## 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 |
---|---|

Article number | 21 |

Journal | Granular Matter |

Volume | 21 |

Issue number | 2 |

DOIs | |

State | Published - 1 May 2019 |

Externally published | Yes |

## Keywords

- Beverloo law
- Granular flow
- Size scaling relation

## ASJC Scopus subject areas

- General Materials Science
- Mechanics of Materials
- General Physics and Astronomy