Particle Migration due to Non-Uniform Laminar Flow

Abstract Using methods of granular mechanics in the quasi-static limit, with inter-particle interactions derived from the lubrication limit, the intensity of velocity fluctuations in the slurry is associated with fluctuations in the local distribution of inter-particle distances. These are shown to consist of a vector intensity and a scalar intensity; the former couples to the first velocity gradient, the latter (which is associated with solidosity fluctuations) couples to the second velocity gradient. Rheologies for both are presented, as is the rheology that links the particle pressure to the intensity of the velocity fluctuations (also known as the ‘granular temperature’) to the dispersive pressure. The rheologies are informed by experimental results. The granular temperature profile, modified from previous work, is responsible for axial particle migration (Bagnold effect). Two broad categories are assessed: symmetrical vertical and non-symmetrical lateral flow. For the latter the roughness of the boundary walls and a non-zero density contrast are important; this case is studied for a system in which flow effects are confined to the immediate vicinity of the boundary. Sensitivity analysis reveals several key variables including the parameters that control a slipping boundary condition and the mean solidosity in the conduit. For lateral flow, a sedimentary deposit with a solidosity profile may develop near the upper or lower boundary. The theory predicts an approximate relation between the fluid-particle density contrast and sediment thickness as a function of the mean flow rate, conduit width, the mean particle diameter and fluid viscosity that has utility in a range of engineering and geological situations where particulate matter is transported in the laminar flow regime.