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.^[1]

The Bagnold number is defined by

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

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

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

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