μ(I) rheology
In granular mechanics, the μ(I) rheology is one model of the rheology of a granular flow.
Details
[edit]The inertial number of a granular flow is a dimensionless quantity defined as
where is the shear rate tensor, 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:
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 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
[edit]The μ(I) rheology was developed by 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
[edit]References
[edit]- ^ a b 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.
- ^ 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.
- ^ Holyoake, Alex (December 2011). Rapid Granular Flows in an Inclined Chute (PDF). Retrieved 21 July 2015.
- ^ 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. ISSN 0022-1120.
- ^ 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.