The Bagnold number (Ba) is the ratio of grain collision stresses to viscous fluid stresses in a granular flow with interstitial Newtonian fluid, first identified by Ralph Alger Bagnold.
is the dynamic viscosity of the interstitial fluid.
is the maximum possible concentration (see random close packing).
In flows with small Bagnold numbers (Ba < 40), viscous fluid stresses dominate grain collision stresses, and the flow is said to be in the "macro-viscous" regime.
Grain collision stresses dominate at large Bagnold number (Ba > 450), which is known as the "grain-inertia" regime.
A transitional regime falls between these two values.