# 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.

## References

1. Bagnold, R. A. (1954). "Experiments on a Gravity-Free Dispersion of Large Solid Spheres in a Newtonian Fluid under Shear". Proc. R. Soc. Lond. A. 225 (1160): 49–63. Bibcode:1954RSPSA.225...49B. doi:10.1098/rspa.1954.0186.
2. Hunt, M. L.; Zenit, R.; Campbell, C. S.; Brennen, C.E. (2002). "Revisiting the 1954 suspension experiments of R. A. Bagnold". Journal of Fluid Mechanics. 452: 1–24. CiteSeerX 10.1.1.564.7792. doi:10.1017/S0022112001006577.