A kinetic theory derivation of the stress tensor for granular material that includes normal stress effects

An expression for the stress tensor of a granular material that can exhibit the normal-stress effects caused by a solids fraction gradient was derived from both continuum and kinetic models. The continuum model motivates and develops the form of the stress tensor, but introduces undetermined coefficients. The kinetic model evaluates those coefficients using Enskog's dense gas theory. The dependence of the granular stress tensor on the solids fraction gradient arises by requiring that the correlating factor that links the two-particle distribution function to the two single-particle distribution functions be the contact value for the radial distribution function of a nonhomogeneous, hard-sphere fluid. A representation for that contact value was found by developing the generalized van der Waals expression for the stress tensor element of a nonhomogeneous fluid (a fluid that exhibits a density gradient) in equilibrium, and comparing it to the exact expression. That representation of the contact value was introduced into the two-particle distribution function, and its contribution to the stress tensor was found. The resulting stress tensor expression was applied to a simple shear flow problem in which a linear, solids-fraction profile is transverse to the flow. The resulting normal-stress effects increased with an increase in the solids-fractionmore » and its gradient. 35 refs., 5 figs.« less