The flow of granular materials through a vertical channel is examined using the discrete element method (DEM) and the recent continuum models of Henann & Kamrin (Proc. Natl Acad. Sci. USA, vol. 110, 2013, pp. 6730–6735), Barker et al. (Proc. R. Soc. Lond. A, vol. 473, 2017, p. 20160846), Schaeffer et al. (J. Fluid Mech., vol. 874, 2019, pp. 926–951) and Dsouza & Nott (J. Fluid Mech., vol. 888, 2020, p. R3). The channel is bounded by walls separated by a distance $2 \, W$ in the $x$ -direction. For the DEM, periodic boundary conditions are used in the $z$ - and $y$ - (vertical) directions with no exit at the bottom of the channel. The governing equations reduce to ordinary differential equations in the $x$ -direction. There is a plug layer near the centre and a shear layer near the wall, as observed in experiments. There is a decrease in the solids fraction $\phi$ in the shear layer, except for the models of Barker et al. and Henann & Kamrin. A modification of the latter gives more realistic $\phi$ profiles. The thickness of the shear layer depends on $2\,W$ and the bulk solids fraction $\bar {\phi }$ . For all the models, solutions could not be obtained for some parameter values. An example is the negative fluidity in the model of Henann & Kamrin. The model of Dsouza & Nott predicts much higher normal stresses, possibly because of large contributions from the non-local terms. None of the models specify a complete set of boundary conditions (b.c.). The DEM results suggest that the slip velocity and the wall friction b.c. lead to a slip length and an angle of wall friction that are independent of $2\,W$ . The models are based on extensions of the equations for slow, rate-independent flow. A model that includes collisional effects, such as kinetic theory, should be combined with the present models. A preliminary analysis of the kinetic theory model of Berzi et al. (J. Fluid Mech., vol. 885, 2020, p. A27), shows that it may have undesirable feature.

The steady dense granular flow in a vertical channel bounded by flat frictional walls in one horizontal direction and with periodic boundary conditions in the other horizontal and vertical directions is studied using the discrete element method. The shape of the scaled velocity profile is characterized quantitatively by a universal exponential function, and the ratio of the maximum and slip velocities is independent of the average volume fraction $\bar {\phi }$ and the channel width $W$ . For sufficiently wide channels, the maximum and slip velocities increase proportional to $\sqrt {W}$ , and the thickness of the shearing zones increases proportional to $W$ . There are four zones in the flow, each with distinct dynamical properties. There is no shear in the plug zone at the centre, but there is particle agitation, and the volume fraction $\phi$ is lower than the random close packing volume fraction $\phi _{rcp}$ . In the adjoining dense shearing zone, $\phi$ is greater than the volume fraction for arrested dynamics $\phi _{ad} = 0.587$ , and the granular temperature and shear rate depend on the particle stiffness. Adjacent to the dense shearing zone is the loose shearing zone where $\phi < \phi _{ad}$ . Here, the properties do not depend on the particle stiffness, and the constitutive relations are well described by hard-particle models. The rheology in the loose shearing zone is similar to the dense flow down an inclined plane. There is high shear and a sharp decrease in $\phi$ in the wall shearing zone of thickness about two particle diameters, where the particle angular velocity is different from the material rotation rate due to the presence of the wall.

The discrete element method has been used to study the lift F L on a stationary disc immersed coaxially in a slowly rotating cylinder containing a granular material. In a tall granular column, F L rises with the immersion depth h, but reaches a roughly constant asymptote at large h, in agreement with previous studies. Our results indicate that the argument in some earlier studies that F L is proportional to the static stress gradient is incorrect. Instead, our results show that the lift is caused by an asymmetry in the dilation and shear rate between the regions above and below the disc. We argue that the cause of the lift is similar to that in fluids, namely that it arises as a result of the disturbance in the velocity and density fields around the body due to its motion relative to the granular bed.Positive and negative values of ðz2hÞ correspond to the regions below and above the disc, respectively. Here the data of (a) and (b) correspond to H562 d p and H5222 d p , respectively, for a disc of thickness t510 d p and different depths of immersionh. Panel (c) shows the difference in the solids fraction (averaged in the same manner) between the lower and upper surfaces of the disc; the blue triangles and red asterisks are data for the short and tall columns, respectively. [Color figure can be viewed at wileyonlinelibrary.com] AIChE Journal

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