Size scaling relation of velocity field in granular flows and the Beverloo law

Abstract

In a hopper with cylindrical symmetry and an aperture of radius R, the vertical velocity of granular flow $v_z$ depends on the distance from the hopper’s center r and the height above the aperture z and $v_z = v_z (r,z;\,R)$. We propose that the scaled vertical velocity $v_{z}(r,z;\,R)/v_{z} (0,0;\,R)$ is a function of scaled variables $r/R_r$ and $z/R_z$, where $R_{ r}=R- 0.5 d$ and $R_{ z}=R-k_2 d$ with the granule diameter d and a parameter $k_2$ to be determined. After scaled by $v_{ z}^2 (0,0;\,R)/R_z$, the effective acceleration $a_{\mathrm{eff}} (r,z;\,R)$ derived from $v_z$ is a function of $r/R_r$ and $z/R_z$ also. The boundary condition $a_\mathrm{eff} (0,0;\,R)=-\,g$ of granular flows under earth gravity g gives rise to $v_{ z} (0,0;\,R) \propto \sqrt{g}\left( R -k_2 d\right) ^{1/2}$. Our simulations using the discrete element method and GPU program in the three-dimensional and the two-dimensional hoppers confirm the size scaling relations of $v_{ z} (r,z;\,R)$ and $v_{ z} (0,0;\,R)$. From the size scaling relations, we obtain the mass flow rate of D-dimensional hopper $W \propto \sqrt{g } (R-0.5 d)^{D-1} (R-k_2 d)^{1/2}$, which agrees with the Beverloo law at $R\gg d$. It is the size scaling of vertical velocity field that results in the dimensional R-dependence of W in the Beverloo law.