Clogging and avalanches in quasi-two-dimensional emulsion hopper flow

We experimentally and computationally study the flow of a quasi-two-dimensional emulsion through a constricting hopper shape. Our area fractions are above jamming such that the droplets are always in contact with one another and are in many cases highly deformed. At the lowest flow rates, the droplets often clog and thus exit the hopper via intermittent avalanches. At the highest flow rates, the droplets exit continuously. The transition between these two types of behaviors is a fairly smooth function of the mean strain rate. The avalanches are characterized by a power-law distribution of the time interval between droplets exiting the hopper, with long intervals between the avalanches. Our computational studies reproduce the experimental observations by adding a flexible compliance to the system (in other words, a finite stiffness of the sample chamber). The compliance results in continuous flow at high flow rates, and allows the system to clog at low flow rates leading to avalanches. The computational results suggest that the interplay of the flow rate and compliance controls the presence or absence of the avalanches.

Figure 1

Schema of our sample chamber (left) and raw image of the emulsion flowing in the $+x$ direction (right). The hopper angle $\theta ={54}^{\circ }\pm 5^{\circ }$.

Figure 2

Description of the three flow behaviors. (a)–(c) Images of the samples at a particular time, with the color indicating the time when the droplet exits (see color bars). The red droplets exit earlier and blue droplets exit later. (d)–(f) The cumulative number of droplets that have exited the hopper as a function of time. (g)–(i) Histograms of the number of droplets exiting the hopper within a short time window $\mathrm{\Delta }t$, chosen such that the mean of the histogram is ten droplets. The flow conditions are: (a), (d), (g) smooth flow, $\varphi =0.87, r_{\mathrm{expt}}=0.17{\mathrm{s}}^{-1}$. (b), (e), (h) Intermediate, $\varphi =0.96, r_{\mathrm{expt}}=0.085{\mathrm{s}}^{-1}$. (c), (f), (i) Avalanche, $\varphi =0.96, r_{\mathrm{expt}}=0.020{\mathrm{s}}^{-1}$.

Figure 3

Schema of the definition of interval $\mathrm{\Delta }t$. The left figure is at time $t_1$ when the black droplet exits the hopper. The right figure is at time $t_2$ when the red droplet exits the hopper.

Figure 4

Typical examples of three types of probability distribution functions of $\mathrm{\Delta }t$: (a) exponential distribution, (b) intermediate case, (c) power-law distribution. The area fraction $\varphi$ and mean flux rate $r_{\mathrm{expt}}$ are as indicated in each panel. In (a) the line shows an exponential fit $P\left(\mathrm{\Delta }t\right)\sim e^{-\mathrm{\Delta }t/\tau }$ with $\tau =6.3$ s. In (b) the straight line is a power-law fit $P\left(\mathrm{\Delta }t\right)\sim \mathrm{\Delta }{t}^{-\alpha }$ with $\alpha =2.1$ and the curved line is an exponential fit with $\tau =12.7$ s. In (c) the line is a power-law fit with $\alpha =1.6$.

Figure 5

Phase diagram of fitting patterns of $P\left(\mathrm{\Delta }t\right)$ in terms of area fraction $\varphi$ and flux rate $r_{\mathrm{expt}}$ for our 45 experiments. Red circle: power law; blue triangle: intermediate; black cross: exponential.

Figure 6

(a) The power-law exponent $\alpha$ as a function of the experimental flux rate $r_{\mathrm{expt}}$. For the power-law fits, only data in the tail are used for the fit. Different choices of the minimum $\mathrm{\Delta }t$ used for the fit give rise to different values of $\alpha$, reflected in the error bars shown.

Figure 7

Image of the simulation data, colored similarly to Fig. 2 with color indicating the time when the droplet exits. For this simulation, the exit width is $w/d=1.4$ (width divided by mean droplet diameter), the flux rate is $r={10}^{-3}$, and the stiffness is $s=3\times {10}^{-5}$. The exit time distribution appears power law with exponent $\alpha =1.7$. The total time shown is $5.53\times {10}^5$, during which given $r={10}^{-3}$ we would expect 553 droplets to flow out; only 500 droplets flow out for this particular time interval.

Figure 8

The value of the simulation driving parameter $g$ as a function of time. Time is normalized by the rate $r$ such that on average one droplet should exit per unit normalized time. Stretches of data where $g$ increases are due to periods where the system is clogged. The values of the rate $r$ are as indicated.

Figure 9

Typical examples of three types of probability distribution functions of $\mathrm{\Delta }t$ from simulation data: (a) exponential distribution, (b) intermediate case, (c) power-law distribution. The flux rate $r$ is as indicated, and for these data the stiffness is $s={10}^{-4}$. The time is normalized by $r$, so that the mean time between droplets exiting is 1. In (b) the straight line is a power-law fit $P\left(\mathrm{\Delta }t\right)\sim \mathrm{\Delta }{t}^{-\alpha }$ with $\alpha =6.5$. In (c) the straight line is a power-law fit with $\alpha =2.1$. The dashed lines in (a) and (b) are exponential fits.

Figure 10

(a) Symbols: the mean value of the gravitational driving parameter $g$ as a function of the desired mean flux rate $r$, for $w/d=1.4$ (circles) and $w/d=2.0$ (triangles). The thin vertical lines indicate the spread from the tenth percentile to 90th percentile of $g$. Thick lines: the mean value of the measured flux rate $r$ from simulations with constant $g$, omitting moments when the simulation permanently clogged. (b) The horizontal axis shows the probability per droplet of a clogging event from simulations with constant gravity $g$ (vertical axis).

Figure 11

(a) Power-law exponent $\alpha$ as a function of flux rate $r$ from simulation data, for $w/d=1.4$ (circles) and $w/d=2.0$ (triangles). The stiffness is $s={10}^{-4}$. (b) $\alpha$ as a function of stiffness $s$ for fixed flux rate $r={10}^{-3}$ and the symbols indicating $w/d$ as in (a). For both panels, the error bars indicate the uncertainty of $\alpha$. Where not shown, the uncertainty is smaller than the symbol size.