Bubbles breaking the wall: Two-dimensional stress and stability analysis

Submerged granular material exhibits a wide range of behavior when the saturating fluid is slowly displaced by a gas phase. In confined systems, the moving interface between the invading gas and the fluid/grain mixture can cause beads to jam, and induce intermittency in the dynamics. Here, we study the stability of layers of saturated jammed beads around stuck air bubbles, and the deformation mechanism leading to air channel formations in these layers. We describe a two-dimensional extension of a previous model of the effective stress in the jammed packing. The effect of the tangential stress component on the yield stress is discussed, in particular how arching effects may impact the yield threshold. We further develop a linear stability analysis, to study undulations which develop under certain experimental conditions at the air-liquid interface. The linear analysis gives estimates for the most unstable wavelengths for the initial growth of the perturbations. The estimates correspond well with peak to peak length measurements of the experimentally observed undulations.

Figure 1

The Hele-Shaw cell as seen from above ($20\times 30$ cm). Air (white region) is injected into a liquid mixture with a layer of sedimented beads on the bottom plate (gray region). The front is the accumulated region of beads along the air interface (the dark rim around the white regions). The different images correspond to different normalized filling fractions $\varphi$, i.e., the height of the sedimented layer relative to the cell gap. (a) $\varphi =0.35$, (b) $\varphi =0.49$, and (c) $\varphi =0.53$. We see a gradual transition from frictional fingers (a) to bubble dynamics (c) as $\varphi$ increases.

Figure 2

A closeup view of connected bubbles of air, which displace a liquid containing glass beads. The front is the accumulated region of beads adjacent to the air interface, and is identified as the dark region. The white dashed line indicates parts of the separation path between the front and the sedimented region. This path can develop cusps, as front segments from different bubbles merge. The front thickness ($L$) is only defined where this separation path runs parallel to the air-front interface. Grids of 1 mm spacing, are superposed on the image to reveal the scales. The front thickness ($L$) is $\simeq 3$ mm thick. The channels that connect the bubbles are $\simeq 1$ mm. The cell gap is $0.5$ mm, and the bead diameter is $0.1$ mm. The numbers refer to the order in which the bubbles are formed. The air-front interface of the bubble develops undulations. The arrows in bubble 4 points to peaks of these undulations.

Figure 3

Schematics of a section of the interface, with the adjacent front, seen from above. At every point along the interface we introduce a coordinate system $(u,v)$, such that the point is placed in the origin. The unit vectors ${\vec{e}}_u$ and ${\vec{e}}_v$ point respectively perpendicular and parallel to the interface.

Figure 4

Schematic examples of different configurations. (a) Convex interface, positive curvature. (b) Concave interface, negative curvature. (c) Negative curvature with radius of curvature which is smaller than the front thickness.

Figure 5

Schematic cross section of the cell at the front. The front thickness ($L$) is defined to be the length of the region of beads which fills the whole cell gap.

Figure 6

Effective yield stress, ${\sigma }_Y(\kappa ,L)$, defined in Eq. (15), for values of $K_2$, less than and greater than 1. (a) $K_2=0.6$. (b) $K_2=1.4$. The contour lines are logarithmically spaced. The threshold increases as Eq. (8), along the dashed line ($\kappa =0$). The white region in the top left corner corresponds to $\kappa L<-1$, and is not accounted for by the theory. The numerical values of the other parameters are presented in Table 1.

Figure 7

Illustration of the perturbation. The front is originally enclosed between the dashed lines. After the perturbation, it is enclosed between $g_q\left(x\right)$ and $f_q\left(x\right)$. A perturbation is considered unstable if the yield threshold at the peaks of the perturbations is lower than the threshold at the troughs, and otherwise stable.

Figure 8

Schematic representation of the cross section of the cell. The front is assumed to be incompressible. A volume associated to a displacement of a straight air-front interface ($\kappa =0$), shown by the white striped pattern, is therefore coupled to an equal volume associated to the displacement of the front-liquid boundary (the outer boundary in Fig. 7). The regions which are marked as “liquid” and “sedimented beads” corresponds to the “sed. beads” in Fig. 7. The accumulation of the sedimented beads results in the increased displacement of the front-suspension boundary; $\delta u^{\prime }=\delta u/(1-\varphi )$ and $\delta L=\delta u\varphi /(1-\varphi )$.

Figure 9

Stability criterion function $\mathrm{\Gamma }$ [Eq. (29)] versus the product of the wave number and the front length $qL$ for various values of $\alpha$. The normalized filling fraction is set to $\varphi =0.5$. The gray shaded region corresponds to $qL<-ln(1-\varphi )$, which is a stable region, independent of $\alpha$ [see Eq. (31)].

Figure 10

(a) Stability criterion function $\mathrm{\Gamma }$, as defined in Eq. (29), vs the wavelength $\lambda =2\pi /q$. A wavelength is unstable if $\mathrm{\Gamma }\left(\lambda \right)<0$. $\mathrm{\Gamma }\left(\lambda \right)$ is drawn 15 times, to visualize the sensitivity to the parameters ${\sigma }_T$, $\xi$, $\gamma$, and $L$. For each realization, the parameters are drawn from uncorrelated uniform distributions on the interval defined by $\pm 15\%$ of the mean value, $L=0.3$ cm, $\gamma =60$ mN/m, $\xi =0.06$ cm, and ${\sigma }_T=10$ Pa in accordance with Table 1. The filling fraction is fixed at $\varphi =0.5$, and $K_2=1.0,0.9$, and $0.8$, for the green dashed, red dotted, and blue solid lines respectively. The thick black dashed/dotted/solid lines correspond to the mean values of the parameters, for each value of $K_2$. (b) Histograms of the theoretically estimated wavelengths ${\lambda }^{*}$, which minimize $\mathrm{\Gamma }$ (i.e., the most unstable wavelength), based on ${10}^5$ realizations similar to the one plotted in (a), and for the three values of $K_2$. (c) Experimental observations of the wavelength ${\lambda }_e$ of the undulations. Estimated by measuring the linear peak to peak distance in the experimental pictures (see arrows in Fig. 2). This histogram is based on 214 measurements.

Figure 11

Compaction of beads in two dimensions. Black connections indicate contacts with more than double the average contact force. Contacts are overlaid for 50 consecutive time steps of the simulation. The beads are compacted as (a) the inner boundary moves outwards or as (b) the outer boundary slowly moves inwards. Note how the chains of contacts tend to orient radially in (a), which suggests that the average stress in the radial direction ${\sigma }_{rr}$ is bigger than the stress in the orthoradial direction ${\sigma }_{\theta \theta }$, i.e., $K_2<1$. In contrast, the chains tend to orient orthoradially in (b), which suggests that ${\sigma }_{\theta \theta }>{\sigma }_{rr}$ and $K_2>1$.

Figure 12

Schematic of the channeling of air through the front. Consider two regions of the front. A: The front adjacent to the tip of the channel is compacted as an interface segment of high curvature $\kappa$ moves outwards (towards the sedimented beads). B: The front at the shoulders of the channel are compacted as the negatively curved interface moves outwards. Undulations along the interface are not shown in the figure.