Self-similar breakup of near-inviscid liquids

The final stages of pinchoff and breakup of dripping droplets of near-inviscid Newtonian fluids are studied experimentally for pure water and ethanol. High-speed imaging and image analysis are used to determine the angle and the minimum neck size of the cone-shaped extrema of the ligaments attached to dripping droplets in the final microseconds before pinchoff. The angle is shown to steadily approach the value of 18.0 $\pm {0.4}^{\circ }$, independently of the initial flow conditions or the type of breakup. The filament thins and necks following a ${\tau }^{2/3}$ law in terms of the time remaining until pinchoff, regardless of the initial conditions. The observed behavior confirms theoretical predictions.

Figure 1

Breakup regions found in dripping. The breakup at region A forms the primary drop, the breakup in region B detaches the filament from the pendant drop, and region C is found at one or several points of breakup within the filament.

Figure 2

Imaging of the pinchoff process in region A for a dripping water droplet; the time shown corresponds to the time to breakup. For the conditions observed, the flow is $Q_1^{\text{water}}=0.092$ ml s${}^{-1}$. The videos from which these images were extracted were recorded at two camera speeds: The frame rate was (a) $23\times {10}^3$ frames/s and (b) $220\times {10}^3$ frames/s. Image analysis was used to extract the angle of the conical filament $\theta$ from which the droplet is breaking up. Two microseconds prior to breakup ($t=2\mu$s), $\theta =16.9\pm {0.2}^{\circ }$ degrees; the asymptotic analysis shows that at the point of breakup ($t=0\mu$s) $\theta =17.9\pm {0.2}^{\circ }$.

Figure 3

Imaging of the pinchoff process in region C (filament breakup). The frame rate used in this experiment was $23\times {10}^3$ frames/s. For the condition observed, the flow is $Q_4^{\text{water}}=0.750$ ml s${}^{-1}$.

Figure 4

Experimental measurements of the minimum neck diameter $h_{\text{min}}$ in terms of the dimensionless time $\tau$ to breakup for different fluids and different flows. Regardless of the condition of flow, within simple dripping the thinning follows a self-similar behavior. The dotted line represents the potential flow scaling law. The measurement of $h_{\text{min}}$ is limited by the overturning of the droplet surface.

Figure 5

Small pinchoff angle for different flow conditions, breakup regions, and fluids. The mean of the convergent angle value for all the cases is $\theta =18.0\pm {0.4}^{\circ }$. According to the theory in Ref. [18], approaching pinchoff, all solutions tend towards self-similarity with the same limiting exponent: This exponent appears both numerically and experimentally (see the inset) to be about 0.5. The angles do not approach the asymptotic value of ${18.1}^{\circ }$ along the same curve because the path depends on the initial conditions.