The flow of granular materials such as sand, snow, coal, and catalyst particles is a common occurrence in natural and industrial settings. They are important since a large fraction of the materials handled and processed in the chemical, metallurgical, pharmaceutical, and food-processing industries are granular in nature. The mechanics of these materials' flows is not well understood. This book describes the theories for granular flow based mainly on continuum models, although alternative discrete models are also discussed briefly. The level is appropriate for advanced undergraduates or beginning graduate students. The goal is to inform the reader about observed phenomena and some available models and their shortcomings and to visit some issues that remain unresolved. There is a selection of problems at the end of the chapters to encourage exploration, and extensive references are given.

Summary. A rigid-plastic Cosserat model has been used to study dense, fully developed flow of granular materials through a vertical channel. Frictional models based on the classical continuum do not predict the occurrence of shear layers, at variance with experimental observations. This feature has been attributed to the absence of a material length scale in their constitutive equations. The present model incorporates such a material length scale by treating the granular material as a Cosserat continuum. Thus localised couple stresses exist and the stress tensor is asymmetric. The velocity profiles predicted by the model are in close agreement with available experimental data. The predicted dependence of the shear layer thickness on the width of the channel is in reasonable agreement with data. In the limit of small ǫ (ratio of the particle diameter to the half-width of the channel), the model predicts that the shear layer thickness scaled by the particle diameter grows as ǫ −1/3 .

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.

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Climate change can be considered to be the greatest environmental challenge our world is facing today. Along with the need to ensure long-term assurance of energy supply, it imposes an obligation on all of us to consider ways of reducing our carbon footprint and sourcing more of our energy from renewable sources. Wind energy is one such source and forecasting methods for the prediction of wind speed are becoming increasingly significant due to the penetration of wind power as an alternative to conventional energy sources. This paper proposes time series models for short-term prediction of wind speed. The predictions are done for 1 day ahead using different time series models. For each model, these predicted values are compared with the actual values for the next day. Basic exponential smoothing for different duration of data was tested. A hybrid model with decomposition and exponential smoothing is proposed. A multiplicative decomposition is carried out for the measured data. Separate models were developed for seasonal and trend series and then combined to carry out the forecast. The models were tested for different durations of samples and different weather conditions. It is observed from the results that the prediction with decomposition model for 4 months data gave the least error.