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تنش اصطکاکی
In dense granular flows, in addition to the kinetic and collisional stresses (described by the kinetic theory), the model should account for the frictional stresses as well, which is dominant in flow regimes denser than the bubbling regime.
, kinetic theory), the model should account for the frictional stresses as well, which could be calculated based on soil mechanics principles.
The most common approach to consider the effect of frictional stresses is the kinetic frictional model based on the addition of stress from the two limiting regimes at a critical solid volume fraction (ecr) (Johnson and Jackson 1987; Syamlal et al. 1993; Ocone et al. 1993)
Most of the reported frictional stress models in the literature (Makkawi and Ocone 2005) and CFD codes such as Ansys and MFIX are based on the critical state theory of soil mechanics, where the shear stress tfr is described in terms of a frictional viscosity based on the work of Schaeffer (1987).
Schaeffer (1987) derived the following expression for the frictional stress, by assuming the system to be perfectly rigid-plastic, incompressible, non-cohesive, Coulomb powder with a yield surface of von Mises type, and the eigenvectors of the strain rate and stress tensors are parallel, as
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