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arXiv 1708.09647cond-mat.soft
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颗粒流速度场的尺度标度

Size Scaling of Velocity Field in Granular Flows through Apertures

Gaoke Hu, Ping Lin, Yongwen Zhang, Liangsheng Li, Lei Yang, Xiaosong Chen

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5 pages, 8 figures
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摘要：

对于颗粒流通过半径为$R$的小孔的垂直速度场$v_{\rm z} (r,z;R)$，我们提出了小孔上方区域的尺寸标度形式$v_{\rm z}(r,z;R)=v_{\rm z} (0,0;R)f (r/R_{\rm r}, z/R_{\rm z})$。长度标度为$R_{\rm r}=R- 0.5 d$和$R_{\rm z}=R+k_2 d$，其中$k_2$为待测定参数，$d$为颗粒直径。从$v_{\rm z}$导出的有效加速度也遵循尺寸缩放形式$a_{\rm eff} = v_{\rm z}^2(0,0;R)R_{\rm z}^{-1} θ(r/R_{\rm r}, z/R_{\rm z})$。对于重力作用下的颗粒流$g$，存在边界条件$a_{\rm eff} (0,0;R)=-g$，该边界条件产生具有$λ=-1/θ(0,0)$的$v_{\rm z} (0,0;R)= \sqrt{ λg R_{\rm z}}$。利用垂向速度场的尺度形式及其边界条件，可以得到流速$W =C_2 ρ\sqrt{g } R_{\rm r}^{D-1} R_{\rm z}^{1/2} $，当$R \gg d$。采用离散元法（DEM）和GPU程序对三维和二维料斗内的垂直速度场$v_z (r,z;R)$进行了数值模拟。模拟数据证实了$v_{\rm z} (r,z;R)$的尺寸缩放形式和$v_{\rm z} (0,0;R)$的$R$依赖性。

英文摘要：

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 $a_{\rm eff} = v_{\rm z}^2(0,0;R)R_{\rm z}^{-1} θ(r/R_{\rm r}, z/R_{\rm z})$. For granular flow under gravity $g$, there is a boundary condition $a_{\rm eff} (0,0;R)=-g$ which gives rise to $v_{\rm z} (0,0;R)= \sqrt{ λg R_{\rm z}}$ with $λ=-1/θ(0,0)$. Using the size scaling form of vertical velocity field and its boundary condition, we can obtain the flow rate $W =C_2 ρ\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 $v_{\rm z} (0,0;R)$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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