Μ(I) rheology explained

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 = (• || \gamma|| d)/(\sqrt{P/\rho) },

where

• \gamma is the shear rate tensor, • || \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:

\sigma _{ij} = - P\delta _{ij} + \mu(I)P (• \gamma _{ij})/(• || \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

• \mu(I)P/|| \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 Jop et al. in 2006.^[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. A constitutive law for dense granular flows. Nature. 8 June 2006. 441. 7094. 727–730. 10.1038/nature04801. 16760972 . cond-mat/0612110. 2006Natur.441..727J.
2. Forterre. Yoël. Pouliquen. Olivier. Flows of Dense Granular Media. Annual Review of Fluid Mechanics. January 2008. 40. 1. 1–24. 10.1146/annurev.fluid.40.111406.102142. 2008AnRFM..40....1F.
3. Book: Holyoake. Alex. Rapid Granular Flows in an Inclined Chute. December 2011. 21 July 2015.
4. Barker . T. . Gray . J. M. N. T. . October 2017 . Partial regularisation of the incompressible (I)-rheology for granular flow . Journal of Fluid Mechanics . en . 828 . 5–32 . 10.1017/jfm.2017.428 . 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 . 10.1016/j.powtec.2023.118843 . 0032-5910. free .