Bagnold number
# 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

    $\mathrm {Ba} ={\frac {\rho d^{2}\lambda ^{1/2}{\dot {\gamma }}}{\mu }}$,

where $\rho$ is the particle density, $d$ is the grain diameter, ${\dot {\gamma }}$ is the shear rate and $\mu$ is the dynamic viscosity of the interstitial fluid. The parameter $\lambda$ is known as the linear concentration, and is given by

    $\lambda ={\frac {1}{\left(\phi _{0}/\phi \right)^{\frac {1}{3}}-1}}$,

where $\phi$ is the solids fraction and $\phi _{0}$ 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.

