A Granular Mixture Model with Goodman–Cowin-Type Microstructure and its Application to Shearing Flows in Binary Solid–Fluid Bodies

Abstract

A continuum theory of a granular mixture is formulated. In the basic balance laws, we introduce an additional balance of equilibrated forces to describe the microstructural response according to Goodman and Cowin (Arch Rational Mech Anal, 44:249–266,1972, [10]) and Passman et al. (Rational Thermodynamics, Springer, New York, pp. 286–325, 1984, [18]) for each constituent. Based on the müller –Liu form of the second law of thermodynamics, a set of constitutive equations for a viscous solid–fluid mixture with microstructure is derived. These relatively general equations are then reduced to a system of ordinary differential equations describing a steady flow of the solid–fluid mixture between two horizontal plates. The resulting boundary value problem is solved numerically and results are presented for various values of parameters and boundary conditions. It is shown that simple shearing generally does not occur. Typically, for the solid phase, in the vicinity of a boundary, if the solid volume fraction is small, a layer of high shear rate occurs, whose thickness is nearly between 5 and 15 grain diameters, while if the solid volume fraction is high, an interlock phenomenon occurs. The fluid velocity depends largely on the drag force between the constituents. If the drag coefficient is sufficiently large, the fluid flow is nearly the same as that of the solid, while for a small drag coefficient, the fluid shearing flow largely decouples from that of the solid in the entire flow region. Apart from this, there is a tendency for solid particles to accumulate in regions of low shear rate.

This chapter is heavily based on the paper by Wang and Hutter [26].