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 isotropic pressure and &rho; is the density. It is a local quantity and may take different values at different locations in the flow.

The &mu;(I) rheology asserts a constitutive relationship between the stress tensor of the flow and the rate of strain tensor: $$ \sigma_{ij} = -P\delta_{ij} + \mu(I)P \frac{\dot\gamma_{ij}}{||\dot\gamma||} $$ where the eponymous &mu;(I) is a dimensionless function of I. As with Newtonian fluids, the first term -P&delta;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 &mu;(I). This is therefore a generalisation of the standard Coulomb friction model. The multiplicative term $$\mu(I)P/||\dot\gamma||$$ 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.

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

Development
The μ(I) rheology was developed by Jop et al. in 2006. Since its initial introduction, many works has been carried out to modify and improve this rheology model. This model provides an alternative approach to the Discrete Element Method (DEM), offering a lower computational cost for simulating granular flows within mixers.