Dense non-Brownian suspensions undergo a viscous to inertial transition with increasing shear rate ($\dot{\gamma }$), from a regime of constant viscosity at low shear rates to a regime where the viscosity varies linearly with the shear rate. The shear rate associated with this transition has been shown to vary dramatically between different experimental systems. Using the discrete element method, we show that in the regime where frictional contacts are absent, the characteristic shear rate for the transition to an inertial state is sensitive to the volume fraction ($\varphi$) of the suspension and exhibits a critical behavior at the jamming volume fraction (${\varphi }_m$) for the system. By uncovering the presence of a microstructural length scale ($L_c$) that controls the emergence of inertial effects in the suspension, it is shown that the criticality emerges as a consequence of $L_c$ diverging at jamming. We further show that in contrast to the conventional Stokes number $St=\rho d^2\dot{\gamma }/{\eta }_f$, the definition of a modified Stokes number $S{t}_{\mathrm{cluster}}=\rho L_c^2\dot{\gamma }/{\eta }_f$ correctly quantifies the inertial effects within the suspension and leads to a collapse of the rheological data.

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• Jamming
• Lubrication theory
• Rheology
• Multiphase flows
• Suspensions
• Shear thickening