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

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.