Effect of interstitial fluid on the fraction of flow microstates that precede clogging in granular hoppers

We report on the nature of flow events for the gravity-driven discharge of glass beads through a hole that is small enough that the hopper is susceptible to clogging. In particular, we measure the average and standard deviation of the distribution of discharged masses as a function of both hole and grain sizes. We do so in air, which is usual, but also with the system entirely submerged under water. This damps the grain dynamics and could be expected to dramatically affect the distribution of the flow events, which are described in prior work as avalanche-like. Though the flow is slower and the events last longer, we find that the average discharge mass is only slightly reduced for submerged grains. Furthermore, we find that the shape of the distribution remains exponential, implying that clogging is still a Poisson process even for immersed grains. Per Thomas and Durian [Phys. Rev. Lett. 114, 178001 (2015)], this allows for an interpretation of the average discharge mass in terms of the fraction of flow microstates that precede, i.e., that effectively cause, a stable clog to form. Since this fraction is barely altered by water, we conclude that the crucial microscopic variables are the grain positions; grain momenta play only a secondary role in destabilizing weak incipient arches. These insights should aid ongoing efforts to understand the susceptibility of granular hoppers to clogging.

Figure 1

When time advances by one sampling time, ${\tau }_o$, there is a new configuration of grains near the hole and a new chance to clog. In this illustration, $N$ distinct flow states are sampled prior to the formation of a stable clog. The fraction of microstates that similarly cause clogging is $F=1/\langle N\rangle$, where $\langle N\rangle$ equals the average flow duration divided by ${\tau }_o$.

Figure 2

Normalized distribution of particle diameters for glass beads, measured by a Retch Technology Camsizer and labeled by average and standard deviation.

Figure 3

Schematic illustration of the clogging apparatus, shown in vertical cross section. The hopper (gray) is cylindrical and hangs from a digital balance (black). An orifice with adjustable diameter, $D$, fits into a depression in the bottom plate. The walls and bottom plate are made of polycarbonate, 6 and 13 mm thick, respectively. The flowing grains are indicated with brownish shading. The dry case is identical, except that the fluid (light blue) is absent. Note that the top of the hopper is open, so that there is no backflow of air or water into the hopper as the grains exit.

Figure 4

Standard deviation vs mean for the distribution of discharge masses, for all measured combinations of hole and grain sizes, under both dry and submerged conditions. The data fall on the line $y=x$, which implies that the distributions are exponential and that clogging is a Poisson process. This figure has two fewer data points than seen in later figures, where $\langle m\rangle$ but not ${\sigma }_m$ were measured by collecting multiple events into a cup and weighing.

Figure 5

One minus the cumulative distribution function vs scaled discharge event mass for $d=1$ mm diam grains. Solid circles represent the submerged dataset with the largest number of events (1172); crosses and diamonds represent all datasets for submerged and dry cases, respectively. The solid line labeled $y=exp(-x)$ is the expectation for an exponential distribution of discharge masses; its good agreement with the data demonstrates that clogging is a Poisson process.

Figure 6

The average discharge mass vs hole diameter (a), and scaled by grain mass and plotted vs the cube of hole diameter (b). In both plots, the solid curves represent fits to Eq. (1), while the dashed curves represent fits to a diverging power law Eq. (2) with exponent taken to be $\gamma =5$. The former is exponential in $D^3$ and hence comes out as a straight line in the bottom plot. Fitting parameters are given in Table 1.

Figure 7

The fraction of flow configurations that cause a clog vs the cube of hole diameter divided by grain diameter. Experimental results are for three grain sizes, and under both dry and submerged conditions. The solid lines represent fits to $F=exp\{-C[{(D/d)}^3-1]\}$, and the dashed lines represent the range of fitting functions given by the quoted value and uncertainty in the fitting parameter $C$.