Size Scaling of Velocity Field in Granular Flows through Apertures

For vertical velocity field $v_{rm z} (r,z;R)$ of granular flow through an aperture of radius $R$, we propose a size scaling form $v_{rm z}(r,z;R)=v_{rm z} (0,0;R)f (r/R_{rm r}, z/R_{rm z})$ in the region above the aperture. The length scales $R_{rm r}=R- 0.5 d$ and $R_{rm z}=R+k_2 d$, where $k_2$ is a parameter to be determined and $d$ is the diameter of granule. The effective acceleration, which is derived from $v_{rm z}$, follows also a size scaling form $a_{rm eff} = v_{rm z}^2(0,0;R)R_{rm z}^{-1} theta (r/R_{rm r}, z/R_{rm z})$. For granular flow under gravity $g$, there is a boundary condition $a_{rm eff} (0,0;R)=-g$ which gives rise to $v_{rm z} (0,0;R)= sqrt{ lambda g R_{rm z}}$ with $lambda=-1/theta (0,0)$. Using the size scaling form of vertical velocity field and its boundary condition, we can obtain the flow rate $W =C_2 rho sqrt{g } R_{rm r}^{D-1} R_{rm z}^{1/2} $, which agrees with the Beverloo law when $R gg d$. The vertical velocity fields $v_z (r,z;R)$ in three-dimensional (3D) and two-dimensional (2D) hoppers have been simulated using the discrete element method (DEM) and GPU program. Simulation data confirm the size scaling form of $v_{rm z} (r,z;R)$ and the $R$-dependence of $v_{rm z} (0,0;R)$.