We use particle simulations to investigate the mass flow in two-dimensional hopper flow and to analyze the dependency of the flow rate with the bottleneck width and the particle diameter. A flow rate law is derived from self-similar velocity and density profiles at the neck. The resulting relation is an enhancement of the Beverloo relation that incorporates the dependency of the density with the neck width. The parameters of the Beverloo relation are interpreted by coupling the hourglass theory with the free-fall arch theory using non-zero arch velocity as accounted by the hourglass theory.

Graphic abstract

中文翻译：

---

自相似轮廓导出料斗流量的Beverloo定律

我们使用粒子模拟研究二维料斗中的质量流，并分析流速与瓶颈宽度和粒径的相关性。流量定律是从颈部的自相似速度和密度曲线得出的。所得到的关系是Beverloo关系的增强，其中包括了密度与颈部宽度的依赖性。Beverloo关系的参数通过沙漏理论与自由落体拱理论的结合来解释，其中使用了由沙漏理论解释的非零拱速。

图形摘要