Inertial number

The Inertial number $$I$$ is a dimensionless quantity which quantifies the significance of dynamic effects on the flow of a granular material. It measures the ratio of inertial forces of grains to imposed forces: a small value corresponds to the quasi-static state, while a high value corresponds to the inertial state or even the "dynamic" state. It is given by:

$$I = \frac{\dot\gamma d}{\sqrt{P/\rho}},$$

where $$\dot\gamma$$ is the shear rate, $$d$$ the average particle diameter, $$P$$ is the pressure and $$\rho$$ is the density.

Generally three regimes are distinguished:
 * $$I<10^{-3}$$: quasi static flow
 * $$10^{-3}10^{-1}$$: collisional flow

One model of dense granular flows, the μ(I) rheology, asserts that the coefficient of friction μ of a granular material is a function of the inertial number only.