Edge effect: Liquid sheet and droplets formed by drop impact close to an edge

Asymmetric liquid sheet fragmentation is ubiquitous in nature and potentially shapes critical phenomena such as rain-induced propagation of foliar diseases. In this experimental study, we investigate the formation and fragmentation of a liquid sheet upon impact of a drop close to the edge of a solid substrate. Both the impact Weber number and the offset, the distance from the impact point to the edge, are systematically varied. Their influence on the kinematics of the liquid sheet and the subsequent statistics of droplet ejection are rationalized. Three major asymmetry scenarios are identified and linked to distinct droplet ejection patterns. Scaling laws are proposed to rationalize these scenarios based on impact parameters.

Figure 1

Liquid sheet formation and fragmentation into droplets, in three configurations. (a) In the field, a raindrop impacts onto a potato leaf on which dyed fluid (red) was deposited. Times are, from left to right, $-3$, 2, 6, 13, and 29 ms with respect to impact. The sheet is fully three dimensional. (b) A water drop impacts close to a dyed sessile drop on a flat horizontal dry plexiglass substrate. Times are, from left to right, $-1$, 3, 8, 11, and 23 ms with respect to impact. The sheet is still fully three dimensional. (c) Impact of a dyed drop close to the edge of a flat dry plexiglass substrate. Times are, from left to right, 0, 2.5, 8, 10.5, and 14 ms from impact. The sheet remains largely in the plane of the substrate. Scale bars are 5 mm.

Figure 2

Drop impact on a flat surface close to its edge, from top and side views. The radius of the impacting drop is $R_0\simeq 2.4$ mm and the Weber number $\text{We}=1340$. The offset $d$ is defined as the distance between the impact point and the edge. The scale bar is 5 mm and the times are (a) $t=0$, (b) $t=1.6$, (c) $t=3$, (d) $t=4.4$, (e) $t=7$, (f) $t=10.6$, (g) $t=12.4$, and (h) $t=19$ ms after impact.

Figure 3

Inclination $\phi$ of the sheet with respect to the vertical as a function of the offset $d/R_0$ for different We (see Table 1 for symbols of $\text{We})$. The solid line is $\phi ={87}^{\circ }$ and the dashed line represents Eq. (1). The inset shows the side-view time sequence of a drop impacting near an edge at $\text{We}=680$ for (a) $d/R_0=2.2$ and (b) $d/R_0=1$. Images are taken at $-0.2$, 0.4, 1, 2.2, and 6.4 ms from the time of impact, respectively, from top to bottom. The scale bar is 4 mm.

Figure 4

Time evolution of the sheet in the air for $d/R_0=4.4$, 3.2, 2, and 1.3 (top to bottom) and $\text{We}=1340$. Rows 1, 2, 3, and 4 correspond to the retraction scenarios I, II, II, and III, respectively. Snapshots in the same column are taken at the same time $t$ post impact, with $t=0$, 3, 6, 8, 11, 12, and 14 ms from left to right. Column (a) illustrates the position of the drops right before impact, with respect to the edge. The scale bar is 5 mm. The dashed frames highlight three different droplet ejection mechanisms illustrated in Fig. 16. See videos 1–4 in the Supplemental Material in Ref. [61].

Figure 5

Time evolution of the sheet in the air for $\text{We}=367$, 700, and 1340, from top to bottom, for $d/R_0=1.3$, all leading to scenario III. Snapshots in the same column are taken at the same time $t$ post impact, with $t=3$, 6, 8, 11, 12, 13, and 14 ms from left to right. The scale bar is 5 mm. See videos 4–6 in the Supplemental Material in Ref. [61].

Figure 6

Phase diagram We vs $d/R_0$, in which different sheet asymmetry scenarios are colored differently: I, black; II, red (gray); and III, blue (clear). Symbols correspond to different We, according to Table 1. The data corresponding to the examples of Figs. 4 and 5 are circled. The shaded region $d/R_0<1.3$ corresponds to experiments for which the sheet is not planar. The solid and dashed lines correspond to Eqs. (15) and (16), respectively.

Figure 7

Main variables that characterize the kinematics of the spreading on a solid, of a liquid sheet in the air, and of ejected droplets. The contours (blue lines) are detected by image processing, after thresholding and morphological removal of the corrugations. This particular image is taken 10.5 ms after impact, at offset $d/R_0=1.28$ and $\text{We}=367$. The scale bar is 4 mm. The radius $R_s$ of the spreading liquid on the solid, the extension of the liquid sheet tangent to the edge $l_t$, and the normal extension of the sheet $l_n$ are represented. The latter is taken to be the quantile 85% in distance to the edge of the contour of the sheet located in a sector of $\pm {10}^{\circ }$ (fine dotted lines). Several properties are recorded for each droplet at the time they separate from the sheet: the angular position ${\theta }_x$ (measured from the impact point), the radius $r$ of a sphere of equivalent volume (here circled in blue), and the magnitude $v$ and direction ${\theta }_v$ of the ejection velocity $\mathbf{v}$.

Figure 8

(a) Time evolution of the spreading radius $R_s/R_0$ for increasing We (from bottom to top, $\text{We}=186$, 367, 700, 1340, and 2435). The left inset shows the maximum spreading $R_{sM}/R_0$ as a function of We. The solid line corresponds to Eq. (3). The right inset shows the time $t_{sM}/t_i$ of the maximum spreading as a function of We. The solid line corresponds to Eq. (4). The dashed line is the scaling $t_{sM}/t_i\simeq {\text{We}}^{3/10}{\text{Oh}}^{-1/10}$ from [68]. (b) Rescaled time evolution of the spreading radius $R_s/R_{sM}$ vs $t/t_{sM}$, for the five We values in Table 1. The dashed line shows Eq. (2). The inset shows dimensionless offset $\delta =d/R_{sM}$ as a function of $t_d/t_{sM}$. The solid line corresponds to Eq. (5). Symbols correspond to different We, all collapsed from 186 to 2435.

Figure 9

(a) Normalized time evolution of the sheet extension normal to the edge $l_n\left(t\right)$ for the six examples of Figs. 4 and 5. The colors represent the different scenarios corresponding to these examples: I, black; II, light and dark red (light and dark gray); and III, blue (clear). The solid black line is Eq. (7). The blue lines correspond, respectively, to the theoretical results from [38] (solid) and from [39] (dashed) and to a harmonic oscillator suggested in Refs. [69, 70] (dotted). (b) Maximum normal extension of the liquid sheet $l_{nM}$ normalized by $R_0{\text{We}}^{1/2}$ as a function of $\delta$. The solid line shows Eq. (8), which is fitted on all the data points (with $d/R_0>1.3)$. The values for different We are $186(◯)$, $367(▹)$, $700\left(★\right)$, $1340(\square )$, and $2435(◇)$. The top inset shows the slope $l_{nM}/R_0$ vs $\delta$, as a function of We. The bottom inset shows the late spreading on solid, close to reaching $R_{sM}$, with and without the presence of an edge (bottom and top lines, respectively), for $\delta =0.73$ and $\text{We}=180$ (black circle in the main graph). The time interval is 1.5 ms from $t=5$ ms after impact until reaching $R_{sM}$, from left to right. The cross and the dotted line are positioned at distances $R_{sM}$ and $l_{nM}+d$ from the impact point, respectively. The cross height is 1 mm.

Figure 10

(a) Nondimensional acceleration $a_n$ of the sheet normal to the edge as a function of offset $\delta$. The inset shows the absolute value of $a_n$, averaged over $\delta$, as a function of We. The solid line is Eq. (10). (b) Normalized time of maximum normal extension $(t_{nM}-t_d)/t_c$ as a function of the offset $\delta$. The solid line corresponds to Eq. (11). The values of We are $186(◯)$, $367(▹)$, $700\left(★\right)$, $1340(\square )$, and $2435(◇)$.

Figure 11

(a) Schematics of the geometrical similarity between the sheet extension for the edge and pole configurations. (b) Comparison of the maximum radial distance from the impact point reached by the liquid for three configurations: full spreading on a solid (closed green symbols, $R_{sM}/R_0)$, liquid sheet from a flat edge [shaded area, $(l_{nM}+d)/R_0]$, and liquid sheet from a pole [open triangles, $(l_{nM}+d)/R_0]$. Two ratios of the pole to drop radius were considered: $d/R_0=1.5\left(△\right)$ and $d/R_0=2.4\left(▽\right)$. The crossed $▽$ and $△$ represent the maximal distance reached in edge experiments with the same offsets $d/R_0$. The solid and dashed lines correspond to Eqs. (3) and (12), respectively.

Figure 12

(a) Maximum tangential extension $l_{tM}$ of the liquid sheet along the edge normalized by the maximum spreading on solid $R_{sM}$, as a function of offset $\delta$. The solid line is $\sqrt{1-{\delta }^2}$, in agreement with Eq. (14). (b) Time evolution of the sheet extension along the edge $l_t\left(t\right)$, normalized by its maximum value $l_{tM}$ for the six examples of Figs. 4 and 5. The scenarios are represented as follows: I, black; II, light and dark red (light and dark gray); and III, blue (clear). The solid line for ${\tau }_s<1$ represents Eq. (13). The values of We are $186(◯)$, $367(▹)$, $700\left(★\right)$, $1340(\square )$, and $2435(◇)$.

Figure 13

Normalized time of collapse ${\tau }_r=(t_r-t_d)/(t_{nM}-t_d)$ of the liquid sheet along the edge as a function of the offset $\delta$. Symbols corresponds to We in Table 1. The solid line is the average across We and $\delta$ and the gray area corresponds to one standard deviation.

Figure 14

Scenarios of liquid sheet expansion and retraction, in a $(\delta ,l_{nM}/l_{tM})$ diagram. The symbols correspond to different We given in Table 1 and the colors to different scenarios: I, black; II, red (gray); and III, blue (clear). The solid and dashed lines correspond to Eqs. (15) and (16), respectively. The horizontal line is $l_{nM}/l_{tM}=0.85$ and the vertical line is $\delta =0.3$.

Figure 15

Time superposition of snapshots from a given experiment, with $t\in [t_{sM},t_{nM}]$. The curved red solid lines represent the reconstructed envelope region accessible to the sheet, predicted by Eq. (17). The top row shows, from left to right, $\text{We}=1340$ and $\delta =0.79$, 0.54, and 0.34. The bottom row shows, from left to right, $\text{We}=1340$, 700, and 367 and $d/R_0=1.3$, with $\delta =0.22$, 0.24, and 0.28. Scale bars are 4 mm.

Figure 16

Mechanisms of droplet ejection. (a) Radial ejection from the rim. Successive snapshots are separated by 0.4 ms, from 6.2 ms after impact. The highlighted droplet is ejected at ${\tau }_n=1.02$, with a mass $m=0.23$ mg and a speed $v=0.28V_0$. Its ejection angles are ${\theta }_x=4^{\circ }$ and ${\theta }_v=-9^{\circ }$. (b) Tangential ejection from the rim. Successive snapshots are separated by 0.6 ms, from 4 ms after impact. The highlighted droplet is ejected at ${\tau }_n=1.17$, with a mass $m=0.29$ mg and a speed $v=0.46V_0$. Its ejection angles are ${\theta }_x={48}^{\circ }$ and ${\theta }_v=7^{\circ }$. (c) Collapse of the sheet. Snapshots are taken at 11.8, 12.2, and 13 ms after impact. Scale bars are 2 mm. The orange and blue solid lines join the center of mass of the ejected droplets across frames. The vertical black lines are at fixed position, so they highlight the left-to-right motion of the droplets. These three snapshots correspond to close-up of the droplet ejections framed in Fig. 4. Panels (b) and (c) display also a small part of the solid substrate. The edge is then the horizontal black line.

Figure 17

Angle of the ejection velocity ${\theta }_v$ as a function of the angular position of ejection ${\theta }_x$ measured from the impact point (defined in Fig. 7), for $\text{We}=1340$. Symbols without (with) black contour correspond to $\delta \in [0.2,0.3]\left(\delta \in [0.7,0.8]\right)$. The color refers to the ejection time: dark blue, during sheet expansion, i.e., ${\tau }_n<1$; light blue, during sheet retraction, i.e., $1<{\tau }_n<{\tau }_r$; and red, after sheet collapse, i.e., ${\tau }_n>{\tau }_r$. The inclined solid line is the bisector ${\theta }_v={\theta }_x$. The vertical lines correspond to the maximum sheet angle ${\theta }_M$ (defined in Fig. 15), for $\delta =0.25$ (dotted) and $\delta =0.75$ (solid).

Figure 18

Time evolution of the droplet ejection speed $v$ normalized by (a) the impact speed $V_0$ and (b) $v_T$, for the six examples of Figs. 4 and 5. The time ${\tau }_n$ is normalized according to the normal extension of the sheet. Each data point corresponds to a single droplet. Symbols correspond to the Weber numbers $\text{We}=367(▹)$, $700\left(★\right)$, and $1340(\square )$, while colors indicate the degree of asymmetry from lowest (I) to highest (III) with I, black; II, light and dark red (light and dark gray); and III, blue (clear). The gray area indicates the time ${\tau }_r$ at which the sheet has fully retracted from the edge (average across We and $\delta$, plus or minus the standard deviation). The numbers 1, 2, and 3 indicate periods of time of the sheet (1, sheet expansion; 2, sheet retraction; and 3, after full sheet retraction along the edge). In (a), horizontal rectangles represent the duration of the sheet retraction. The darker blue rectangle in scenario III corresponds to the retraction of the sheet after it has pinched from the edge. The inclined solid lines correspond to Eq. (18) for the six examples (one line per scenario). The dotted line represents Eq. (19). Circled data points correspond to the snapshots of Fig. 16.

Figure 19

Average (large symbols) and standard deviation (small symbols) of $v/v_T$ for each $(\text{We},\delta )$. (a) Over the sheet expansion $\left({\tau }_n<1\right)$. The dotted line is the average over all $(\text{We},\delta )$. The inset shows the PDF of $v/v_T$ during the sheet expansion. The solid line is a fit with a Gaussian distribution of mean 1.05 and standard deviation 0.2. (b) After sheet expansion $\left({\tau }_n>1\right)$. The dotted line is the average over all $(\text{We},\delta )$ for ${\tau }_n<1$. The values of We are $186(◯)$, $367(▹)$, $700\left(★\right)$, $1340(\square )$, and $2435(◇)$.

Figure 20

(a) Time evolution of the mass $m$ of the ejected droplets (normalized by the mass $M_0$ of the impacting drop) as a function of their normalized time of ejection ${\tau }_n$, for the six examples of Figs. 4 and 5. Each data point corresponds to a single droplet. Symbols correspond to Weber numbers $\text{We}=367(▹)$, $700\left(★\right)$, and $1340(\square )$, while colors indicate scenarios I, black; II, light and dark red (light and dark gray); and III, blue (clear). The gray area indicates the time ${\tau }_r$ at which the sheet has fully retracted from the edge (average across We and $\delta$, plus or minus the standard deviation). The inclined solid line represents Eq. (21). The horizontal lines correspond to the cutoff mass $\mathrm{\Phi }(m/M_0)$ during retraction with $\text{We}=367$ (dotted), $\text{We}=700$ (dashed), and $\text{We}=1340$ (solid). (b) Probability distribution function of the normalized mass $m/\left(M_0{\tau }_n^{5/2}\right)$ of droplets ejected during sheet expansion $\left({\tau }_n<1\right)$, pooled per We (all $\delta$ together). The symbols correspond to the values $\text{We}=186(◯)$, $367(▹)$, $700\left(★\right)$, $1340(\square )$, and $2435(◇)$. The inset shows the saturation value $\mathrm{\Phi }$ of the PDF, for each We and $\delta$. The dotted line corresponds to the coefficient of Eq. (21).

Figure 21

(a) Maximum horizontal distance $\mathrm{\Phi }(\mathrm{\Psi }/R_0)$ traveled by the ejected droplets, pooled together per We and $\delta$. Symbols correspond to different We (Table 1) while colors indicate scenarios I, black; II, red (gray); and III, blue (clear). The dashed lines correspond to Eq. (22). (b) $\mathrm{\Phi }(\mathrm{\Psi }/R_0)$ during the whole ejection process compared to the one dictated by the droplets ejected during the sheet extension. The solid line is the bisector. Symbols correspond to the values $\text{We}=186(◯)$, $367(▹)$, $700\left(★\right)$, $1340(\square )$, and $2435(◇)$.

Figure 22

Mass $m$ of the ejected droplets, normalized by the mass of the impacting drop, as a function of their traveled distance $\mathrm{\Psi }$ for $\text{We}=1340(\square )$ and (a) $\delta \in [0.2–0.3]$ and (b) $\delta \in [0.5–0.6]$. The color indicates the ejection time: during sheet expansion $({\tau }_n<1$, dark blue), during sheet retraction $(1<{\tau }_n<{\tau }_r$, light blue), and after sheet collapse $({\tau }_n>{\tau }_r$, red). The vertical dotted lines represent Eq. (22).

Figure 23

(a) Non-normalized cumulative distribution function (expressed in number of ejected droplets) of the normalized time ${\tau }_n$ for the six examples of Figs. 4 and 5. Symbols correspond to Weber numbers $\text{We}=367(▹)$, $700\left(★\right)$, and $1340(\square )$, while colors indicate scenarios I, black; II, light and dark red (light and dark gray); and III, blue (clear). The gray area indicates the time ${\tau }_r$ at which the sheet has fully retracted from the edge (average across We and $\delta$, plus or minus the standard deviation). The horizontal rectangles represent the duration of the sheet retraction. The darker blue rectangle in scenario III corresponds to the retraction of the sheet after it has pinched from the edge. (b) Average number $N$ of droplets ejected per impact, as a function of $\delta$, for $\text{We}=186(◯)$, $367(▹)$, $700\left(★\right)$, $1340(\square )$, and $2435(◇)$. Solid lines are Eq. (23). The inset shows the second derivative of $N$ with respect to $\delta$, as a function of We. The solid line corresponds to $d^{2}N/d{\delta }^2=0.04{\text{We}}^{1.4}$ in Eq. (23).

Figure 24

Maximum traveled horizontal distance $\mathrm{\Psi }$, as a function of the dimensionless droplet size $\beta$. Each symbol corresponds to a different ejection speed $v$: $◯,v={10}^{-3}$ m/s; $▹,v={10}^{-2}$ m/s; $★,v={10}^{-1}$ m/s; $\square ,v=1$ m/s; and $◇,v=10$ m/s. The solid line corresponds to $\mathrm{\Psi }\left(\beta \right)/\mathrm{\Psi }\left(0\right)\simeq 12{\beta }^{-1/2}$.

Figure 25

Cutoff $\mathrm{\Phi }$ of the mass distribution of ejected droplets, pooled per We and $\delta$, (a) during the retraction of the sheet and (b) after collapse of the sheet, for the values $\text{We}=186(◯)$, $367(▹)$, $700\left(★\right)$, $1340(\square )$, and $2435(◇)$. The insets show the dependence on We of the cutoff $\mathrm{\Phi }$ (a) after pooling all $\delta$ [the solid line is Eq. (C1)] and (b) after pooling scenarios II and III [the solid line is Eq. (C2)].