Bagnold number

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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

B a = ρ d 2 λ 1 / 2 γ ˙ μ {\displaystyle \mathrm {Ba} ={\frac {\rho d^{2}\lambda ^{1/2}{\dot {\gamma }}}{\mu }}} {\displaystyle \mathrm {Ba} ={\frac {\rho d^{2}\lambda ^{1/2}{\dot {\gamma }}}{\mu }}},[2]

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

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

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

See also

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. S2CID 98030586.
  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): 1–24. Bibcode:2002JFM...452....1H. CiteSeerX 10.1.1.564.7792. doi:10.1017/S0022112001006577. S2CID 9416685.