Theory for the rheology of dense non-Brownian suspensions: divergence of viscosities and $\mu$-$J$ rheology

A systematic microscopic theory for the rheology of dense non-Brownian suspensions characterized by the volume fraction $\phi$ is developed. The theory successfully derives the critical behavior in the vicinity of the jamming point (volume fraction ${\phi }_J$), for both the pressure $P$ and the shear stress ${\sigma }_{xy}$, i.e. $P \sim \sigma_{xy} \sim \dot\gamma \eta_0 \delta\varphi^{-2}$,where$\dot\gamma$istheshearrate,$\eta_0$ is the shear viscosity of the solvent, and $\delta \phi ={\phi }_J-\phi >0$ is the distance from the jamming point. It also successfully describes the behavior of the stress ratio $\mu ={\sigma }_{xy}\mathrm{/}P$ with respect to the viscous number $J=\dot{\gamma }{\eta }_0\mathrm{/}P$.