Two jets during the impact of viscous droplets onto a less-viscous liquid pool

It is generally accepted that a Worthington jet occurs when a droplet impacts onto a liquid pool. However, in this experimental study of the impact of viscous droplets onto a less-viscous liquid pool, we identify another jet besides the Worthington jet, forming a two-jet phenomenon. The two jets, a surface-climbing jet and the Worthington jet, may appear successively during one impact event. By carefully tuning the impact conditions, we find that the two-jet phenomenon is jointly controlled by the droplet-pool viscosity ratio, the droplet Weber number, and the droplet-pool miscibility. The mechanism of the surface-climbing jet is completely different from that of the Worthington jet: the liquid in the pool climbs along the surface of the droplet and forms a liquid layer which converges at the droplet apex and produces the surface-climbing jet. This surface-climbing jet has a very high speed, i.e., an order of magnitude higher than the droplet impact speed. The effects of the impact speed, droplet viscosity, droplet size, and surface tension on the surface-climbing jet are also analyzed. This study not only provides physical insights into the mechanism of droplet and jet dynamics but also will be helpful in the optimization of the droplet impact process in many relevant applications.

Figure 1

Schematic diagram of the experimental setup.

Figure 2

Two-jet phenomenon during one impact event. (a)–(i) Sequences of images showing the impact of a viscous droplet (glycerol-water mixture, $D=2.61\mathrm{mm}, V=3.37\mathrm{m}/\mathrm{s}, \text{We}=429, {\mu }_d=784\mathrm{mPa}\mathrm{s}, \overline{\mu }=871$) onto a water pool. The two jets are marked by black and red arrows. (a1)–(i1) Images taken from the horizontal view; (a2)–(i2) images taken from the aerial view. (j)–(k) Schematic drawings of droplets in dashed rectangles in (b1) and (f1), respectively. See Movies 1 and 2 in the Supplemental Material [31].

Figure 3

(a) Schematic diagram of droplet impact on a liquid pool. (b)–(c) Magnification of the triple-line region. (b) The impact speed is lower than the critical threshold speed. (c) The impact speed is higher than the threshold speed. ${\theta }_d$ is the dynamic contact angle, which is larger than the static contact angle ${\theta }_0$ for a moving triple line.

Figure 4

Regime map for the surface-climbing jet and the two types of Worthington jets (induced by bubble pinch-off or by surface wave). The lines in the diagram are just to guide the eyes. The green solid line and the red solid line indicate thresholds for the surface-climbing jet and for the Worthington jet, respectively. The blue dashed line indicates the region for capturing large bubbles, while the black dashed line indicates the region for capturing tiny bubbles without the formation of any jet [5]. When categorizing the phenomena, we regard the liquid column as a jet when there is a breakup of droplets. $\overline{\mu }\equiv {\mu }_d/{\mu }_p, D=2.35\mathrm{mm}$, and the pool liquid is pure water.

Figure 5

Sequences of images showing the effect of increasing the impact speed for a glycerol droplet impacting a pool of water, $D=2.35\mathrm{mm}, {\mu }_d=1186\mathrm{mPa}\mathrm{s}, \overline{\mu }=1318$. (a) $V=2.33\mathrm{m}/\mathrm{s}, \text{We}=257$; (b) $V=2.62\mathrm{m}/\mathrm{s}, \text{We}=324$; (c) $V=2.91\mathrm{m}/\mathrm{s}, \text{We}=400$; and (d) $V=3.20\mathrm{m}/\mathrm{s}, \text{We}=484$.

Figure 6

Diameter of the rim of the liquid layer ($D_0$) obtained from high-speed images corresponding to Fig. 5.

Figure 7

Sequences of images showing the effect of increasing the viscosity of the droplet, for droplets of various viscosities impacting a pool of water, $D=2.35\mathrm{mm}, V=3.13\mathrm{m}/\mathrm{s}, \text{We}=429–463$. (a) ${\mu }_d=96\mathrm{mPa}\mathrm{s}, \overline{\mu }=107$; (b) ${\mu }_d=314\mathrm{mPa}\mathrm{s}, \overline{\mu }=349$; (c) ${\mu }_d=645\mathrm{mPa}\mathrm{s}, \overline{\mu }=717$; (d)–(e) ${\mu }_d=1186\mathrm{mPa}\mathrm{s}, \overline{\mu }=1318$. (a)–(d) Images taken from the horizontal view; (e) images taken from the aerial view.

Figure 8

Diameter of the rim of the liquid layer obtained from high-speed images corresponding to Fig. 7.

Figure 9

Sequences of images showing the effect of changing the surface tension of the pool fluid. $D=2.35\mathrm{mm}, V=2.62\mathrm{m}/\mathrm{s}, \text{We}=309, {\mu }_d=194\mathrm{mPa}\mathrm{s}, \overline{\mu }=216$. (a) The liquid pool is pure water, ${\sigma }_p=71.3$ mN/m. (b) Surfactant SDS is added to the pool at a concentration of 2 wt %, ${\sigma }_p=44.2$ mN/m.

Figure 10

Sequences of images showing the effect of changing droplet size, for droplets impacting on a pool of pure water, $V=2.62\mathrm{m}/\mathrm{s}, {\mu }_d=1186\mathrm{mPa}\mathrm{s}, \overline{\mu }=1318$. (a) $D=2.32\mathrm{mm}, \text{We}=320$; (b) $D=2.70\mathrm{mm}, \text{We}=373$; (c) $D=3.24\mathrm{mm}, \text{We}=447$.

Figure 11

Diameter of the rim of the liquid layer obtained from high-speed images corresponding to Fig. 10.

Figure 12

(a) Relationship between the speed of the surface-climbing jet ($V_{\text{jet}}$) and the impact speed of droplets ($V$). ${\mu }_d=314\mathrm{mPa}\mathrm{s}$. (b) Relationship between the speed of the surface-climbing jet ($V_{\text{jet}}$) and the droplet-pool viscosity ratio ($\overline{\mu }$). $V=2.62\mathrm{m}/\mathrm{s}$. For each point in the plots, the experiment was repeated ten times, and the error bars show the standard deviations.