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

B Debnath and V Kumaran and KK Rao, JOURNAL OF FLUID MECHANICS, 937, A33 (2022).

DOI: 10.1017/jfm.2022.119

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 2W 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 2W and the bulk solids fraction (phi) over bar. 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 2W. 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.

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