Dense granular flow down an inclined plane: from kinetic theory to granular dynamics
Authors: KUMARAN V. 
Published: March 2008

The hydrodynamics of the dense granular flow of rough inelastic particles down an inclined plane is analysed using constitutive relations derived from kinetic theory. The basic equations are the momentum and energy conservation equations, and the granular energy conservation equation contains a term which represents the dissipation of energy due to inelastic collisions. A fundamental length scale in the flow is the ‘conduction length’ δ=(d/(1-en)1/2), which is the length over which the rate of conduction of energy is comparable to the rate of dissipation. Here, d is the particle diameter and en is the normal coefficient of restitution. For a thick granular layer with height h ≫ δ, the flow in the bulk is analysed using an asymptotic analysis in the small parameter δ/h. In the leading approximation, the rate of conduction of energy is small compared to the rates of production and dissipation, and there is a balance between the rate of production due to mean shear and the rate of dissipation due to inelastic collisions. A direct consequence of this is that the volume fraction in the bulk is a constant in the leading approximation. The first correction due to the conduction of energy is determined using asymptotic analysis, and is found to be O(δ/h)2 smaller than the leading-order volume fraction. The numerical value of this correction is found to be negligible for systems of practical interest, resulting in a lack of variation of volume fraction with height in the bulk.

The flow in the ‘conduction boundary layers’ of thickness comparable to the conduction length at the bottom and top is analysed. Asymptotic analysis is used to simplify the governing equations to a second-order differential equation in the scaled cross-stream coordinate, and the resulting equation has the form of a diffusion equation. However, depending on the parameters in the constitutive model, it is found that the diffusion coefficient could be positive or negative. Domains in the parameter space where the diffusion coefficients are positive and negative are identified, and analytical solutions for the boundary layer equations, subject to appropriate boundary conditions, are obtained when the diffusion coefficient is positive. There is no boundary layer solution that matches the solution in the bulk for parameter regions where the diffusion coefficient is negative, indicating that a steady solution does not exist. An analytical result is derived showing that a boundary layer solution exists (diffusion coefficient is positive) if, and only if, the numerical values of the viscometric coefficients are such that volume fraction in the bulk decreases as the angle of inclination increases. If the numerical values of the viscometric coefficients are such that the volume fraction in the bulk increases as the angle of inclination increases, a boundary layer solution does not exist.

The results are extended to dense flows in thin layers using asymptotic analysis. Use is made of the fact that the pair distribution function is numerically large for dense flows, and the inverse of the pair distribution function is used as a small parameter. This approximation results in a nonlinear second-order differential equation for the pair distribution function, which is solved subject to boundary conditions. For a dissipative base, it is found that a flowing solution exists only when the height is larger than a critical value, whereas the temperature decreases to zero and the flow stops when the height becomes smaller than this critical value. This is because the dissipation at the base becomes a larger fraction of the total dissipation as the height is decreased, and there is a minimum height below which the rate of production due to shear is not sufficient to compensate for the rate of dissipation at the base. The scaling of the minimum height with dissipation in the base, the bulk volume fraction and the parameters in the constitutive relations are determined. From this, the variation of the minimum height on the angle of inclination is obtained, and this is found to be in qualitative agreement with previous experiments and simulations.