## Abstract

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 vz=vz(r,z;R). We propose that the scaled vertical velocity vz(r,z;R)/vz(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 vz2(0,0;R)/Rz, the effective acceleration aeff(r,z;R) derived from v _{z} is a function of r/ R _{r} and z/ R _{z} also. The boundary condition aeff(0,0;R)=-g of granular flows under earth gravity g gives rise to vz(0,0;R)∝g(R-k2d)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 vz(r,z;R) and vz(0,0;R). From the size scaling relations, we obtain the mass flow rate of D-dimensional hopper W∝g(R-0.5d)D-1(R-k2d)1/2, which agrees with the Beverloo law at R≫ d. It is the size scaling of vertical velocity field that results in the dimensional R-dependence of W in the Beverloo law.

Original language | English |
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Article number | 21 |

Journal | Granular Matter |

Volume | 21 |

Issue number | 2 |

DOIs | |

State | Published - 1 May 2019 |

Externally published | Yes |

### Bibliographical note

Funding Information:Acknowledgements This work is supported by the National Magnetic Confinement Fusion Science Program of China under Grant No. 2014GB104002, the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No. XDA03030100, and the National natural Science Foundation of China under Grant No. 11421063.

Funding Information:

This work is supported by the National Magnetic Confinement Fusion Science Program of China under Grant No. 2014GB104002, the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No. XDA03030100, and the National natural Science Foundation of China under Grant No. 11421063.

Publisher Copyright:

© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

## Keywords

- Beverloo law
- Granular flow
- Size scaling relation