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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.
The Bagnold number is defined by
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
where ϕ {\displaystyle \phi } is the solids fraction and ϕ 0 {\displaystyle \phi _{0}} is the maximum possible concentration.