Flow of granular materials with slip boundary condition: A continuum-kinetic theory approach

We study the steady fully developed flow of granular materials between two horizontal flat plates, subject to slip at the walls. The constitutive model for the stress tensor is based on ideas in continuum mechanics and kinetic theory. The constitutive equation used in our study is a model proposed by Rajagopal et al. (1994) [24], and the material properties such as viscosity and the normal stress coefficients are derived using the kinetic theory approximation proposed by Boyle and Massoudi (1990) [2] which includes the effect of the gradient of volume fraction. The slip boundary condition is based on the particle dynamics simulation results of Rosato and Kim (1994) [30]. The governing equations are non-dimensionalized, and the resulting system of non-linear differential equations is solved numerically. The results for the velocity profiles and the volume fraction profiles are presented. The earliest study of granular materials can be dated perhaps to the Middle Ages, where the hourglass or the sand clock were used by scholars to regulate their studies and by the clergy to time their sermons [31]. After industrialization, the basic issues were the need of predicting the stresses in container walls to avoid failure, minimizing the erosion in solids transport, and promoting the mixing in fluidized bed, etc. Furthermore, many natural phenomena require an understanding of flow of bulk solids, such as debris flow [7], and snow or avalanches [32]. Other examples of bulk solids include coal, sand, ore, grains, cereals, and so on. Granular materials can be treated as the limiting case of two-phase flow at high solid concentration and high solid-to-fluid density ratios. For two-phase flows, the interstitial fluid interacts with the solid component, but for bulk solids, the solid components dominate, especially when the interstitial fluid is a gas, the influence of fluid component on the solid component is negligible. A powder is composed of particles up to 100 lm (diameter) with further subdivision into ultrafine (0.1–1 lm), superfine (1–10 lm), or granular (10–100 lm) particles. A granular solid consists of materials ranging from about 100 to 3000 lm [3]. It is well known that granular materials can sustain shear stress without deformation, and the critical shear stress at which the shearing begins to occur depends on the normal stress. It is also widely observed that granular materials can exhibit normal stress differences under shearing motion, related to the phenomenon known as dilatancy. Also it is known …