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 sand͒ in quasi-twodimensional 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 (Frϭ 2 R/g, where is the angular speed, R is the cylinder radius, and g the acceleration due to gravity͒ and with increase in size ratio (sϭd/R, where d is the particle diameter͒. 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 (). 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 (␤), 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 ͑1998͔͒ and Makse ͓Phys. Rev. Lett. 83, 3186 ͑1999͔͒ show good agreement at low Fr while that of Khakhar et al. ͓Phys. Fluids, 9, 31 ͑1997͔͒ gives good predictions over the entire range of parameters considered. An analysis of the data indicate that the velocity gradient (␥ ) is nearly constant along the layer at low Fr, and the value calculated at the layer midpoint varies as ␥ 0 ϰ͓g sin(␤ 0 Ϫ␤ s )/d cos ␤ s ͔ 1/2 for all the experimental data, where ␤ s is the static angle of repose and ␤ 0 is the interface angle at the layer midpoint. An extension of ''heap'' models ͑BCRE, BRdG͒ is used to predict the interface angle profiles, which are in reasonable agreement with experimental measurements.

Foams in a rotating drum: An analogy with granular materials

“…As observed with foams, the deformations of the granular assembly interface depend on the cell angular velocity ω. The inclination angle as well as the S-shape amplitude increase with ω [7,24,25]. These observations clearly evidence the similarities between foams and granular materials.…”

“…Depending on the drum angular velocity, several flow regimes occur at the surface. For example, at relatively low angular velocities, the surface inclines and the grains flow along a flat interface, while at large angular velocities, the inclined surface is deformed into an S-shape [6,7]. These phenomena have essentially been studied for dry granular media.…”

“…In the last few years, efforts have been made to unify the rheology of dry and immersed granular materials [7,[10][11][12]. Dry granular flows can be phenomenologically described by a frictional rheology based on only one dimensionless control parameter, the inertial number I [3].…”

Kinematics of monodisperse and bidisperse granular flows in quasi-two-dimensional bounded heaps

“…du/dx = 0, where u is the streamwise velocity and x is the streamwise direction; figure 1c-d). While flow in rotating tumblers is similar to bounded heap flow in that u varies in the streamwise direction [2][3][4][5], the location of the free surface in the rotating tumbler remains fixed, as in unbounded heap and inclined chute flows (figure 1b-d). Thus, bounded heap flow is unique in comparison with these other flows in that it is never fullydeveloped and its free surface continually rises.…”

Sshape of a granular pile in a rotating drum

“…Depending on the angular velocities, two different regimes occur. At low rotation speed the free surface of the pile is inclined and flat, but a significant curvature appears at high rotation speed: the so called S shape [4,5,6]. Up until now this transition remains relatively unexplored [1,7].…”