Bursting dynamics of viscous film without circular symmetry: The effect of confinement

We experimentally investigate the bursting dynamics of a confined liquid film suspended in air and find a viscous dynamics distinctly different from the unconfined counterpart due to lack of circular symmetry in the shape of the expanding hole: the confined-viscous bursting proceeds at a constant speed and a rim formed at the bursting tip does not grow. We find a confined-viscous to confined-inertial crossover, as well as an unconfined-inertial to confined-inertial crossover, at which bursting speed does not change although the circular symmetry in the hole shape breaks dynamically.

Figure 1

(a) Experimental setup. (b) Dynamics of a bubble in the cell well before the film bursting starts $(\eta =9.7$ Pa s). (c) Image sequence of a bubble bursting $(\eta =4.8$ Pa s). (d) Magnified snapshots of a bursting tip taken at a fixed time interval $(\eta =9.7$ Pa s). See Fig. 4 for $h_{max}$. (e) Bursting tip position $r$ vs time $t$ obtained from films with different thickness $h(D=0.97$ mm, $\eta =9.7$ Pa s). The shapes of tips in (c) and (d) look different because focal points of the photograph are different (compare SM movie 1 with 2).

Figure 2

(a) Example images for analysis of the flow inside the film by adding a small number of particles to oil $(D=1.0$ mm, $\eta =4.8$ Pa s, $h=44\mu \mathrm{m})$. The thick vertical arrows indicate the horizontal position of the particle in question. (b) The velocity $v$ normalized by the bursting velocity $V$ vs distance from the tip $x(D=1.0$ mm, $\eta =4.8$ Pa s, $h=44\mu \mathrm{m})$, obtained by the velocity change of a single particle. Inset: Semilogarithmic plot of the same data showing an exponential decay of $v$. (c) Decay length $L$ vs cell thickness $D$. Each point shows the average of four data points, and error bars indicate maximum and minimum values. The straight line shows $L=k_{1}D$ with $k_1=0.409\pm 0.026$. (d) Radius of curvature at the bursting tip $R$ vs thickness $h$ on a log-log scale. The straight line shows $R=k_{2}h$ with $k_2=2.18\pm 0.05$.

Figure 3

(a) Bursting velocity $V$ vs film thickness $h$. (b) Normalized velocity $\eta V/\gamma$ vs normalized thickness $D/h$.

Figure 4

(a) The maximum rim thickness in the $z$ direction during bursting $h_{max}$ [see Fig. 1] as a function of the tip position $r$ from respective reference positions for various $h(D=0.97$ mm, $\eta =9.7$ Pa s). The dashed line indicates the size of a rim in the case in which the rim grows under the volume conservation for $h=53\mu \mathrm{m}$. See the text for details. (b) Side view of a film bursting. The bursting proceeds in the direction perpendicular to the paper. After the oil film bursts, the liquid “shrinks” toward the walls, and then it is absorbed into the oil films that cover the surfaces of the walls.

Figure 5

The relation between the velocity in the $x$ direction and the position in the $z$ direction for particles whose $x$ coordinate of the position is in the range suggested by the narrow rectangle in (b) for $D=1.4$ mm, $\eta =4.8$ Pa s, and $h=147\mu \mathrm{m}$. $z_{max}$ is the position of the upper film surface at the central position of the narrow rectangle, i.e., at $x=0.46$ mm $(z_{max}=0.24$ mm), and the origin of the $z$ coordinate is placed at the center of the film.

Figure 6

(a) Sequential images of a bursting bubble taken from above $(D=6$ mm, $\eta =0.0965$ Pa s). The vertical arrows in the first shot indicate the nucleation point of bursting. The horizontal yellow arrows define the half width of the hole, $w$. The small bubble observed at the center of each snapshot is irrelevant to the bursting in question. (b) Half width of the hole $w$ (indicated by horizontal arrows in snapshots) vs time $t$. The constant velocity $(\simeq 26.1$ m/s) is maintained during the change in the hole shape from circular to rectangular at $t=0.0729$ ms, indicated by the dashed line. This dimensional crossover occurs before $w$ becomes $D/2$, because films exist on the sidewalls; the hole size in the direction of cell thickness never reaches $D/2$.