Stability of gas channels in a dense suspension in the presence of obstacles

We investigate experimentally the influence of a fixed obstacle on gas rising in a dense suspension. Air is injected at a constant flow rate by a single nozzle at the bottom center of a Hele-Shaw cell. Without obstacles, previous works have shown that a fluidized zone is formed with a parabolic shape, with a central air channel and two granular convection rolls on its sides. Here, we quantify the influence of the obstacle's shape, size, and height on the location and dynamics of the central air channel. Different regimes are reported: the air channel can simply deviate (stable), or it can switch sides over time (unstable), leading to two signatures not only above the obstacle, but sometimes also below it. This feedback also influences the channel deviation when bypassing the obstacle. A wake of less or no motion is reported above the largest obstacles as well as the maximum probability of gas location, which can be interesting for practical applications. The existence of a critical height $h_c\simeq 7$ cm is discussed and compared with the existence of an air finger that develops from the injection nozzle and is stable in time. A dimensionless number describing the transition between air fingering and fracturing makes it possible to predict the channel's stability.

Figure 1

Experimental setup and image processing (circular obstacle, $D=1.5$ cm, $h=10$ cm). (a) Example of experimental snapshot. We can distinguish the obstacle (black circle at height $h$), the air rising through the granular bed, bubbles trapped in the upper part, and the central fluidized zone, marked by a slightly darker contour. (b) Cumulated motion $M$ computed with Eq. (1) (threshold 1%). The color bar indicates the probability of motion in the system over the total experiment duration. The white dashed line is the contour of the fluidized zone determined after binarization of the cumulated motion $M$ (threshold 20%). (c) Binarized image representing the fluidized zone (in white) and the area without motion (in black). The contour of the fluidized zone is extracted from this image (see the text). (d) Cumulative image $C$ computed from Eq. (2). The color bar represents the probability of the presence of bubbles in the system.

Figure 2

Cumulated motion $M$ for different obstacle size $D$ and heights $h$ [symbol, $D$ (cm)]. (a) Circular obstacle [($◯$,1.5), from left to right, $h=2.5$, 5, 7.5, 10, 12.5, and 15 cm]. (b) Different shapes of obstacles for a fixed height $[h=10$ cm, from left to right, ($\square$,1); ($\square$,4); ($\diamond$,1.4); ($\diamond$,5.7); ($◯$,1); ($◯$,4)]. The figures show the deviation of the air channel by the obstacle and the formation of a wake above large obstacles. In some cases, two channels are clearly distinguished [(b), second and third image, cf. Sec. 4].

Figure 3

(a) Sketch of the possible air pathways. The upper labels indicate the number of channel pathways before $⟶$ after the obstacle, and the symbols represent the obstacle shape. To quantify the deviation of the central air channel induced by the obstacle, we define the distance between the vertical of the injection nozzle and the air channel below (${\mathrm{\Delta }}_{\mathrm{down}}$) and above (${\mathrm{\Delta }}_{\mathrm{up}}$) the obstacle. (b) Phase diagram ($D,h$) for the number of air channels before $\to$ after the obstacle. The symbols represent the obstacle shape. The light blue region indicates the parameters for which a single channel is observed before and after the obstacle.

Figure 4

Normalized wake area $\mathcal{W}/\left(eD\right)$ as a function of the obstacle height $h$. [The symbols in Figs. 4, 5, and 6 are representative of the obstacle shape and size, (symbol, regime): (white, stable); (gray, unstable)].

Figure 5

Distance between the maximum of the air channel (from the cumulated image, see Fig. 2) and the vertical of the injection nozzle (a) above, ${\mathrm{\Delta }}_{\mathrm{up}}$ and (b) below, ${\mathrm{\Delta }}_{\mathrm{down}}$ the obstacle. (c) Main deviation of the air channel due to the obstacle $({\mathrm{\Delta }}_{\mathrm{up}}-{\mathrm{\Delta }}_{\mathrm{down}})$ normalized by the obstacle radius, $D/2$. The gray zone indicates a deviation larger than the typical obstacle radius [(symbol, regime): (white, stable); (gray, unstable)].

Figure 6

Location of the maximum probability ($x_{\max },h_{\max }$) for the presence of bubbles, computed from the cumulated image $C$ (Sec. 3b), as a function of the obstacle height $h$. (a) Absolute value of the horizontal position $|x_{\max }|$. The low value for the large square at $h=10$ cm is due to a bubble trapped right below the obstacle. (b) Vertical position $h_{\max }$ [(symbol, regime): (white, stable); (gray, unstable); the solid line is the 1:1 relationship; the dashed line indicates the value without obstacle].

Figure 7

Normalized phase diagram $(D/w,N_f^{*})$, where $N_f^{*}$ denotes the modified fracturing number describing the transition between fingering and fracturing (see the text). The colors indicate the number of air channels before $\to$ after the obstacle, and the symbols represent the obstacle shape. The gray zone displays the transition region, $N_f^{*}\simeq 0.45\pm 0.07$.