Bagnold number

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.

The Bagnold number is defined by

where ρ is the particle density, d is the grain diameter, γ ˙ is the shear rate and μ is the dynamic viscosity of the interstitial fluid. The parameter λ is known as the linear concentration, and is given by

where ϕ is the solids fraction and ϕ 0 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.