Dimensional analysis for clogging of grains in two and three dimensions

We conduct standard dimensional analysis (Vaschy-Buckingham $\mathrm{\ensuremath{\Pi}}$ theorem) for the mean avalanche size $\ensuremath{\langle}s\ensuremath{\rangle}$ when particles flow through, and clog at, a small orifice on the base of a flat-bottomed silo. We consider the effect of particle diameter $d$, orifice diameter $D$, particle density $\ensuremath{\rho}$, particle Young's modulus $E$, and acceleration of gravity $g$. We both perform discrete element method simulations and compile available data in the literature in order to sample the parameter space. We find that our simulations and data across many experiments and simulations of frictional grains are consistent with the scaling equation $ln(\ensuremath{\langle}s\ensuremath{\rangle}+1)={A}_{\ensuremath{\alpha}}[{(D/d)}^{\ensuremath{\alpha}}\ensuremath{-}1]+{B}_{\ensuremath{\alpha}}\sqrt{\frac{\ensuremath{\rho}gd}{E}\phantom{{}^{1}}}$, where ${A}_{\ensuremath{\alpha}}$ and ${B}_{\ensuremath{\alpha}}$ are empirical constants and $\ensuremath{\alpha}$ is the dimensionality of the system ($\ensuremath{\alpha}=2$ and $\ensuremath{\alpha}=3$ for two dimensions and three dimensions, respectively). This expression successfully synthesizes the clogging behavior of a number of related clogging systems and motivates future extensions to more complex configurations involving, for example, very low friction particles or external vibrations.