Lifetime of a two-dimensional air bubble

Air bubbles created in many viscous liquids rise up to the liquid-air interface and stay there for a while before exploding and disappearing. The lifetimes of such bubbles are governed by the thinning dynamics of the hemispherical liquid film separating the bubble from bulk air. Here, the lifetime of bubbles confined by two separated wetting plates is experimentally studied as the distance apart, viscosity, and bubble size are changed. Although the film is not hemispherical but takes a nontrivial shape, a relatively simple hydrodynamic model accounts for the observations.

Figure 1

(Color online) (a) Two-dimensional air bubble sitting at the liquid-air interface of a Hele-Shaw cell. This quasistatic shape could be maintained for a few minutes. (b) Magnified view of the upper wall of the bubble in (a). The film of a well-defined thickness $h$ possesses sharp edges at the top and bottom, followed by fringes. A short movie is available [5].

Figure 2

Change of the film thickness $h$ (mm) with time $t$ (s). Experimental parameters $\nu$, $D$, and $R$ for the labels employed in this figure are summarized in Table .

Figure 3

(Color online) (a) Definition of the $(r,\theta ,z)$ coordinates for a bubble. The $z$ axis is perpendicular to this paper. (b) Possible side view (section) of the film. (c) Actual side view. One of the rounded corners is indicated by the arrow. (d) Direct view, from the direction and of the point, which direction and point are indicated by the white arrow in (a). Two white dashed lines correspond to the surfaces of the acrylic boards. Two black arrows indicate the liquid-air interface (the other interface on the left is out of focus).

Figure 4

(Color online) A portion of the liquid film in question (oblique perspective view) with two possible modes of velocity distribution of the flow inside it: (a) Poiseuille flow in the $z$ direction and (b) Plug flow in the $r$ direction. (c) Slab approximation of this portion.

Figure 5

Result of fitting of the data points in Fig. 2 by Eq. (3) with $\theta =0$ to confirm the exponential law and to extract experimental values of $\tau$ (and $h_0$) for different $\nu$, $D$, and $R$.

Figure 6

Relations between $1∕\tau$ and (a) $\nu$, (b) $D$, and (c) $R$. In all the plots, we use $R$ measured from snapshots and $\tau$ obtained by experiments. The dotted curve in each plot corresponds to Eq. (4), which is drawn without any fitting parameters. The labelings A, B, and C correspond to (A1, A2), (B1, B2), and (C1, C2) in Fig. 2 and Table .