In this article, we try to understand the underlying physical mechanisms important for the fully-stratified and partially-stratified flow of granular solids in a Newtonian carrier fluid. For fully-stratified flow, the particles are concentrated in the lower portion of the pipe and will come into either continuous or sporadic contact with each other and with the pipe wall. The method of analysis for continuous contact by Coulomb (1773) considers the normal stress or pressure to be composed of the hydrostatic pressure p, carried by interstitial fluid, and the intergranular normal stress σs, which is based on force carried by intergranular contacts. Together with intergranular pressure σs, there is intergranular shear stress τs.
In applying granular mechanics to particles, Du Boys (1879) noted that the shear stress applied to the bed by the flow could move the whole upper portion of the bed, possibly comprising several layers each of a thickness equal to the particle diameter. It was proposed that the normal granular stress σs acting on each layer be equal to the sum of the submerged weight of the grains in all the higher layers. At the bottom of the moving layers, the product σstanφ just equals to the applied shear stress τo, and for higher layers, the resting capacity of the granular mass, σstanφ, will be less than the applied shear stress.
Bagnold (1956) continued to employ the concepts of granular pressure and shear stress, noting that the lower non-sheared portion of a granular bed must have a value of σstanφ in excess of the applied shear stress τo. Shearing motion begins when normal intergranular pressure σs equals τo/tanφ, and this intergranular pressure must be produced by the submerged weight of the moving bed-load, transferred downward by intergranular contacts. In considering conditions within the sheared layer where the bed-load is moving, Bagnold, like Du Boys, utilized the Coulombic equality:
where Φ' is the dynamic friction angle. The remaining shear stress τo – τs is carried by the fluid, and hence should be associated with the velocity gradient.
Bagnold’s concept of the two mechanisms of particle support — fluid action and intergranular contact — provides the basis for understanding the heterogeneous-flow results. The presence of particles suspended by the fluid increases the fluid pressure at the bottom of the pipe, but this pressure has no direct influence on fluid shear, and thus should not affect the pressure gradient.
The situation is different for particles whose submerged weight is carried by granular contact. The weight produces normal granular stress against the bottom of the pipe, and the shear stresses proportional to the normal stresses must be provided in order to set the particles in motion.
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