Formation, Rupture, and Healing of an Annular Viscous Film

Unlike conventional jet and drop breakup, which forms a cylindrical liquid column, the dripping of a viscous film will form a liquid annulus, which then ruptures and heals due to surface tension and the inner surface forms a retracting tip. We apply a one-dimensional model to analyze the thinning dynamics, which predicts a universal thinning curve and shows good agreement with experimental measurements. The shape of the tip is documented to be conical and the retraction speed is determined by the balance of viscous and capillary stresses. We also show that the drop contains an entrained bubble and possible formation of satellite bubbles in the healed thread.

Figure 1

Observations of dripping golden syrup (a),(c),(d) and silicone oil (b) films: (a) the developing annulus, (b) the retracting tip and entrained bubble, (c) the retracting tip, and (d) the entrained bubble in the drop. The annulus develops from a liquid film attached to the end of a hollow tube, as indicated by the dotted line in (a). The scale bars are 1 cm.

Figure 2

(a) Sketch of an axisymmetric dripping annulus. (b) The fixed boundary conditions used in the hanging-annulus simulation in dimensionless units.

Figure 3

(a) Numerical simulations of a hanging annulus, which documents the film thinning that is a precursor to rupture. The snapshots are taken at $T=5$, 11, and 12. The outer and inner surface of the annulus are plotted by the solid and dotted lines, respectively. (b) The enlargement of the inner surface. The dotted box around the inner surface is also amplified for $T=11$ to show the local minimum of $R_i$ in the annular region. The parameters of golden syrup are used in this simulation: $\rho =1.39\times {10}^3\mathrm{kg}/{\mathrm{m}}^3$, $\mu =6.8\mathrm{Pa}\mathrm{s}$, $\gamma =67\mathrm{mN}/\mathrm{m}$, $r_c=0.5\mathrm{mm}$, $\mathrm{Re}=0.39$, and $\epsilon =0.05$.

Figure 4

Numerical results of $H_{\min }$ and $R_{i,\min }$ in the annulus region for $\tau >0$ for (a) golden syrup and (b) silicone oil. Experimental data $H_{\min }$ are first nondimensionalized and then rescaled by a factor $H_0$, which is 31 for (a) and 19 for (b). All numerical curves are normalized by the corresponding boundary values $H_0$ or $R_{i,0}$, except for the inset of (a), where the subscripts (1, 2, 3) in the legends correspond to the three boundary conditions: (1) $H_0>R_{i,0}$, (2) $H_0=R_{i,0}$, and (3) $H_0<R_{i,0}$. Note that for case 2, the curves of $H_2$ and $R_{i,2}$ collapse. The parameters of silicone oil are $\rho =953\mathrm{kg}/{\mathrm{m}}^3$, $\mu =1.06\mathrm{Pa}\mathrm{s}$, $\gamma =19\mathrm{mN}/\mathrm{m}$, $r_c=0.4\mathrm{mm}$, $\mathrm{Re}=1.04$ and $\epsilon =0.08$.

Figure 5

The local tip shape. (a) The inner-surface profiles are taken from experiments with golden syrup. The origin is chosen to be the tip location at $\overline{\tau }=0$. Profiles at $\overline{\tau }>0$ are shifted (along the negative $z$ direction) to coincide with the tips. The distance of shifting $L$ are recorded in the inset. The straight line in the inset is a linear fitting of the first five data points and the origin, the slope of which is the initial tip speed. (b) Data with log-log scales. The horizontal error bars of the film profile are shown in (a) and the vertical error bars are about the marker size. A linear fitting of the tip is plotted in (b) with slope $K=1/27$.

Figure 6

Satellite bubbles form after the annulus heals: (b) is a enlargement of the labeled square region in (a).