Sink Flow Deforms the Interface Between a Viscous Liquid and Air into a Tip Singularity

In our experiment, an interface between a viscous liquid and air is deformed by a sink flow of constant flow rate to form a sharp tip. Using a microscope, the interface shape is recorded down to a tip size of $1\mu \mathrm{m}$. The curvature at the tip is controlled by the distance $h$ between the tip and the sink. As a critical distance $h^{*}$ is approached, the curvature diverges like $1/(h-h^{*}{)}^3$ and the tip becomes cone shaped. As the distance to the sink is decreased further, the opening angle of the cone vanishes like $h^2$. No evidence for air entrainment was found, except when the tip was inside the orifice.

Figure 1

Sketch of the experimental setup.

Figure 2

Photographs of the air-liquid interface (silicone oil, $\eta =60\mathrm{Pa}\mathrm{s}$ and $Q=3.9\times {10}^{-3}\mathrm{ml}{\mathrm{s}}^{-1}$). (a) $h>h^{*}$, (b) $h=h^{*}$, (c) $h<h^{*}$. (d) A Gaussian fit to the tip just before the singularity. (e) the tip as it is about to enter the orifice.

Figure 3

Tip curvature $C$ as a function of $h$ for silicone oil ($\eta =30\mathrm{Pa}\mathrm{s}$, $\gamma =2.13\times 1.0\times {10}^{-2}{\mathrm{Nm}}^{-1}$), $a=1\mathrm{mm}$ and $Q=1.7\times {10}^{-3}\mathrm{ml}{\mathrm{s}}^{-1}$ (○), $5.1\times {10}^{-3}\mathrm{ml}{\mathrm{s}}^{-1}$ (⊞), ${10}^{-2}\mathrm{ml}{\mathrm{s}}^{-1}$ (●); and for $a=3\mathrm{mm}$ and $Q=3.9\times {10}^{-2}\mathrm{ml}{\mathrm{s}}^{-1}$ (▲). Errors in the curvature represent typical deviations between Gaussian and parabolic fits, whereas errors of 1.5% in $h$ come from the finite range of focus. Inset: log-log plot of $C$ versus $h-h^{*}$ for $Q={10}^{-2}\mathrm{ml}{\mathrm{s}}^{-1}$.

Figure 4

Critical $h=h^{*}$ with $a=1\mathrm{mm}$ for silicone oil ($\eta =30\mathrm{Pa}\mathrm{s}$) and $Q/\mathrm{ml}{\mathrm{s}}^{-1}=1.7\times {10}^{-3}$ (○), $5.1\times {10}^{-3}$(⊞), $6.8\times {10}^{-3}$ (⊞), $9\times {10}^{-3}$ (×), ${10}^{-2}$ (●); silicone oil ($\eta =60\mathrm{Pa}\mathrm{s}$) and $Q/\mathrm{ml}{\mathrm{s}}^{-1}=1.4\times {10}^{-3}$ (◇), $2.5\times {10}^{-3}$ (⊲), $3.9\times {10}^{-3}$ (◆); castor oil and $Q/\mathrm{ml}{\mathrm{s}}^{-1}=0.23$ (■). Errors in $h^{*}$ come from the absolute measurement of the orifice position: parameters for castor oil are $\gamma =3.53\pm 0.1\times {10}^{-2}{\mathrm{Nm}}^{-1}$, $\eta =0.93\pm 0.01\mathrm{Pa}\mathrm{s}$. Inset: capillary number ${\mathrm{Ca}}^{*}$ for all experiments [same symbols, ▲: $a=3\mathrm{mm}$ with silicone oil ($\eta =30\mathrm{Pa}\mathrm{s}$) and $Q/\mathrm{ml}{\mathrm{s}}^{-1}=3.9\times {10}^{-2}$].

Figure 5

Master curve, collapsing all data according to (3). Symbols are the same as in Fig. 4. Castor oil data with $C^{-1}\le 25\mu \mathrm{m}$ are not shown, owing to dust perturbing the tip.

Figure 6

Slope $R^{\prime }(h)$ versus Ca for all experiments but the one with castor oil. Same symbols as in Fig. 4. Line: best fit with $R^{\prime }(h)=(0.43\pm 0.1)/\mathrm{Ca}$. In the shaded region the slope is defined as the minimum value of $R^{\prime }(z)$.