Droplet Splashing by a Slingshot Mechanism

When a drop impacts onto a liquid pool, it ejects a thin horizontal sheet of liquid, which emerges from the neck region connecting the two liquid masses. The leading section of this ejecta bends down to meet the pool liquid. When the sheet touches the pool, at an “elbow,” it ruptures and sends off microdroplets by a slingshot mechanism, driven by surface tension. High-speed imaging of the splashing droplets suggests the liquid sheet is of submicron thickness, as thin as 300 nm. Experiments in partial vacuum show that air resistance plays the primary role in bending the sheet. We identify a parameter regime where this slingshot occurs and also present a simple model for the sheet evolution, capable of reproducing the overall shape.

Figure 1

(Top) Typical ejecta sheet and sketch showing the slingshot mechanism. (Bottom sequence) Ejecta sheet emerges, bends downwards, ruptures as it touches the pool, and then slingshots droplets horizontally. Frames are 50, 70, 80, 90, 100, 110, 120, 130, 140, and $170\mu \mathrm{s}$ after impact. See Ref. 19.

Figure 2

Characterization of the ejecta sheet dynamics. (▪) Irregular broken sheets and spray. (Blue ○) Ejecta sheets do not bend towards the pool. (Green ⋄) Sheets bend downwards but do not reach the pool. For both triangles, the sheet bends downward and reaches the pool. (Red ▵) Sheet breaks before impacting the pool and no slingshot. (Yellow ▽) Slingshot of fine droplets.

Figure 3

The height of the ejecta sheet $H$ when the elbow touches the pool, normalized by the drop radius $R$, vs $K$. Data for $\mathrm{Oh}=0.017$ ($\mu =10\mathrm{cP}$, $D=4.4\mathrm{mm}$) (○), 0.023 ($\mu =14\mathrm{cP}$, $D=4.4\mathrm{mm}$) (▪), 0.044 ($\mu =30\mathrm{cP}$, $D=5.7\mathrm{mm}$) (▽), and 0.049 ($\mu =30\mathrm{cP}$, $D=4.4\mathrm{mm}$) (▴). Lower inset: The ejecta sheet for large value of $K$, where it touches the pool $\sim 50\mu \mathrm{s}$ after impact, when the drop has penetrated only 7% of $R$ into the pool. The bar is 0.5 mm.

Figure 4

Slingshot velocity $U_s$ of droplets and the corresponding film thickness $\delta$ based on the relative Taylor-Culick law. For $\mu =75\mathrm{cP}$ and $\mathrm{Oh}=0.049$ (open symbols). Filled symbols show results for the largest impact velocity $U=7.8\mathrm{m}/\mathrm{s}$ and $\mu =109\mathrm{cP}$ ($\mathrm{Oh}=0.18$).

Figure 5

Slingshotting sheets break up into tendrils and then droplets, $\mathrm{Re}\simeq 3500$. Times between image pairs ($dt$) are (a) 50, (b) 35, (c) 50, and (d) $100\mu \mathrm{s}$. The arrow in lower panel of (a) points out the rupture of the film as it touches the pool. The bar is 1 mm.

Figure 6

The effect of air pressure on the ejecta shapes. Impact of a 73% glycerin–water drop ($D=5\mathrm{mm}$) onto a liquid layer, for different air pressures, for $\mathrm{Re}\simeq 820$ and $\mathrm{We}=2700$. (a) $P_g=0.21\mathrm{bar}$ and (b) $P_g=1.0\mathrm{bar}$. Times shown are $\simeq 0$, 150, 300, 450, 600, 850, and $1250\mu \mathrm{s}$. The scale bar is 1 mm.

Figure 7

The effect of air pressure on the trajectory of the edge of the ejecta sheet. We follow the elbow just behind the tip, as this point touches the pool first. The curves correspond to air pressures of (from bottom to top curve) $P_g=1.0$, 0.74, 0.47, 0.34, 0.28, 0.14 bar (broken line). Data are for 58% glycerin–water mixture at $\mathrm{Re}=2450$ and $\mathrm{We}=2700$. The distances are normalized by the drop radius.

Figure 8

(a) Kinematic model of the sheet evolution for $C=14$, plotted for $\Delta \tau =0.015$. (b) Evolution of the curved liquid sheet, subjected to air resistance, but ignoring viscous stress and surface tension, for $R_{\mathrm{sp}}=2.5\mathrm{mm}$, $V_{\mathrm{sp}}=4\mathrm{m}/\mathrm{s}$, and $\delta =30\mu \mathrm{m}$. The drag coefficient is fixed at $C_d=2$. The tip is kept slightly thicker to mimic the observed bead produced by surface tension. Shapes shown until $200\mu \mathrm{s}$.