the pressure P: When integrated together, The flows are assumed to be so dense that n is greater than 0.49 everywhere within the flow. Then, in the expressions for the stress, we retain only terms that result from collisional transfer of momentum and ignore those that result from transport. Then, [1] ρ s , where (5) in which the critical volume fraction, nc, at which the collisional interactions become singular, is given by [12] ( 6) and (7) where [1] (8) The dependance of the shear stress on the volume fraction may be eliminated in favor of a dependance on the pressure:

Energy balance
The balance of fluctuation energy in the flow relates the gradient of the energy flux, Q, the working of the shear stress, and the rate of collisional dissipation, G: Upon retaining only the collisional transfers, [1] where (12) or (13) and [1] where is the effective restitution [12] and L is the correlation length [12] (15) with (16) With Eqs. (9), (13), and (14), the energy equation becomes (17) When written in terms of the fluctuation velocity, this is (18)

Boundary conditions
At the free surface, we assume that and make the rough approximation that At the rigid, bumpy base, the flow momentum balance gives the slip velocity, [13] (19) where the slip coefficient, F, is given by (20) in which q is the bumpiness of the boundary, defined in terms of the diameter of the boundary spheres and the average separation of their edges by The energy balance at the base relates the energy flux, the rate of working of the shear stress through the slip velocity, and the rate of collisional dissipation, D: [13] (21) where P = 2ρ s ν(1+ e)GT , (11− e) .

Approximate analytical solutions
We make lengths dimensionless by d, velocities by stress by , and prescribe the mass holdup: To obtain the solution for w, we assume that the pressure vanishes at the top of the flow and approximate the integration of the normal momentum balance and the dimensionless pressure by (27) where is the depth-averaged volume fraction, and We regard this as a nonliner equation for h and use the Matlab function fsolve to determine h as a function of M and the other known parameters. Fig. (1) shows h as a function of M for the parameters indicated. The introduction of the correlation length and its dependence on volume fraction permits the extended kinetic theory to predict the observed differences in height at a given inclination (e.g., [14]), in contrast to the single κ I 1 (κ h) = BI 0 (κ h). so, In this, we make the rough approximation that (43) This determination of w0 completes the solutions.

Comparisons and conclusions
In Fig. 1, we show profiles predicted for n , w, and u by the aproximate analytical solutions and by the numerical integration of the system of differential equations and boundary conditions using the Matlab code bvp4c. The agreement is generally good, with the greatest error at the rigid base. This error is associated with the approximation made to determine w0. These results are encouraging and support an attempt to improve upon them.