Chemistry:Inertial number

The Inertial number [math]\displaystyle{ I }[/math] 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:

[math]\displaystyle{ I = \frac{\dot\gamma d}{\sqrt{P/\rho}}, }[/math]

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

Generally three regimes are distinguished:

• [math]\displaystyle{ I\lt 10^{-3} }[/math]: quasi static flow
• [math]\displaystyle{ 10^{-3}\lt I\lt 10^{-1} }[/math]: dense flow
• [math]\displaystyle{ I\gt 10^{-1} }[/math]: 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.