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
B a = ρ d 2 λ 1 / 2 γ ˙ μ {\displaystyle \mathrm {Ba} ={\frac {\rho d^{2}\lambda ^{1/2}{\dot {\gamma }}}{\mu }}} , where ρ {\displaystyle \rho } is the particle density, d {\displaystyle d} is the grain diameter, γ ˙ {\displaystyle {\dot {\gamma }}} is the shear rate and μ {\displaystyle \mu } is the dynamic viscosity of the interstitial fluid. The parameter λ {\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}}} , where ϕ {\displaystyle \phi } is the solids fraction and ϕ 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 Bingham plastic
References
External links Granular Material Flows at N.A.S.A