Breakup of asymmetric liquid ligaments

The evolution of a finite liquid ligament into either a single drop or several droplets is driven by the competition of two simultaneous processes: the recoil and the pinch-off. Scalings providing the kinetics of each process can be compared to predict the breakup of cylindrical symmetric ligaments. Here, asymmetric ligaments formed by a main drop connected to a cylindrical tail and commonly produced by inkjet printing are considered. Using two print heads, 16 inks, and various printing parameters, we show the limits of commonly admitted scalings for asymmetric ligaments. The recoil is governed by a modified Taylor-Culick velocity and a complex drainage, possibly leading to pinch-off, develops at the junction of the ligament with the main drop. The ligament aspect ratio affects this process for which two regimes must be accounted for. Finally, the ratio between the recoil and pinch-off timescales successfully predicts asymmetric ligament breakup.

Figure 1

Drop formation pictures showing (a) ligament recoil for ${\mathrm{Oh}}_n=0.261$, with printer parameters 99 V, $33\mu \mathrm{s}$, and 100 Hz; and (b) ligament pinch-off for ${\mathrm{Oh}}_n=0.084$, with printer parameters 105 V, $36\mu \mathrm{s}$, and 100 Hz. The initial dimensions of the ligament ($l_0$ and $d$) are indicated in both cases.

Figure 2

(a) Temporal evolution of the ligament length obtained from the pictures of Fig. 1 showing a constant recoil velocity (full points and fit). (b) $V_{\mathrm{recoil}}^{\mathrm{expt}}$ as a function of $V_{\mathrm{Taylor}}$. (c) $V_{\mathrm{recoil}}^{\mathrm{expt}}$ as a function of $V_{\mathrm{recoil}}^{\mathrm{theo}}$ [Eq. (1)]. (d) ${\tau }_{\mathrm{recoil}}^{\mathrm{expt}}$ as a function of ${\tau }_{\mathrm{recoil}}^{\mathrm{theo}}$ [Eq. (2)]. Squares correspond to device 1 and diamonds to device 2. ${\mathrm{Oh}}_n$ is varied as follows: 0.124 (green), 0.148 (yellow), 0.155 (orange), 0.216 (red), 0.232 (purple), and 0.261 (brown).

Figure 3

(a) ${\tau }_{p-o}^{\mathrm{expt}}$ as a function of $\lambda$ for different liquids and devices. The vertical lines correspond to Eq. (3) evaluated with $A=18.5$, $\alpha =3.32$ and $\mathrm{Oh}$ corresponding to each liquid. Inset: ${\tau }_{p-o}^{\mathrm{expt}}$ as a function of ${\tau }_{p-o}^{\mathrm{visc}}$. (b) ${\tau }_{p-o}^{\mathrm{expt}}$ as a function of ${\tau }_{p-o}^{\mathrm{short}}$ [Eq. (4)]. (c) ${\tau }_{p-o}^{\mathrm{expt}}$ as a function of ${\tau }_{p-o}^{\mathrm{long}}$ [Eq. (6), with $\tilde{\beta }=1.63\times {10}^{-4}$ and $\tilde{\gamma }=5\times {10}^{-6}$ s]. For all graphs, ${\mathrm{Oh}}_n$ varies as follows: 0.074 (dark blue), 0.084 (light blue), 0.097 (green), 0.124 (yellow), and 0.219 (orange). Triangles indicate data from device 1, circles from device 2. For (b) and (c), the points encircled (not encircled) by a black line correspond to short (long) ligaments.

Figure 4

(a) Our data in the form of ${\tau }_{p-o}^{\mathrm{theo}}/{\tau }_{\mathrm{recoil}}^{\mathrm{theo}}$ as a function of $\lambda$. Inset: Images evidencing the neck reopening, taken at $t=97$, 117, and $137\mu \mathrm{s}$ for ${\mathrm{Oh}}_n=0.124$, 88 V, $25\mu \mathrm{s}$, 100 Hz with device 2. (b) Same data plotted as a ($\mathrm{Oh}$;$\lambda$) map. Additional data from asymmetric (blue) [27] and symmetric (red) [2] ligaments. The red dashed line corresponds to the linear stability theory [19], the orange continuous line to the one of [28], the gray one to the criterion of [27], and the dashed (continuous) black lines to our criterion for short (long) ligaments (gray and black symbols). Full symbols correspond to pinch-off, empty ones to recoil; circles filled with orange correspond to observed neck reopening.