Fig. 4: Surface grain flow measurements.
a Measured local surface grain velocity \({v}_{{y}^{{\prime} }}\), and solid fraction ϕ with calculated local grain flux \(\phi {v}_{{y}^{{\prime} }}\) in the steady state for 125–150 μm grains at a total flow rate of 2.5 ml/min, plotted as a function of depth perpendicular to the surface flow \({x}^{{\prime} }\). b Free surface angle θ, c instantaneous surface flow rate Qs, and d average central silo velocity \({\bar{v}}_{{{{{\rm{mid}}}}}}\) 1 cm below the surface, plotted as a function of time from wormhole formation t − tc. Vertical dashed lines represent the start of wormhole formation at t = tc. The red solid line represents average velocity in the central region of the silo 1 cm below the surface according to Eq. (5). e Steady state free surface angle θ* as a function of silo grain flow rate Qg for each grain size fitted with Eq. (2). Inset: θ* normalised by \({\theta }_{\max }^{* }\) and fitted with Eq. (2) with the approximation \({\theta }_{{{{{\rm{r}}}}}}/{\theta }_{\max }^{* }\approx 0.45\). Colours and symbols indicate different grain size intervals; error bars represent the standard deviation of multiple angle measurements.
