A THEORY FOR ANTIGRANULOCYTES MATERIALS EXHIBITING NORMAL STRESS EFFECTS BASED ON ENSKOGS DENSE GAS THEORY

Granular materials exhibit phenomena such as normal stress differences, which are typical of materials whose response is non-linear. For example, when a non-linearly elastic slab is sheared, its motion is not determined by the shear force but by the normal forces that manifest themselves due to the shearing (Poynting effect). Another example is a non-linear fluid which exhibits normal stress differences that lead to phenomena like "die-swell" or "rod-climbing," which is again a manifestation of the stresses that develop orthogonal to planes of shear. In this paper, an expression for the stress tensor of a granular material that can exhibit normal-stress effects due to a solids fraction gradient is 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 non-homogeneous, hard-sphere fluid. A representation for that contact value is found by developing the generalized van der Waals theory expression for a 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 is introduced into the two-particle distribution function, and its contribution to the stress tensor is found. The resulting stress tensor expression is applied to a simple shear flow problem in which a linear, solids-fraction profile is transverse to the flow. The resulting normal-stress effects increase with the solids-fraction and its gradient.