Drop impact close to the edge of an inclined substrate: Liquid sheet formation and breakup

Raindrop impacts on plant leaves are responsible for the dispersal of several crop diseases. In this experimental study, we propose a generic impact configuration that shares several qualitative features with more complex impacts on leaves. We investigate the formation and breakup of a liquid sheet upon impact of a drop close to the horizontal edge of an inclined substrate. The impact Weber number, the distance from the impact point to the edge, and the substrate inclination are systematically varied. We analyze their influence on the kinematics of both the liquid spreading on substrate and the liquid sheet beyond the edge, as well as on the subsequent statistics of droplet ejection (radius, speed, and time of ejection).

Figure 1

(a) Crescent-moon impact where a raindrop impacts next to and pushes a sessile drop of dyed water, which is then stretched in a liquid sheet and finally fragmented in droplets. The red dye represents potential pathogens that would be present in the contaminated liquid residue on the leaf. (b) Crescent-moon impact where the dyed contaminated liquid sheet reaches the edge of the leaf. In both (a) and (b) the scale bar is 4 mm and the time $t$ from impact is indicated on each snapshot [16].

Figure 2

(a) A clear water drop impacts on a graminea leaf on which a dyed drop has been deposited. Upon impact, the leaf deflects downward and stretches the liquid sheet. The time $t$ from impact is indicated on each snapshot [16]. The yellow crosses and the double arrow indicate the original position of the leaf and the deflection from impact, respectively. (b) Schematic of the stretching of the crescent-moon liquid sheet due to the deflection of a leaf. The sheet is formed by the impact of a drop of radius $R_0$ and speed $V_0$ at a distance $d$ from another sessile drop. (c) Schematic of the stretching of the liquid sheet following the impact of a single drop of radius $R_0$ and speed $V_0$ on an inclined substrate, at a distance $d$ from its edge.

Figure 3

Side (top row) and top (bottom row) views of the impact of a drop of radius $R_0$ at speed $V_0$ on a substrate of inclination $\alpha$, at a distance $d$ from its edge. The dimensionless parameters are $\alpha ={60}^{\circ }$, $\text{We}\simeq 565$, and $d/R_0=2.5$. The scale bars are 5 mm and the time $t$ from impact is indicated on each snapshot [16].

Figure 4

Time evolution of the liquid sheet for $d/R_0=12$, 7.3, and 1.6 (left to right). The substrate inclination is $\alpha ={60}^{\circ }$ and the Weber number is $\text{We}\simeq 565$. Each column corresponds to an experiment illustrating a different sheet scenario (I, II, and III, from left to right). Snapshots in the same row are taken at the same time $t$ from impact indicated in the leftmost snapshot [16]. Row (a) illustrates the position of the drops right before their impact, with respect to the edge. The scale bars are 5 mm. The arrow in (c), column, III indicates the position of a tiny air bubble trapped at early impact and then advected with the sheet.

Figure 5

Time evolution of the sheet for inclination (a) $\alpha ={60}^{\circ }$, (b) ${40}^{\circ }$, (c) ${20}^{\circ }$, (d) $0^{\circ }$, and (e) $-{20}^{\circ }$ with similar $d/R_0\in [2,3.6]$ and $\text{We}\simeq 2115$. Scenario III is observed in rows (a)–(c), while scenarios II and I are observed in rows (d) and (e), respectively. Snapshots in the same column are taken at the same time $t$ from impact indicated in the top row [16]. The scale bars are 5 mm.

Figure 6

(a) Time superposition of the spreading following an impact at coordinates $(X_0,Y_0)$ on an inclined substrate ($\alpha ={40}^{\circ }$, $\text{We}\simeq 565$, and $d/R_0\simeq 7.0$). The spreading envelope is approximated by an ellipse of minor axis $W$ and major axis $L$, with one of the foci in $(X_F,Y_F)$. Polar coordinates $(\varrho ,\theta )$ are centered on the impact point. The scale bar is 5 mm. (b) Parity plot showing that $X_F$ and $X_0$ are almost identical for both $\text{We}\simeq 565$ ($▾$) and $\text{We}\simeq 2115$ ($▵$) and for $\alpha \in [-{40}^{\circ },{60}^{\circ }]$ (colors defined in the legend).

Figure 7

(a) Semiminor axis $W$ of the ellipse (normalized by $R_0$) obtained by fitting the envelope of the spreading liquid rim on the substrate as a function of $\alpha$ for both $\text{We}\simeq 565$ ($▾$) and $\text{We}\simeq 2115$ ($▵$). The solid (dashed) lines correspond to Eq. (8) for $\text{We}=565$ ($\text{We}=2115$). (b) Time $t_{sM}$ (normalized by $t_c$) at which spreading in the direction tangent to the edge reaches its maximum value $W$, as a function of ${\text{We}}_{\perp }$. The symbols have the same meaning as in (a), with the color representing $\alpha$. The solid line is Eq. (9) with $c_1=1.5$. In both (a) and (b), error bars represent the standard deviation over all experiments at a given $(\text{We},\alpha )$.

Figure 8

Ratio between the average translation speed $V_c$ of the center of the circular arc described by the rim as it spreads on the substrate and the tangential component of the impact speed $V_{0}sin\alpha$. Symbols correspond to different Weber numbers: $\text{We}\simeq 565$ ($▾$) and $\text{We}\simeq 2115$ ($▵$). The solid horizontal line corresponds to $V_c/V_{0}sin\alpha =1/c_1$.

Figure 9

(a) Aspect ratio $W/L$ of the ellipse made by the spreading envelope and (b) maximal spreading distance $R_{snM}$ in the direction normal to the edge, both as functions of $\alpha$, for $\text{We}\simeq 565$ ($▾$) and $\text{We}\simeq 2115$ ($▵$). The solid lines in (a) and (b) are Eqs. (11) and (12), respectively.

Figure 10

Main variables $l_t$ and $l_n$ that characterize the liquid sheet kinematics in directions tangent and normal to the edge, respectively. The scale bar is 5 mm and the snapshot is taken at 14.5 ms after impact, at $\text{We}\simeq 565$ and $\alpha ={60}^{\circ }$.

Figure 11

(a) Extension $l_n$ of the liquid sheet normal to the edge, as a function of the time $t-t_d$ since the liquid reached the edge. These five experiments at $\text{We}\simeq 2115$, $\delta =0.2$, and various $\alpha$ (see the legend) correspond to the snapshots of Fig. 5. Symbols correspond to experimental measurements, while the solid lines are the approximations by Eq. (15). (b) Snapshots of a collapse of the sheet in the direction tangent to the edge at $\text{We}\simeq 565$ and $\alpha ={60}^{\circ }$. The normal extension $l_n\left(t\right)$ experiences a sharp decrease. Only values of $l_n\left(t\right)$ before such collapse are considered. The scale bar is 5 mm and the time $t$ from impact is indicated on each snapshot.

Figure 12

(a) Normalized speed $V_d$ of the liquid rim at sheet birth, as a function of $\delta$. (b) Same speed $V_d$, now normalized by $V_0\sqrt{1-\delta }$ and then averaged over $\delta$. In both (a) and (b), the solid line corresponds to Eq. (16) with $c=1.28$. Symbols represent different Weber numbers $\text{We}\simeq 565$ ($▾$) and $\text{We}\simeq 2115$ ($▵$). Each symbol color corresponds to a different $\alpha$, as given in (b).

Figure 13

(a) Normalized deceleration $a_n$ of the rim, as a function of $\delta$. (b) Average of $a_n{t}_c^2/2R_0$ over $\delta$ as a function of $\alpha$, normalized by ${\text{We}}^{0.5}$. The solid line is Eq. (18). The error bar is the standard deviation over the same population. Symbols correspond to different Weber numbers $\text{We}\simeq 565$ ($▾$) and $\text{We}\simeq 2115$ ($▵$). Each symbol color corresponds to a different $\alpha$, as given in (b).

Figure 14

(a) Maximum extension $l_{tM}$ of the liquid sheet along the edge, normalized by the maximum spreading on solid $R_{sM}$, as a function of dimensionless offset $\delta$, for $\text{We}\simeq 565$ ($▾$) and $\text{We}\simeq 2115$ ($▵$) and for various $\alpha$ (color in legend). The solid lines correspond to Eq. (22). (b) Time evolution of the sheet extension $l_t\left(t\right)$, along the edge, normalized by its maximum value $l_{tM}$ for $\text{We}\simeq 2115$ and various $\alpha$ (colors defined in the legend). Different offsets are not distinguished. The solid line represents Eq. (24).

Figure 15

Parity plot of ${\tau }_r$ (i.e., the measured time of full retraction along the edge $t_r-t_d$, normalized by the characteristic time of extension normal to the edge $V_d/a_n$) vs the ratio ${\tau }_{tM}$ (i.e., the time of maximum extension along the edge, normalized similarly) for $\text{We}\simeq 565$ ($▾$) and $\text{We}\simeq 2115$ ($▵$) and for various $\alpha$ (colors defined in the legend) and $\delta$ (not distinguished). The ratio ${\tau }_{tM}$ is calculated from $\text{We}$, $\delta$, and $\alpha$ according to Eq. (25). The parameters $V_d$ and $a_n$ involved in the normalization of ${\tau }_r$ are also calculated, from Eqs. (16) and (17), respectively. The solid line corresponds to ${\tau }_r=2{\tau }_{tM}$.

Figure 16

Scenarios of the liquid sheet (illustrated in Fig. 4): (a) in a diagram $(\alpha ,d/R_0)$ and (b) in a diagram representing the competition of the timescale ${\tau }_{tM}$ and length scale $2a_n{l}_{tM}/V_d^2$. Closed symbols (open symbols) correspond to $\text{We}\simeq 565$ ($\text{We}\simeq 2115$), while color indicates the scenario (see the legend). In (b) the horizontal solid line is in $2a_n{l}_{tM}/V_d^2=1.3$. The vertical dashed line is in ${\tau }_{tM}=0.5$.

Figure 17

Properties of ejected droplets detected by image processing: angular position of ejection (from impact point) ${\theta }_x$, velocity $\mathbf{v}$, velocity direction ${\theta }_v$, and radius $r$.

Figure 18

Mechanisms of droplet ejection, illustrated for $\alpha ={40}^{\circ }$, $\delta =0.5$, and $\text{We}\simeq 2115$. (a) Radial ejection from the rim. The ejection direction of the highlighted droplet is ${\theta }_v\simeq 6^{\circ }$, which is here perpendicular to the rim. (b) Tangential ejection from the rim. The ejection direction of the highlighted droplet is ${\theta }_v\simeq {15}^{\circ }$, which is here much more aligned with the rim. (c) Ejections during the collapse of the sheet. In (a)–(c) the scale bars are 2 mm and the time $t$ from impact is indicated on each snapshot. 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 help visualize the left-to-right motion of the droplets.

Figure 19

(a) Formation of fingers on inclined surfaces, leading to (b) an additional droplet ejection mechanism for substrates inclined downward. The time superposition of a drop impacting at $\text{We}\simeq 2115$, taken at 0, 4, and 9 ms from impact, is shown for (a) $\alpha =-{40}^{\circ }$ and $\delta \simeq 1$ and (b) $\alpha ={40}^{\circ }$ and $\delta \simeq 0.9$. The scale bar is 5 mm.

Figure 20

(a) Droplet ejection speed $v$, normalized by the speed $V_0$ of the impacting drop, as a function of the normalized time $\tau <{\tau }_r$. Each data point corresponds to a single ejected droplet, issued from six experiments (one per $\alpha$; see the legend for color definitions) at $\text{We}\simeq 2115$ and $\delta \in [0.1,0.3]$. The vertical gray line indicates the time of maximum sheet extension normal to the edge. The solid curve represents Eq. (28). Sheet scenarios I, II, and III are represented by circles, squares, and diamonds, respectively. (b) Average (large symbols) and standard deviation (small symbols) of $v/v_T$ over the sheet expansion, i.e., $\tau <min(1,{\tau }_r)$, as a function of $\delta$ for $\text{We}\simeq 565$ ($▾$) and $\text{We}\simeq 2115$ ($▵$) and various $\alpha$ [colors as defined in legend of (a)]. The solid line is $v=v_T$, according to Eq. (28). The dashed line is the average of the standard deviations (small symbols) over all $\text{We}$, $\alpha$, and $\delta$.

Figure 21

(a) and (b) Normalized time ${\tau }_e$ of first ejection along the rim as a function of $\delta$, for $\text{We}\simeq 565$ ($▾$) and $\text{We}\simeq 2115$ ($▵$) and for various $\alpha$ (color defined in legend). In (b) the time is further multiplied by ${\text{We}}^{0.5}$. The black solid line corresponds to Eq. (29).

Figure 22

(a) Radius $r$ of the ejected droplets, normalized by the radius $R_0$ of the impacting drop, as a function of the time since birth of the sheet ($t-t_d$), normalized by the capillary time $t_c$. Each data point corresponds to a single ejected droplet, issued from six experiments (one per $\alpha$, colors defined in the legend) at $\text{We}\simeq 2115$ and $\delta \in [0.1,0.3]$. These six experiments are the same as in Fig. 20. The solid line represents Eq. (31). (b) Coefficient $c_4$ of Eq. (31), calculated from the quantile 99% of $r{t}_c^{2/3}/R_0/{(t-t_d)}^{2/3}$ for each $\delta$ and $\alpha$ (colors as in other figures), for $\text{We}\simeq 565$ ($▾$) and $\text{We}\simeq 2115$ ($▵$). The dashed line corresponds to $c_4\simeq 0.4$.

Figure 23

Maximum distance traveled by the droplets ejected at $\text{We}\simeq 2115$, for various $\alpha$, and all $\delta$ pooled together. Circles represent the quantile 99% of the distance traveled by droplets ejected at $\tau <{\tau }_r$. Triangles represent the quantile 99% of the distance traveled by droplets ejected at $\tau >{\tau }_r$. The solid line is the theoretical prediction corresponding to Fig. 24.

Figure 24

Colormap of the maximum horizontal distance $\mathrm{\Psi }$ (in m) traveled by the ejected droplets, estimated theoretically for various $\text{We}$ and $\alpha$, for an incoming raindrop of radius $R_0=2.36$ mm with $\delta =0$. The dashed line corresponds to the angle $\alpha ={30}^{\circ }$ that yields the maximum distance for a given $\text{We}$.

Figure 25

Droplet ejection speed $v$ normalized by $V_0$, as a function of the droplet radius $r$ normalized by $R_0$, for droplets ejected after the full retraction of the sheet along the edge ($\tau >{\tau }_r$). Each data point corresponds to a single ejected droplet, issued from the six experiments of Fig. 20 (one per $\alpha$, with colors defined in the legend) at $\text{We}\simeq 2115$ and $\delta \in [0.1,0.3]$. Sheet scenarios I, II, and III are represented by circles, squares, and diamonds, respectively.