Flow of anisometric particles in a quasi-two-dimensional hopper

The stationary flow field in a quasi-two-dimensional hopper is investigated experimentally. The behavior of materials consisting of beads and elongated particles with different aspect ratio is compared. We show, that while the vertical velocity in the flowing region can be fitted with a Gaussian function for beads, in the case of elongated grains the flowing channel is narrower and is bordered with sharper velocity gradient. For this case, we quantify deviations from the Gaussian velocity profile. Relative velocity fluctuations are considerably larger and slower for elongated grains.

Figure 1

(a) Schematic view of the experimental geometry. Dashed line indicates the observation area. [(b)–(g)] Photographs of the granular samples.

Figure 2

(a) Sample images of the flow for (a) silica gel beads and (b) glass rods with $L/d=3.5$. [(c) and (d)] The flowing region visualized by image differencing.

Figure 3

(a) Time-averaged vertical velocity as a function of $x$, at height $z=60d^{*}$, for all six materials. The curves are normalized by the integral of the fitted functions [Eq. (1)], symbols are displayed on each curve for a better distinction. The thin black lines represent the fitted curves. [(b) and (c)] The two fitting parameters: the exponent $\eta$ and the half width of the flowing channel $x_0$ [defined in Eq. (1)] at different heights. Both $x_0$ and the height coordinate $z$ are in units of the effective particle diameter $d$*. For each material the outlet size was $D\approx 7.5d^{*}$ and the angle of the wedge shaped walls was set to $\mathrm{\Phi }={80}^{\circ }$.

Figure 4

The two parameters describing the velocity profile: The exponent $\eta$ and the normalized half width $x_0/d^{*}$ [see Eq. (1)] as a function of the hopper angle $\mathrm{\Phi }$, at a height of $z=60d^{*}$, for three of the investigated materials. The lines are a guide to the eye.

Figure 5

(a) Temporal fluctuations in the velocity values above the orifice at $z=60d^{*}$. (b) Fourier power spectrum of the curves presented in panel (a), normalized by their integral across the whole frequency range (from 0 to $70\mathrm{Hz}$). The inset shows the standard deviation of the velocity data as a function of the particle elongation $L/d$.

Figure 6

The form of velocity profiles during fluctuations for rods with $L/d=1.4$ at (a) $z=60d^{*}$ and (b) $z=150d^{*}$ and (c) rods with $L/d=6$ at $z=60d^{*}$. Different colors represent different instant velocity profiles, the dashed lines show the time-averaged velocity profile for the whole run.