Bretherton's buoyant bubble

When a buoyant bubble is inserted into a closed capillary that is slightly smaller than the capillary length, it appears stuck; exactly why this is so is a puzzle that has remained unanswered over the past 50 years. Recent calculations suggest that the bubble's motion is critically dependent on the hydrodynamics of the surrounding liquid film; however, quantitative measurements of these dynamics are lacking. We provide direct measurements of the dynamics of the liquid film surrounding a “stuck” bubble, recorded using interference microscopy. The film slowly relaxes to a constant thickness, and is stabilized by disjoining pressure at long times. The film's stability at this thickness is demonstrated by recovery after applied thermal perturbations; thus, we confirm that Bretherton's buoyant bubble is not pinned at a contact line, but is instead ostensibly stuck by extraordinarily slow flow in the surrounding liquid film whose thickness is set by a balance of capillary stress and disjoining pressure.

Figure 1

Visualizing the thin film around the bubble with Fizeau interferometry. (a) Schematic of the experimental setup: A collimated LED source is focused onto the capillary's internal surface by a microscope objective. Reflection from the glass-liquid interface interferes with the reflection from the liquid-air interface, forming an interference pattern. (b) Typical image of Bretherton's buoyant bubble in a sealed capillary, $2R=0.33$ mm ($\text{Bo}=0.009$). The bubble appears stationary over a timescale of days. (c) Interferogram of a section of the film around the bubble recorded several hours after its formation; here, $2R=1$ mm ($\text{Bo}=0.082$). The intensity profile is converted to the film thickness as described in the main text. Thickness is averaged over the $y$ coordinate, yielding the bubble profile shown at right; raw data (red) are smoothed in $z$ using local averaging to show the shape of the bubble (blue). Averaging in this manner controls for imaging noise and submicron dirt on the capillary's surface.

Figure 2

Different gas volume bubbles at steady state. Bubbles with increasing volumes of gas were inserted into capillaries with $\text{Bo}=0.123$. The ratio of the radius of the equivalent-volume spherical bubble to the capillary radius $\alpha$ is shown with five example bubbles (inset). A waiting period of 48 h transpired before the average value of the steady-state film thickness was measured over the region of the liquid film between the bubble menisci ${\langle h_{\text{film}}\rangle }_z$. The steady-state heights are identical within the measured variability in $z$, as can be seen by the average value of these data points for all values of $\alpha$, ${\langle h_{\text{film}}\rangle }_{\alpha }$, represented by the dashed red line.

Figure 3

Thermocapillary perturbations to the film in a $2R=2$ mm ($\text{Bo}=0.329$) capillary show four distinct timescales. (a) A thermal perturbation at $34{}^{\circ }\mathrm{C}$ is applied in succession to opposite sides of the capillary, which is held at an ambient temperature of $23{}^{\circ }\mathrm{C}$. A heat probe is applied first to the side nearest the camera for a fraction of a second (1), and then removed while the film recovers; the heat probe is then applied to the side of the capillary opposite from the camera for a second (2), and then removed. (b) Film profile evolution in response to the thermal perturbation with a temporal resolution of 1 s. The mean of the height in $z$, shown below the height map, reveals persistent recovery to a constant thickness from both perturbations, suggesting that the film attains a finite thickness at steady state. The schematic representations of the bubble illustrate the film profile during the experiment. The gray curved lines indicate time elapsed after the second step of the perturbation protocol.

Figure 4

Initial drainage dynamics immediately after formation of the bubble in a capillary with $2R=1.58$ mm ($\text{Bo}=0.205$). The minimum height corresponding to the bottom meniscus transition region (orange) and the mean height (blue) are plotted as a function of time after the bubble is inserted into the capillary. Initially the film is nonuniform, and not cylindrically symmetric, with a gravity biased dimple as can be seen in the interferogram and corresponding height map in (b) for $t_0$. Over 2 h later, the dimple has drained on average, and the film has become flatter, as can be seen by the interferogram and the height map shown at $t_1$. The film breaks azimuthal symmetry, perhaps due to residual submicron scale dirt on the glass surface remaining after the meticulous cleaning protocol. After over 8 h have elapsed, the film has reached an apparent steady state at a symmetric, uniform thickness of approximately 70 nm as shown in (b) at $t_2$. The drainage dynamics are shown on a log-log scale (inset), suggesting that $h_{\text{min}}\sim t^{-0.5}$; this scaling may be self-similar, as shown by the collapse of the $h\text{−}x$ profiles, as shown in the Supplemental Material [35]. The calculated drainage scaling for a 2D bubble $h_{\text{min}}\sim t^{-\frac{4}{5}}$ [19] using the experimentally observed initial value (gold) is shown for comparison. A scaling of film thickness to the power 1/2 is expected at the base of the bubble, however [19].