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

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

where ${\displaystyle {\dot {\gamma }}}$ is the shear rate tensor, ${\displaystyle ||{\dot {\gamma }}||}$ 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:

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

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. The multiplicative term ${\displaystyle \mu (I)P/||{\dot {\gamma }}||}$ 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 Pierre Jop et al. in 2006.[1][3] Since its initial introduction, many works has been carried out to modify and improve this rheology model.[4] This model provides an alternative approach to the Discrete Element Method (DEM), offering a lower computational cost for simulating granular flows within mixers.[5]

See also

• Dilatancy (granular material)

References

1. Jop, Pierre; Forterre, Yoël; Pouliquen, Olivier (8 June 2006). "A constitutive law for dense granular flows". Nature. 441 (7094): 727–730. arXiv:cond-mat/0612110. Bibcode:2006Natur.441..727J. doi:10.1038/nature04801. PMID 16760972.
2. Forterre, Yoël; Pouliquen, Olivier (January 2008). "Flows of Dense Granular Media". Annual Review of Fluid Mechanics. 40 (1): 1–24. Bibcode:2008AnRFM..40....1F. doi:10.1146/annurev.fluid.40.111406.102142.
3. Holyoake, Alex (December 2011). Rapid Granular Flows in an Inclined Chute (PDF). Retrieved 21 July 2015.
4. Barker, T.; Gray, J. M. N. T. (October 2017). "Partial regularisation of the incompressible 𝜇(I)-rheology for granular flow". Journal of Fluid Mechanics. 828: 5–32. doi:10.1017/jfm.2017.428. hdl:20.500.11820/834cfa72-7cc7-4f75-b0c4-733ad4ed346c. ISSN 0022-1120.
5. Biroun, Mehdi H.; Sorensen, Eva; Hilden, Jon L.; Mazzei, Luca (October 2023). "CFD modelling of powder flow in a continuous horizontal mixer". Powder Technology. 428: 118843. doi:10.1016/j.powtec.2023.118843. ISSN 0032-5910.