Figure 1

Experimental setup. The experimental device is a silo consisting of two mechanically independent parts: a lateral wall made of an acrylic cylinder and an outlet plane constituted of a pair of rings. A first pair of sensors measures the total vertical force applied to the lateral wall. A second pair of sensors measures the vertical force applied to the inner ring at the bottom. The outer ring is added to ensure a flat bottom surface. A displacement stage is used to avoid the mechanical contact between the lateral wall and the outlet plane, to tune the gap between the two parts, and to mobilize the friction at the wall. The mass of the grains flowing out of the silo is measured by means of an electronic scale.Reuse & Permissions

Figure 2

Mobilization of the friction at the lateral wall. An experimental run consists of 2 or 3 independent steps: the filling, the (optional) mobilization and the discharge. We report measurements of the vertical forces applied to the lateral wall and to the outlet plane, in terms of the equivalent masses $M_{\mathrm{wall}}$ (gray line) and $M_{\mathrm{app}}$ (black line), respectively. During the filling at constant flow rate, we observe an almost linear increase of the forces $M_{\mathrm{wall}}$ and $M_{\mathrm{app}}$. We check that the sum $M_{\mathrm{wall}}+M_{\mathrm{app}}$ indeed equals the mass of grains inside the silo, $M_{\mathrm{in}}$ (dashed line). After the filling, the apparent mass $M_{\mathrm{app}}$ measured at the base is of about 1/3 of the total mass $M_{\mathrm{T}}\simeq 2000$ g of grains inside the container. If the discharge is initiated starting from the latter initial conditions we say that the results correspond to a “partial mobilization.” However, previous to the discharge, supplemental mobilization of the friction at the lateral wall can be achieved by lifting the outer cylinder thanks to the translation stage (in the present example, each plateau corresponds to a step of 10 $\mu$m). The apparent mass $M_{\mathrm{app}}$ decreases toward an asymptotic value of about 80 g. In this case, we refer to the “full mobilization.” Note that $M_{\mathrm{wall}}$ increases such that $M_{\mathrm{wall}}+M_{\mathrm{app}}$ is constant to within the precision of the measurements (dashed line). Finally, during the discharge, a dynamic Janssen effect is clearly at play. Indeed, $M_{\mathrm{app}}$ remains almost constant whereas the remaining part of the grain weight is redirected to the wall ($d=250–425$ $\mu$m, $A=10$ mm, “full mobilization”).Reuse & Permissions

Figure 3

Static Janssen effect. We report the apparent mass ${\mathrm{M}}_{\mathrm{app}}$ measured at the base of the silo as a function of $M_{\mathrm{T}}$, the total mass of grains inside. For the partial mobilization (open diamonds), $M_{\mathrm{app}}$ is only slightly smaller than $M_{\mathrm{T}}$ (dashed line). For the full mobilization (full squares), $M_{\mathrm{app}}$ saturates to $M_{\mathrm{app}}^{\mathrm{sat}}=81$ g when the height of the granular column, $H_g$, exceeds the container radius ${\mathcal{D}}_c/2$, which corresponds to $M_{\mathrm{T}}\gg M_{\mathrm{T}}^c\simeq 97$ g ($d=125–250$ $\mu$m).Reuse & Permissions

Figure 4

Beverloo law–Mass flow rate $W$ as a function of the diameter of the orifice corrected by $kd$ due to the “empty annulus.” The granular flow is observed to obey the Beverloo law (dashed line from Eq. (1) with $k\simeq 3$ and $C\simeq 0.73\pm 0.02$ for the two aperture sizes and all the different grain sizes we considered (Partial mobilization–To within the experimental accuracy, $W$ is the same for the full mobilization). Inset: Temporal evolution of the mass of grains, $M$, deposited in the scale ($d=250-425\mu$m, $A=10$ mm).Reuse & Permissions

Figure 5

Dynamic Janssen effect. We report the apparent mass ${\mathrm{M}}_{\mathrm{app}}$ measured at the base of the silo as a function of $M_{\mathrm{in}}$, the mass of grains inside during the discharge. For the full mobilization (dotted line), $M_{\mathrm{app}}$ is almost constant during the discharge, except when the height $H_g$ of the granular column is of the order of or smaller than the silo diameter ${\mathcal{D}}_c$. We thus observe a clear dynamic Janssen effect and estimate the screening length ${\lambda }_{\mathrm{dyn}}\sim 70$ mm (equivalent mass $M_{\mathrm{in}}\simeq 200$ g). For the partial mobilization (continuous line), $M_{\mathrm{app}}$ continuously decreases from its initial value toward the value measured in the case of the full mobilization. In addition, we report $M_{\mathrm{app}}+M_{\mathrm{wall}}$ (gray dashed line) during the discharge and observe, at all times, a good agreement (to within 20 g) with the mass $M_{\mathrm{in}}$ (thin black line) of grains in the container ($A=5$ mm, $d=250–425$ $\mu$m).Reuse & Permissions

Figure 6

Full mobilization: apparent mass $M_{\mathrm{app}}$ during the discharge. (a) Apparent mass $M_{\mathrm{app}}$ vs inner mass $M_{\mathrm{in}}$ for various disk diameters $D$. (b) Apparent mass $M_{\mathrm{app}}$ vs surface area $S$ for various inner mass $M_{\mathrm{in}}$, or equivalently at different times $t$ ($A=5$ mm, $d=250–425$ $\mu$m).Reuse & Permissions

Figure 7

Full mobilization: pressure profile $P\left(r\right)$ during the discharge. The gray region at the center indicates the outlet ($A=5$ mm, $d=250–425$ $\mu$m).Reuse & Permissions

Figure 8

Partial mobilization: apparent mass $M_{\mathrm{app}}$ during the discharge. Apparent mass $M_{\mathrm{app}}$ vs inner mass $M_{\mathrm{in}}$ for various disk diameters $D$. Note the large difference between the results obtained for $D=50$ mm and $D=51$ mm ($A=5$ mm, $d=250–425$ $\mu$m).Reuse & Permissions

Figure 9

Partial mobilization: pressure profile $P\left(r\right)$ during the discharge. Note the large pressure at the periphery of the base plane, at $r\simeq 25$ mm. The gray region at the center indicates the outlet. Inset: An enlarged view of the pressure profile to highlight the behavior away from the wall ($A=5$ mm, $d=250–425$ $\mu$m).Reuse & Permissions

Figure 10

Grain velocity at the wall. Left: Schematic view of the stream lines during a discharge. Center: Schematic profile of the grain velocity at the wall, $v_z\left(z\right)$, at a given time $t$. Right: Vertical position of the front $z$ above the outlet plane as a function of time $t$. The front reaches a steady value in about 60 s whereas, in the given experimental conditions, the flow rate $W=3$ g/s is such that discharge lasts about 600 s ($A=5$ mm, $d=250–425$ $\mu$m).Reuse & Permissions

Figure 11

Pressures $\overline{P}$ and $P_o$ vs pressure $P_h$. The average pressure in the outlet plane, $\overline{P}$ (lines), and the pressure in the vicinity of the outlet, $P_o$ (symbols), are reported as functions of the hydrostatic pressure, $P_h$ (the weight of the granular matter inside divided by the surface area of the base). The average pressures $\overline{P}$ for partial (dashed line) and full mobilization (continuous line) differ significantly. In contrast, the pressure in the vicinity of the outlet is not drastically altered by the preparation of the system. Whatever the preparation, partial (open squares) or full mobilization (full squares), the pressure near the outlet decreases by at least a factor 2 whereas the flow rate remains constant during the whole discharge. The vertical gray dash points to the pressure at the screening length ($A=5$ mm, $d=250–425$ $\mu$m).Reuse & Permissions