Granular Flows: Boundary Conditions for Slightly Bumpy Walls

An approximate analytical solution of the boundary conditions of Pasquarell and Ackermann is presented for the simple shear Couette flow of identical, smooth, slightly inelastic disks in the limit where a parametric measure of the wall form roughness is small. Excellent agreement is shown between the approximate solution and a numerical solution of the original integral equations. The solution predicts that simple shear Couette flow only occurs for unique combinations of wall and particle properties. As the wall form roughness decreases, so must the wall elasticity in order to sustain simple shear. The theory is applied to previous experimental work in which shear cells were used to study granular flows. Experimental observations comparing soft to hard walls and bumpy to smooth walls are explained in the context of the theory. It is shown that the accuracy of such experimental results is greatly increased when the walls in the shear cells are nearly elastic and roughened with closely spaced elements of dimensions on the same order or larger than the diameter of the sheared particles. These boundary conditions may be employed in problems involving more dissipative disks than those considered herein.