The Thickness of Steady Plane Shear Flows of Circular Disks Driven by Identical Boundaries

We employ balance laws and constitutive relations for rapid, dense, plane flows of identical circular disks together with boundary conditions at a bumpy wall to analyse steady shearing flows maintained by the relative motion of two identical, parallel walls. The disks and the walls are assumed to be frictionless and nearly elastic. Given the properties of the flowing disks, those of the boundary, and the ratio of the tangential and normal tractions applied to the boundary, we determine what the distance between the walls must be for a steady solution to be possible. For these steady solutions we relate the velocity of the walls to the normal and tangential tractions applied to them. We find that in certain circumstances steady motions may be maintained even when the ratio of tangential to normal traction is much less than its value in a homogeneous simple shear. In the Appendix, the corresponding results for spheres are outlined.