Bagnold number

{{Short description|Dimensionless number in fluid dynamics}}

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.{{cite journal |last1=Bagnold |first1=R. A. |year=1954 |title=Experiments on a Gravity-Free Dispersion of Large Solid Spheres in a Newtonian Fluid under Shear |journal= Proc. R. Soc. Lond. A |volume=225 |issue=1160 |pages=49–63 |doi=10.1098/rspa.1954.0186 |bibcode=1954RSPSA.225...49B |s2cid=98030586 }}

The Bagnold number is defined by

: $\backslash mathrm\{Ba\}=\backslash frac\{\backslash rho\; d^2\; \backslash lambda^\{1/2\}\; \backslash dot\{\backslash gamma\}\}\{\backslash mu\}$,{{cite journal |last1=Hunt |first1=M. L. |last2=Zenit |first2=R. |last3=Campbell |first3=C. S. |last4=Brennen |first4=C.E. |year=2002 |title=Revisiting the 1954 suspension experiments of R. A. Bagnold |journal=Journal of Fluid Mechanics |volume=452 |issue=1 |pages=1–24 |doi=10.1017/S0022112001006577 |bibcode=2002JFM...452....1H |citeseerx=10.1.1.564.7792 |s2cid=9416685 }}

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

: $\backslash lambda=\backslash frac\{1\}\{\backslash left(\backslash phi\_0\; /\; \backslash phi\backslash right)^\{\backslash frac\{1\}\{3\}\}\; -\; 1\}$,

where $\backslash phi$ is the solids fraction and $\backslash phi\_0$ 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.