''μ(I)'' rheology

In granular mechanics, the μ(I) rheology is one model of the rheology of a granular flow.

Details

The inertial number of a granular flow is a dimensionless quantity defined as

I={\frac {||{\dot {\gamma }}||d}{\sqrt {P/\rho }}},

where ${\dot {\gamma }}$ is the shear rate tensor, $||{\dot {\gamma }}||$ is its magnitude, d is the average particle diameter, P is the pressure and ρ is the density. It is a local quantity and may take different values at different locations in the flow.

The μ(I) rheology asserts a constitutive relationship between the stress tensor of the flow and the rate of strain tensor:

\sigma _{ij}=-p\delta _{ij}+\mu (I)p{\frac {{\dot {\gamma }}_{ij}}{||{\dot {\gamma }}_{ij}||}}

where the eponymous μ(I) is a dimensionless function of I. As with Newtonian fluids, the first term -pδ_ij represents the effect of pressure. The second term represents a shear stress: it acts in the direction of the shear, and its magnitude is equal to the pressure multiplied by a coefficient of friction μ(I). This is therefore a generalisation of the standard Coulomb friction model.

One deficiency of the μ(I) rheology is that it does not capture the hysteretic properties of a granular material.^[1]

Development

The μ(I) rheology was developed by Jop et al. in 2006.^[2]^[3]

References

↑ Forterre, Yoël; Pouliquen, Olivier (January 2008). "Flows of Dense Granular Media". Annual Review of Fluid Mechanics. 40 (1): 1–24. doi:10.1146/annurev.fluid.40.111406.102142.
↑ Holyoake, Alex (December 2011). Rapid Granular Flows in an Inclined Chute (PDF). Retrieved 21 July 2015.
↑ Jop, Pierre; Forterre, Yoël; Pouliquen, Olivier (8 June 2006). "A constitutive law for dense granular flows". Nature. 441 (7094): 727–730. doi:10.1038/nature04801.

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