Inertial number

The Inertial number ${\displaystyle I}$ 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.^[1][2][3][4] It is given by:

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

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

Generally three regimes are distinguished:

• ${\displaystyle I<10^{-3}}$: quasi static flow
• ${\displaystyle 10^{-3}<I<10^{-1}}$: dense flow
• ${\displaystyle I>10^{-1}}$: 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.