Bubble eruptions in a multilayer Hele-Shaw flow

We study the dynamical rearrangement of gravitationally unstable multilayer fluid inside the narrow vertical gap of a Hele-Shaw cell. Four layers of immiscible fluids are superposed inside the cell, which is subsequently turned over. We vary the fluid properties and the relative thicknesses of the layers. One of the layers is air, the others are immiscible liquids: olive oil, water-glycerin mixture, and perfluorohexane. The concentration of the glycerin-water mixture is used to vary its viscosity. We classify various different dynamics of stirring and breakthrough of adjacent layers. We note a prominent phenomenon, where an air finger breaks through the high-viscosity layer to erupt as a hemisphere into the lower-viscosity perfluorohexane layer above it. These eruptions have a periodic neck pinch-off accompanied with high-speed airflow which breaks up some of the low-viscosity liquid to eject a spray of fine droplets. We use high-speed video to characterize the details of the eruptions and how wetting, contact lines and three-dimensionalities play a key role. We also investigate the center-of-mass trajectories for each layer and notice counterflows, where the center of some layers can temporarily move against buoyancy. The top and bottom layers can interchange by channeling through the intermediate layers, which subsequently overturn on longer timescales. We also point out some unexpected dynamics occurring in the triple- and four-phase interactions. Specifically, droplet motions are as much affected by local viscosity as by the density gradients.

Figure 1

Photograph of the Hele-Shaw device, filled with three liquid layers and air on top, before the overturn. The cell can be rotated about the pivot points at the center of the sides. The video camera is prefocused and mounted on a sturdy tripod.

Figure 2

(a) Photograph of the four hydrostatically stable immiscible fluid layers inside the Hele-Shaw device, following its filling, but before the device is rotated up-side-down around the horizontal pivot. (b) The matrix of 16 different layer thicknesses used in the experiments. The unstable vertical arrangement shows the fluid layers immediately following the overturning of the Hele-Shaw cell. The white region indicates the air layer, red the olive oil layer, blue the glycerin-water mixture, and on top the heaviest gray layer, which is the perfluorohexane (Flutech PP1). Each row shows a particular thickness of the PP1 (gray) layer (15, 20, 25, and 33%), while the different columns show an increasing thickness of the blue glycerin-water layer by the same percentages of 15, 20, 25, and 33%. The air layer becomes thinner to compensate for the increased combined thickness of the glycerin-water and PP1 layers, while the red olive oil layer is the same thickness throughout all experiments, at 25% of the total height. The total input potential energy per unit width and depth, available to drive the motions following the overturn [Eq. (2)], is stated beneath each configuration, where we use the density of pure glycerine for the blue layer.

Figure 3

Schematics showing the vertical distribution of densities and dynamic viscosities of the experimental fluid layers, in the unstable arrangement immediately following the overturning of the cell. (a) Density profile and (b) viscosity profile. The different hues of blue indicate the variability for different mass fraction of the glycerin in the glycerin-water mixtures. The viscosity changes strongly with glycerin content, from 1 to 1296 cP. Note that the pure glycerin viscosity is highly influenced by the temperature. The experiments were conducted in an air conditioned laboratory were only slight temperature variation is to be expected. This also changes, to a lesser degree, the density of the mixture, from 1000 to 1260 kg/${\mathrm{m}}^3$. The fluid properties of the other layers remain constant, as listed in Table I.

Figure 4

Multilayer flow-field, with a prominent first eruption, along the centerline, of an air finger into the heavy PP1 top layer. At this stage the red oil layer has fully overturned, replacing the air at the bottom of the cell. Thin streaks of the blue viscous layer have reached through the air layer to the top of the red oil, partitioning the air. Layer thicknesses here are $\mathrm{\Delta }{H}_i=[0.{25}_a$, $0.{25}_o$, $0.{25}_{97\%g}$, $0.{25}_p]$

Figure 5

Setup sketch for the bubble expansion model.

Figure 6

Snapshots from a full-cell video clip to highlight the overall dynamics after the cell overturn, for pure glycerin and equal layer heights $\mathrm{\Delta }{H}_i=[0.{25}_a,0.{25}_o,0.{25}_g,0.{25}_p]$. See also the Supplemental Video 1 [47].

Figure 7

(a) Time-evolution of the center-of-mass of the glycerin-water layer (solid lines), over a range of different viscosities. The dashed lines show the corresponding center-of-mass for the olive oil layer. These realizations are for cases where all the layers have the same thickness $\mathrm{\Delta }{H}_i=[0.{25}_a, 0.{25}_o, 0.{25}_{g/w}, 0.{25}_p]$. (b) The time it takes to reach the early minimum and maximum in the center-of-mass locations of the glycerin-water layer, as marked by the arrows in panel (a). The vertical dashed line is simply to aid the eye to see where the viscous effects start increasing these timescales.

Figure 8

(a) The earliest instability of the olive oil-air interface, with undulations starting before the overturn is complete. (b) Fingers from the olive oil film remaining under the glycerin, as the air pockets expand horizontally. (c) Glycerin finger tips covered by the red olive oil. The red wetting layer becomes thinner as more generations of fingers have emerged. For $\mathrm{\Delta }{H}_i=[0.{35}_a, 0.{25}_o, 0.{25}_g, 0.{15}_p]$. All scale bars are 50 mm long. See also Supplemental Video 1 [47].

Figure 9

Vertical motion of the air-finger tip during its penetration of the glycerin-water layer (a) and the olive oil layer (b), for different viscosities ${\mu }_g$ of the glycerin-water layer, while all layer are of equal depth, i.e., 25 %, which corresponds to 137.5 mm. The dashed line represent the original location of the the interface between the glycerin-water layer and olive oil or PP1 layer, while the reference zero position is set at the bottom of the lower layer. (c) The average tip velocity within the glycerin-water layer, before the finger hits the PP1 interface vs its viscosity ${\mu }_g$. The dashed line has a slope of $-1$. The right abscissa shows the corresponding value of the capillary number ${\text{Ca}}_g$ from Eq. (7).

Figure 10

The influence of the viscosity of the glycerin-water layer, on the air-finger growth. (a) Finger for the water only case (${\mu }_g=1$ cP). (b) The case of glycerin-water layer with similar viscosity as the oil layer (${\mu }_g\simeq 128$ cP). (c) Start of a finger for pure glycerin (${\mu }_g=1296$ cP). The corresponding Darcy-Rayleigh number $G$ [Eq. (5)], using the speed of the fingers, are given in Table 3.

Figure 11

The changes in the typical shapes of the air fingers, rising through the pure glycerin layer (blue), as they are about to erupt into the top PP1 layer. Shown for the full range of different layer thicknesses. The thickness of the glycerin layer increases from top to bottom, whereas the thickness of the PP1 layer (transparent layer on top of the glycerin layer) is gradually increased from left to right. Only for the thickest $\mathrm{PP}1+\text{glycerin}$ layer vertical depths does the air finger pinch off at its base, owing to the very small height of the air layer of $9\%$.

Figure 12

Example of an air-bubble eruption through the blue glycerin-water layer into the PP1 layer. The frames are spaced by 24 ms. Here we use pure glycerin and the layer thicknesses are $\mathrm{\Delta }{H}_i=[0.{25}_a,0.{25}_o,0.{25}_g,0.{25}_p]$. See also online Supplemental Video 2 [47].

Figure 13

The thinning of the glycerin ribbon above the air bubble along the interface vs time, as the air-finger front erupts trough the interface into the PP1 layer above it. For the case of a pure glycerin layer of ${\mu }_g=$ 1296 cP and with all the layers having the same 25 % thickness.

Figure 14

(a) Close-up image of the air finger at the start of the eruption, when it passes up through the top of the blue glycerin-water into the PP1 layer. This leaves two thin glycerin films on top of the PP1 layers on the glass plates, as sketched in the cross-sectional view in (b). These blue glycerin films rupture to allow the air to come in contact with the lower surface-energy PP1 films along the glass. The darker outline corresponds to the dewetting spot on the front glass plate, whereas the three brighter spots are next to the back plate. The image contrast has been enhanced to allow for better visibility. Realization is for pure glycerin and $\mathrm{\Delta }{H}_i=[0.{25}_a,0.{25}_o,0.{25}_g,0.{25}_p]$ (b) Sketch of the liquid interfaces and film breakup in a plane perpendicular to the glass, i.e., in the plane marked by the red dashed line in (a). Shows how the air penetrates past the glycerin filament, as is clear starting from the fifth panel of Fig. 12.

Figure 15

The expansion rate of the bubble area during the air eruption at fixed glycerin-water-layer thickness, but for different PP1 layer thicknesses. The solid curves are from the video data, while the dashed curves are the best fits using the model in Eq. (12), using $\mathrm{\Delta }p$ as the fitting parameter. (a) The glycerine-water layer viscosity of ${\mu }_g=$ 756 cP. (b) Pure glycerine layer with ${\mu }_g=1296$ cP. (c) The best-fit pressures $\mathrm{\Delta }p$ for each curve in (a) red circles and in (b) black squares.

Figure 16

Close-up imaging of the neck pinching and droplet spraying that occurs after the air finger erupts from the pure glycerin layer into the PP1 layer, for $\mathrm{\Delta }{H}_i=[0.{17}_a,0.{25}_o,0.{25}_g,0.{33}_p]$. The first three neck pinch-offs and spraying are shown, one in each row. See also Supplemental Video 3 [47].

Figure 17

The average velocity of the droplet clouds ejected during many neck pinch-offs. The error bars highlight the range of average velocities for many realizations, each averaged over 4–6 neck pinch-offs. Results for two different glycerin-water viscosities. (a) The thickness of the glycerin-water layer is kept at a quarter of the cell height, while the PP1-layer thickness is varied. (b) The thickness of the PP1 layer is kept at a quarter of the cell height, while the glycerin-water-layer thickness is varied.

Figure 18

The formation of glycerin bags, when the air-channel pinches off below the PP1 layer, inside the glycerin. (a) The formation of the pocket in a narrow passage within the glycerin-water layer. (b) Close-up of the expansion and breakup of a glycerin pocket. Both sequences are for $\mathrm{\Delta }{H}_i=[0.{35}_a, 0.{25}_o, 0.{20}_{97\%g}, 0.{20}_p]$. See also Supplemental Videos 4 and 5 [47].

Figure 19

Pinch-off radius of the neck vs time, for the first pinch-offs, like those in Figs. 12 and 23. For two different viscosities. Subsequent pinch-offs become more irregular, as is clear from the Supplemental Video 2 [47].

Figure 20

The frequency of the neck-pinching during the eruption of the main finger. (a) Pinch-off frequency for different thicknesses of pure glycerin and PP1 layers, with ${\mu }_g=1296$ cP. (b) The influence of the viscosity of the blue glycerin-water layer on the pinch-off frequency, for the case where all the layers have the same thickness.

Figure 21

The downward motion of the tips of the air fingers following the eruption of the first air finger through the top of the glycerin-water layer, for $\mathrm{\Delta }{H}_i=[0.{25}_a,0.{25}_o,0.{25}_g,0.{25}_p]$. The bottom sketch superimposes the traces of the air-glycerin interfaces. The frames are separated from each other by 0.17 s. Note that the tips of the air fingers move downward against buoyancy. See also Supplemental Video 1 [47].

Figure 22

The penetration of the PP1 phase into the other layers after the initial pinch-off for pure glycerin and equal layer depths: $\mathrm{\Delta }{H}_i=[0.{25}_a, 0.{25}_o, 0.{25}_g, 0.{25}_p]$.

Figure 23

The late stage of air-bubble breakup following its eruption through the pure glycerin layer, for $\mathrm{\Delta }{H}_i=[0.{25}_a,0.{25}_o,0.{25}_g,0.{25}_p]$. After the first eruption, sections from the bottom of the mushroom-shaped air finger will break into separated bubbles of different sizes, as highlighted with the ellipse in the fifth frame.

Figure 24

The first air-finger eruptions through the blue glycerin-water layer for a range of greatly reduced viscosities, from those studied above in Figs. 4, 6, 10, etc. The viscosity values are listed on the left side. Note the film ruptures starting in the second panel of the middle row. All of the liquid layers have the same thickness of 25 %. For the water case ${\mu }_w=1$ cP see Supplemental Video 6 [47], and for ${\mu }_g=128$ cP see Supplemental Video 7 [47].

Figure 25

(a) The channeling of a descending PP1 liquid blob into a rising continuous air finger, inside the blue glycerin layer, for $\mathrm{\Delta }{H}_i=[0.{25}_a, 0.{25}_o, 0.{25}_g, 0.{25}_p]$. The black arrows show the descending PP1 and the purple arrows the rising air. See also Supplemental Video 8 [47]. (b) Channeling of air upward through a PP1 finger inside pure glycerin, for $\mathrm{\Delta }{H}_i=[0.{27}_a, 0.{25}_o, 0.{33}_g, 0.{15}_p]$. See also Supplemental Video 9 [47].

Figure 26

(a) Overturning dynamics for the lowest-viscosity pure water in the blue layer. For equal layer depths $\mathrm{\Delta }{H}_i=[0.{25}_a, 0.{25}_o, 0.{25}_w, 0.{25}_p]$. The vertical channels of red olive oil have numerous minute blue water droplets within. Keep in mind that the evolution is far from uniform along the width of the cell due to large overturning blobs of air and PP1. (b) Close-up of the filament formation (arrows) when the low-surface-tension PP1 is stretched around the blue water droplets inside the red olive oil continuous phase. Both scale bars are 10 cm. See also Supplemental Video 6 [47].

Figure 27

The interchange of the two middle layers during the later stages after the overturn, for the two extremes in viscosity of the glycerin-water layer. (a) Blue layer is pure water with 1 cP viscosity, see also Supplemental Video 10 [47]. (b) Blue layer is pure glycerin with 1296 cP viscosity, see also Supplemental Video 11 [47]. The image strips show only the middle two layers. The bottom profiles show the horizontal average of the liquid fraction of the blue layers.

Figure 28

[(a), (b)] Examples of caving for $\mathrm{\Delta }{H}_i=[0.{09}_a, 0.{25}_o, 0.{33}_g, 0.{33}_p]$. The arrows point out the flow of PP1 liquid into the air pockets. See also Supplemental Video 12 [47]. (c) Channeling of the air into excavated channels in the glycerin. This occurs when the PP1 layer is narrower than the air layer $\mathrm{\Delta }{H}_i=[0.{35}_a, 0.{25}_o, 0.{25}_g, 0.{15}_p]$. See also Supplemental Video 13 [47].

Figure 29

Eruption and interchange dynamics for a three-layer system, with bottom air, top PP1 and pure glycerin in the middle blue layer. $\mathrm{\Delta }{H}_i=[0.{34}_a, 0.{33}_g, 0.{33}_p]$. See also Supplemental Video 14 [47].