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
- [math]\displaystyle{ \mathrm{Ba}=\frac{\rho d^2 \lambda^{1/2} \dot{\gamma}}{\mu} }[/math],[2]
where [math]\displaystyle{ \rho }[/math] is the particle density, [math]\displaystyle{ d }[/math] is the grain diameter, [math]\displaystyle{ \dot{\gamma} }[/math] is the shear rate and [math]\displaystyle{ \mu }[/math] is the dynamic viscosity of the interstitial fluid. The parameter [math]\displaystyle{ \lambda }[/math] is known as the linear concentration, and is given by
- [math]\displaystyle{ \lambda=\frac{1}{\left(\phi_0 / \phi\right)^{\frac{1}{3}} - 1} }[/math],
where [math]\displaystyle{ \phi }[/math] is the solids fraction and [math]\displaystyle{ \phi_0 }[/math] 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.
See also
References
- ↑ 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. doi:10.1098/rspa.1954.0186. Bibcode: 1954RSPSA.225...49B.
- ↑ 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): 1–24. doi:10.1017/S0022112001006577. Bibcode: 2002JFM...452....1H.
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