Comparison of the compressible $\mu (I)$ class of models and non-local models with the discrete element method for steady fully developed flow of cohesionless granular materials through a vertical channel. (25th April 2022)

• Record Type: Journal Article
• Authors: Debnath, Bhanjan; Kumaran, V.; Rao, K. Kesava
• Is Part Of: Journal of fluid mechanics. Volume 937(2022)
• Publication Date: 2022-04-25
• Subjects: dry granular material; Fluid mechanics -- Periodicals; 532.005
• DOI: 10.1017/jfm.2022.119
• Languages: English
• ISSNs: 0022-1120

Abstract

Abstract: Abstract : 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.