Μ(I) rheology

Short description: Model of the rheology of a granular flow

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=||γ˙||dP/ρ,

where γ˙ is the shear rate tensor, ||γ˙|| is its magnitude, d is the average particle diameter, P is the isotropic 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:

σij=−Pδij+μ(I)Pγ˙ij||γ˙||

where the eponymous μ(I) is a dimensionless function of I. As with Newtonian fluids, the first term -Pδij 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. The multiplicative term μ(I)P/||γ˙|| can be interpreted as the effective viscosity of the granular material, which tends to infinity in the limit of vanishing shear flow, ensuring the existence of a yield criterion.[1]

One deficiency of the μ(I) rheology is that it does not capture the hysteretic properties of a granular material.[2]

## Development

The μ(I) rheology was developed by Jop et al. in 2006.[1][3]

## References

1. ↑ 1.0 1.1 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. Bibcode: 2006Natur.441..727J.
2. ↑ 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. Bibcode: 2008AnRFM..40....1F.
3. ↑ Holyoake, Alex (December 2011). Rapid Granular Flows in an Inclined Chute. http://alexholyoake.com/academic/RapidGranularFlows_Holyoake.pdf. Retrieved 21 July 2015.

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