Hellenica World

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The Bagnold number 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

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

where $$\rho$$ is the particle density, d is the grain diameter, $$\gamma$$ 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

$$\lambda=\frac{1}{\left(\phi_0 / \phi\right) - 1},$$

where $$\phi$$ is the solids fraction and $$\phi_0 i$$ s the maximum possible concentration (see random close packing).

In flows with small Bagnold numbers $$(\mathrm{Ba}<40)$$ , 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 $$(\mathrm{Ba}>450)$$ , which is known as the 'grain-inertia' regime.
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.
^ 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 (Cambridge University Press) 452: 1–24. doi:10.1017/S0022112001006577.