Scaling relations for granular flow in quasi-two-dimensional rotating cylinders

Abstract

An experimental study of the flow of different materials (steel balls, glass beads, and sand) in quasi-two-dimensional rotating cylinders is carried out using flow visualization. The flow in the rotating cylinder comprises of a thin-flowing surface layer with the remaining particles rotating as a fixed bed. Experimental results indicate that the scaled layer thickness increases with increasing Froude number (Fr=ω²R/g, where ω is the angular speed, R is the cylinder radius, and g the acceleration due to gravity) and with increase in size ratio (s=d/R, where d is the particle diameter). The free surface profile, is nearly flat at low Fr and becomes increasingly S shaped with increasing Fr. The layer thickness profiles, which are symmetric at low Fr become skewed at high values of Fr and small s. The dynamic angles of repose for all the materials studied show a near-linear increase with rotational speed (ω). Scaling analysis of the experimental data shows that the shape of the scaled surface profiles and the scaled layer thickness profiles are nearly identical when Froude number and size ratio are held constant, for each material. The surface profiles and layer thickness profiles are also found to be nearly independent of the material used. The dynamic angle of repose (β), however, does not scale with Fr and s and depends on the particle properties. The experimental results are compared to continuum models for flow in the layer. The models of Elperin and Vikhansky [Europhys. Lett. 42, 619 (1998)] and Makse [Phys. Rev. Lett. 83, 3186 (1999)] show good agreement at low Fr while that of Khakhar et al. [Phys. Fluids, 9, 31 (1997)] gives good predictions over the entire range of parameters considered. An analysis of the data indicate that the velocity gradient (γ) is nearly constant along the layer at low Fr, and the value calculated at the layer midpoint varies as γ₀∝[gsin(β₀-β_s)/dcosβ_s]^½ for all the experimental data, where β_s is the static angle of repose and β₀ is the interface angle at the layer midpoint. An extension of "heap"models (BCRE, BRdG) is used to predict the interface angle profiles, which are in reasonable agreement with experimental measurements.