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.