Slow and intermediate flow of a frictional bulk powder in the Couette geometry

Most current research in the field of dry, non-aerated powder flows is directed toward rapid granular flows of large particles. Slow, frictional, dense flows of powders in the so-called quasi-static regime were also studied extensively using Soil Mechanics principles. The present paper describes the rheological behavior of powders in the “intermediate” regime lying between the slow and rapid flow regimes. Flows in this regime have direct industrial relevance. Such flows occur when powders move relative to solid walls in hoppers, bins and around inserts or are mixed in high and low shear mixers using moving paddles. A simple geometry that of a Couette device is used as a benchmark of more complicated flows. The constitutive equations derived by Schaeffer [J. Differ. Equ. 66 (1987) 19] for slow, incompressible powder flows were used in a new approach proposed by Savage [J. Fluid Mech. 377 (1998) 1] to describe flows in the intermediate regime. The theory is based on the assumption that both stress and strain-rate fluctuations are present in the powder. Using Savage's approach, we derive an expression for the average stress that reduces to the quasi-static flow limit when fluctuations go to zero while, in the limit of large fluctuations, a “liquid-like”, “viscous” character is manifested by the bulk powder. An analytical solution of the averaged equations for the specific geometry of the Couette device is presented. We calculate both the velocity profile in the powder and the shear stress in the sheared layer and compare these results to experimental data. We show that normal stresses in the sheared layer depend linearly on depth (somewhat like in a fluid) and that the shear stress in the powder is shear rate dependent. We also find that the velocity of the powder in the vicinity of a rough, moving boundary, decays exponentially so that the flow is restricted to a small area adjacent to the wall. The width of this area is of the order of 10–13 particle diameters. In the limit of very small particles, this is tantamount to a shear band-type behavior near the wall.