Size Scaling of Velocity Field in Granular Flows through Apertures

Abstract

For vertical velocity field $v_{\mathrm{z}}(r,z;R)$ of granular flow through an aperture of radius $R$, we propose a size scaling form $v_{\mathrm{z}}(r,z;R)=v_{\mathrm{z}}(0,0;R)f(r/R_{\mathrm{r}},z/R_{\mathrm{z}})$ in the region above the aperture. The length scales $R_{\mathrm{r}}=R-0.5d$ and $R_{\mathrm{z}}=R+k_{2}d$, where $k_2$ is a parameter to be determined and $d$ is the diameter of granule. The effective acceleration, which is derived from $v_{\mathrm{z}}$, follows also a size scaling form $a_{\mathrm{eff}}=v_{\mathrm{z}}^2(0,0;R)R_{\mathrm{z}}^{-1}\hat{I}¸(r/R_{\mathrm{r}},z/R_{\mathrm{z}})$. For granular flow under gravity $g$, there is a boundary condition $a_{\mathrm{eff}}(0,0;R)=-g$ which gives rise to $v_{\mathrm{z}}(0,0;R)=\sqrt{\hat{I}»g{R}_{\mathrm{z}}}$ with $\hat{I}»=-1/\hat{I}¸(0,0)$. Using the size scaling form of vertical velocity field and its boundary condition, we can obtain the flow rate $W=C_2\ddot{I}\sqrt{g}{R}_{\mathrm{r}}^{D-1}{R}_{\mathrm{z}}^{1/2}$, 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 $v_{\mathrm{z}}(r,z;R)$ and the $R$-dependence of $v_{\mathrm{z}}(0,0;R)$.

5 pages, 8 figures