3.1. Rising of a liquid lamella

Figure 1(b) shows some experimental snapshots of the impact process, zooming in on the lamella. The liquid sheet is ejected during the impact event and subsequently flows over the plate surface. To understand the behaviour of the liquid lamella flow, we measure its height $h$ and rim thickness $e$ with respect to time. Here, $h$ is defined as the distance from the unperturbed water surface to the tip of the liquid film, and $e$ is the distance between the solid surface and the outer point of the rim, referred to as the maximum rim thickness, as indicated by the yellow arrows in figure 1(b). It should be noted that, since $e$ is obtained near the tip of the liquid film from experimental side-view images, it corresponds to the size of fingers in the case where the finger manifests, where fingers emerge owing to the instability of moving contact lines at an early time and subsequently retract (see inset in figure 3 d).

Figure 2. Time evolution of the liquid film height $h$ for different $U_{o}$ (see legend) and $D_{o}=10\,\rm mm$ . The error bars represent the standard deviation of ten experimental runs.

After the liquid film reaches its maximum height, it descends back into the liquid pool while being dragged by the solid, as depicted in figure 2. Since initially the kinetic energy of impacting objects transfers primarily into the liquid film (Thoroddsen et al. Reference Thoroddsen, Etoh, Takehara and Takano2004), the ejection speed of the liquid film directly depends on the initial speed of the object; the liquid film rises faster and higher with increasing $U_{o}$ for the same $D_{o}$ .

Figure 3. (a) The thickness $e$ of the rim of the liquid film as a function of time $t$ , for $D_{o}=10\,\rm mm$ . (b) To illustrate that $e$ is of the same order as the local capillary length $l_{c}$ defined in (3.1), the time evolution of their ratio $e/l_{c}$ is shown for $D_{o}=10\,\rm mm$ . (c) Time evolution of the ratio $e/l_{c}$ for $D_{o}=20\,\rm mm$ . (d) Time evolution of the averaged wavelength $\,\bar {\!\lambda }$ of the fingers for $D_{o}=10\,\rm mm$ . The inset shows how $\lambda$ is defined for each neighbouring pair of fingers in each image frame, which is subsequently averaged to determine $\,\bar {\!\lambda }$ . The error bar is the standard deviation of ten experimental runs.

In contrast to the film height $h$ , which reaches a maximum and then decreases again, the rim thickness $e$ continues to grow over time, as shown in figure 3(a), due to the liquid phase continuously accumulating in the rim. Here, the data were plotted until the point where the rim could be clearly differentiated from the lamella. From the same figure, one may conclude that, with increasing $U_{o}$ the rim becomes thinner. This can be understood by realising that the time rate of change of the volume of water displaced by the plate entering the pool is given by d $V$ / ${\rm d}t \sim D_{o}U_{o}$ , where the volume itself is expected to be determined by the dimensions of the lamella, i.e. $V \sim eh$ . Since $\dot {h} \equiv \mathrm {d}h$ / $\mathrm {d}t$ is at least an order of magnitude larger than $\mathrm {d}e$ / $\mathrm {d}t$ , $e$ is of the order of $D_{o} U_{o}$ / $\dot {h}$ . From the literature, it is known that the momentum transfer from the impact event to the liquid film becomes larger when $U_{o}$ increases (Thoroddsen et al. Reference Thoroddsen, Etoh, Takehara and Takano2004), and thus the ratio $U_{o}$ / $\dot {h}$ decreases monotonically with $U_{o}$ . Because $e \sim U_{o}$ / $\dot {h}$ , it decreases with increasing $U_{o}$ (as shown in figure 3 a).

The behaviour of the rim (i.e. the top part of the lamella) is governed by a competition between the inertia of the liquid sheet and a capillary restoring force concentrated at the rim (Villermaux & Bossa Reference Villermaux and Bossa2011; Wang et al. Reference Wang, Dandekar, Bustos, Poulain and Bourouiba2018). In addition, we consider the gravitational force $g$ in this situation. The viscosity does not come into play for the rim dynamics because viscosity only has an effect near the solid surface, where we assume a no-slip boundary condition. Since the rim continues to wet new solid surface, the viscous effects are confined to the boundary layer (Eggers et al. Reference Eggers, Fontelos, Josserand and Zaleski2010). Given that we measured the rim thickness close to the contact line, where the contribution of the boundary layer is relatively small, the viscous stress is negligible compared with gravity and capillarity. The capillary–inertial time scale is given by $\tau _{c}=\sqrt {\rho e_{o}^{3}/\sigma }$ (Rayleigh et al. Reference Rayleigh1879; Richard et al. Reference Richard, Clanet and Quéré2002), and we define the viscous time scale as $\tau _{\nu }=e_{o}^{2}/\nu$ , where for both time scales we use the initial value of rim thickness $e_{o}$ . Indeed, for $D_{o} = 10\,\rm mm$ , where we observe that $e_{o} \approx 0.5\,\rm mm$ , $\tau _{c}$ is approximately $1.3\,\rm ms$ whereas $\tau _{\nu } \sim 250\,\rm ms$ , ensuring that interfacial constraint on the rim is predominantly due to capillarity. Therefore, the balance of the involved forces acting on the rim may be written as

(3.1) \begin{equation} e \sim \sqrt {\frac {\sigma }{\rho (\mid -\ddot {h}\mid +g)}} = l_{c}, \end{equation}

which states that the rim thickness is proportional to the local and instantaneous capillary length based on the instantaneous film deceleration $\ddot {h}$ . For $D_{o}=10\,\rm mm$ , $e$ indeed closely follows $l_{c}$ without any fitting parameters, which is corroborated by the ratio $e$ / $l_{c}$ $\approx 1$ , as depicted in figure 3(b). The initial destabilisation of the rim is likely caused by a combination of Rayleigh–Plateau and Rayleigh–Taylor instabilities, with deceleration and interfacial constraints on the rim being crucial factors (Wang et al. Reference Wang, Dandekar, Bustos, Poulain and Bourouiba2018; Wang & Bourouiba Reference Wang and Bourouiba2021). At larger $D_{o}$ in figure 3(c), however, the liquid sheet is decelerated more than in the case of $D_{o}=10\,\rm mm$ , which leads to $\mid -\ddot {h}\mid$ being larger. That is, more liquid is pulled away from the liquid sheet into the rim, resulting in $e$ / $l_{c}$ being slightly larger than $1$ . It is interesting to note that, even though the liquid flows over a moving solid surface, there is an analogy with the rim dynamics for impacting on a small target, where the lamella and rim can expand freely, in the absence of the liquid friction from the solid surface (Villermaux & Bossa Reference Villermaux and Bossa2011; Wang et al. Reference Wang, Dandekar, Bustos, Poulain and Bourouiba2018).

Similar to droplet impact, fingering patterns caused by instability can be observed at the rim. For each time frame, we measure the distance of all observable, neighbouring fingers using the front-view experimental images and subsequently average them to obtain what we define to be the average wavelength $\,\bar {\!\lambda }$ of the pattern. The time evolution of this quantity is plotted in figure 3(d). Note that at early times $\,\bar {\!\lambda }$ is significantly smaller than the capillary length of water $l_{c, w}$ = $\sqrt {\sigma /\rho g} = 2.71\,\rm mm$ , but subsequently increases. Unlike sheet fragmentation after droplet impact on a pole, in which case the liquid can freely expand in the radial direction, in our case the advancing motion of the sheet is constrained geometrically by the width of plate $L$ . Hence, the ejected fingers emerge only at early times before the maximum extension of the lamella is reached, and subsequently the fingers merge with each other during the retraction of the rim due to the moving solid surface, causing $\,\bar {\!\lambda }$ to increase throughout the entire time for each $U_{o}$ (see supplementary movie https://doi.org/10.1017/jfm.2025.170). In addition, it can be seen that $\,\bar {\!\lambda }$ decreases with increasing $U_{o}$ . The number of fingers is proportional to $U_{o}$ , and the wavelength of fingers is proportional to $e$ (Wang et al. Reference Wang, Dandekar, Bustos, Poulain and Bourouiba2018; Wang & Bourouiba Reference Wang and Bourouiba2021). Increasing $U_{o}$ accelerates the rim destabilisation as it results in an increase of $\ddot {h}$ . In other words, with increasing $U_{o}$ , the number of fingers increases, causing $\,\bar {\!\lambda }$ to decrease. Further, as already discussed in figure 3(a), $e$ decreases with increasing $U_{o}$ , so that $\,\bar {\!\lambda }$ decreases with $U_{o}$ .

3.2. Modelling the height of the liquid film

Figure 4. (a) Simplified modelling of the liquid film dynamics, showing the considered control volume ( $CV$ , red dotted line). (b) A representative snapshot from PIV measurements of the airflow field in terms of velocity vectors and contour of vertical velocity components ( $U_{o}=1.3\,\rm m\,s^-{^1}$ and $D_{o} = 10\,\rm mm$ at $t = 9.9\,\rm ms$ ). The velocity vector magnitude ranges from $0$ to $0.5\,\rm m\,s^-{^1}$ , and the vertical velocity lies between $-0.3$ and $+0.35\,\rm m\,s^-{^1}$ . (c) Comparison of the experimentally measured liquid film height (symbols) with the simplified model of equation (3.6) (solid black line) for three typical experimental conditions (see legend). Comparison of the order of magnitude of the four acceleration terms on the right-hand side of (3.6) for the liquid film height predicted by the model for (d) $D_{o}=10\,\rm mm$ , $U_{o}=1.1\,\rm m\,s^-{^1}$ and (e) $D_{o}=10\,\rm mm$ , $U_{o}=2.1\,\rm m\,s^-{^1}$ . Note that $h\,\ddot {h}$ is plotted, rather than $\ddot {h}$ itself, to allow for a fair comparison over the time span.

In order to illustrate and understand the rising motion of the liquid film, we want to predict the time evolution of the film height $h$ with a model. A schematic diagram is shown in figure 4(a), with the liquid pool in light blue and the solid object, descending at a velocity $U_{o}$ , in light grey. For simplicity, we neglect the formation of the rim (i.e. uniform thickness in the $y$ direction) and assume that $e$ is constant throughout time, turning it into a one-dimensional problem. We set the position of the solid surface at $x=0$ and that of the undisturbed free surface at $y=0$ . The initial time is set by the moment when the liquid comes into the control volume ( $\rm CV$ ) in figure 4(a), such that the liquid sheet is located at $x =0, y =0$ when $t= 0$ . Since the frontal part of the solid is hemispherical, it takes some time for the liquid to reach the $\rm CV$ . The corresponding time for the liquid film to travel from the initial moment of impact to the origin point is typically less than 1 ms. The flow inside the liquid film is represented by $v(x,y,t)$ , and we assume there is no pressure gradient along the $y$ direction. Neglecting boundary effects at the sides of the object, the spanwise width of the solid plate $L$ cancels out, meaning the dynamics is considered per unit width in the spanwise direction. The mass conservation of the liquid film can therefore be expressed as

(3.2) \begin{equation} \frac {\mathrm {d}}{\mathrm {d}t}\left (\rho he\right )=\rho e \mathrm {v_{1}}, \end{equation}

where $\mathrm {v_{1}}$ is the velocity of the liquid-phase flow from the pool to the sheet. From (3.2), we have $\mathrm {v_{1}}=\dot {h}$ , taking $e$ to be constant as stated above.

As discussed in § 3.1, we consider that viscous effects are confined to a thin boundary layer close to the solid surface (Eggers et al. Reference Eggers, Fontelos, Josserand and Zaleski2010). This can be motivated by the transient character of the impact, where boundary layers start to form upon the first contact of the water and the body. Thus, viscous dissipation occurs in the area where the liquid sheet contacts the solid surface, which can be considered to be proportional to $h$ . The corresponding viscous force acting on the film can be approximated from Stokes’ first problem, given by

(3.3) \begin{equation} F_{\nu } =\mu \frac {\partial \mathrm {v}}{\partial x}\bigg |_{x=0} h = -\mu h \frac {U_{o}+\dot {h}}{\sqrt {\pi \nu t}}, \end{equation}

where $\nu$ is the kinematic viscosity of water. The thickness of the boundary layer $\delta \sim \sqrt {\nu t}$ does not grow beyond $e$ throughout the entire process. For example, for $U_{o}=1.3\,\rm m\,s^-{^1}$ and $D_{o}=10\,\rm mm$ , the initial value of $e$ is approximately 0.5 mm. This means that the boundary layer would take 250 ms to reach this thickness, which is much longer than the impact time of the object. First it should be noted that $F_\nu$ , as all forces introduced in this section, is understood to be a force per unit length in the spanwise direction of the object. Second, and more importantly, in estimating the velocity it needs to be taken into account that the liquid velocity in the film, $\mathrm {v_{1}}=\dot {h}$ , is measured in the laboratory reference frame where the downward moving plate has a velocity $-U_{o}$ such that the velocity difference between film and plate is equal to $-(\dot {h} + U_{o})$ . Consequently, $F_\nu$ is in the downward direction.

Using PIV and small oil droplets as seeds, we measure the airflow along the solid surface while the object enters the water, a snapshot of which is shown in figure 4(b). It is clear that air is being dragged with the downward-moving plate and moves back upward further away from the plate. For droplet impact, the air film is trapped between the liquid lamella and the solid substrate (Liu, Tan & Xu Reference Liu, Tan and Xu2015; Pack et al. Reference Pack, Kaneelil, Kim and Sun2018). Here, it is expected as well that due to the relatively high impact velocity, the influence of the surrounding gas would affect the beginning of the impact event, which is difficult to observe experimentally in our set-up. We assume the air stress is significant at the beginning as a result of air entrainment in front of the liquid sheet. The corresponding stress $\tau _{a}$ is chosen by the assumption that a thin wedge of air exists between the rim of the liquid and the solid surface so that the stress due to the air phase (de Gennes Reference de Gennes2002; Liu et al. Reference Liu, Tan and Xu2015) and corresponding force may be estimated as

(3.4) \begin{equation} F_{a} = \tau _{a} \, e \sim \frac {\rho _{a}c_a}{\sqrt {2\pi \gamma }}\left (U_{o}+\dot {h}\right ) e\,\,, \end{equation}

where $c_a$ is the speed of sound in air, $\rho _{a}$ is the density of air and $\gamma = 1.4$ is the adiabatic constant of air, which is considered an ideal gas.

Applying vertical momentum balance on the $\rm CV$ gives

(3.5) \begin{equation} \frac {\mathrm {d}}{\mathrm {d}t}\left (\rho V(t) \mathrm {v_{1}}\right ) - \rho L e \mathrm {v_{1}}^{2} = L \sum _{\rm CV} F_i\,, \end{equation}

where the film volume $V(t) = Leh$ and $\mathrm {v_{1}} = \dot {h}$ . Note that for clarity we have now included the spanwise width $L$ in the equation, and it should be remembered that $F_i$ denote forces per unit length. Substituting (3.2), (3.3), (3.4) together with the gravitational force $F_{g}=-\rho ehg$ and capillary force $F_{c}=\sigma \cos \theta _{D}$ with the dynamic contact angle $\theta _{D}$ into equation (3.5), it can be rewritten as

(3.6) \begin{equation} \frac {\mathrm {d}^{2}h}{\mathrm {d}{t}^{2}}=\frac {\sigma \cos \theta _{D}}{\rho e}\frac {1}{h}-\frac {\alpha }{e}\sqrt {\frac {\nu }{\pi t}}\left (\frac {\mathrm {d}h}{\mathrm {d}t}+U_{o}\right )-{\beta }\frac {\rho _{a}}{\rho }\frac {c_a}{\sqrt {2\pi \gamma }}\frac {1}{h}\left (\frac {\mathrm {d}h}{\mathrm {d}t}+U_{o}\right )-g. \end{equation}

Here we have introduced numerical constants $\alpha$ and $\beta$ in front of the viscous and air terms, respectively, since these forces are known only up to a multiplicative constant. These constants will be used as fitting parameters.

In figure 4(c) we compare the experimental results and the model of equation (3.6) for three typical, different combinations of $U_{o}$ and $D_{o}$ . Here, we approximate $e = e_{o}$ from experiments as $e_{o}=0.5\,\rm mm$ for $D_{o}=10\,\rm mm$ and $e_{o}=0.7\,\rm mm$ for $D_{o}=20\,\rm mm$ . The value of $\theta _{D} = 130^\circ$ is also obtained from the experiments as the time average from the experimental images. Furthermore, the initial condition for the velocity $\dot {h}(0)$ was also obtained from experimental data: $\dot {h}(0)=2.1\,\rm m\,s^-{^1}$ for $D_{o}=10\,\rm mm$ and $U_{o}=1.1\,\rm m\,s^-{^1}$ ; $\dot {h}(0)=3.3\,\rm m\,s^-{^1}$ for $D_{o}=10\,\rm mm$ and $U_{o}=2.1\,\rm m\,s^-{^1}$ ; and $\dot {h}(0)=2.5\,\rm m\,s^-{^1}$ for $D_{o}=20\,\rm mm$ and $U_{o}=1.1\,\rm m\,s^-{^1}$ . The fitting parameters $\alpha$ , $\beta$ are determined by the least-squares method, yielding $\alpha \approx 5$ and $\beta \approx 2.5$ , which were rounded for consistency across all three cases presented in figure 4(c).

The present model is able to predict the behaviour of the liquid film until it reaches its maximum height, but it starts to deviate strongly from the experimental result during the downward motion of the film. This might be due to constraints in the present model, such as the assumption that the film thickness $e$ is a constant, which is clearly violated during the downward motion of the film. Also, the descending dynamics of the film is expected to be qualitatively different from that of the rising motion; the interaction of the air with the liquid phase is relevant mostly at the first moments of the contact when the ejecta is emitted. After reaching its maximum height, the initial assumptions for our model, in particular regarding the effect of the air, no longer hold, and hence the model would need to be adapted, e.g. by including an equation for the time evolution of $e$ . Other factors could also have played a role, such as the fact that $\dot {h}(0)$ is likely to be underestimated due to measurement errors in the high-speed imaging footage.

In addition to a reasonable agreement with the experiment up to reaching the maximum film height, what the model can deliver is an estimate of the order of magnitude of each of the relevant forces: in figure 4(d,e), we plot the temporal evolution of (3.6), multiplied by the film height $h(t)$ for better comparison. It is clear that the viscous force provides the dominant term in the dynamics of the liquid film, whereas capillarity and gravity are negligible. Also, the air term is generally smaller, but may in some cases become non-negligible, e.g. in the case of figure 4(d). We may conclude that the initial dynamics of the liquid film is governed by the viscous dissipation in the boundary layer with the object and that this dissipation becomes both larger and more dominant for large impact velocity $U_{o}$ (figure 4 e). It is good to stress at this point that it is difficult to discriminate between the viscous and air terms in (3.6). If we compute the ratio of those two terms, we obtain (including the constants introduced in (3.6))

(3.7) \begin{equation} \frac {F_\nu }{F_a} = \frac {\alpha }{\beta }\frac {\rho }{\rho _a}\frac {h(t)}{e}\sqrt {\frac {2\nu \gamma }{c_a^2t}} \approx 0.6 \frac {\alpha }{\beta }\,, \end{equation}

where the time dependence is provided by $h(t)/\sqrt {t}$ . Since $h(t)$ is (at least initially) similar to $\sqrt {t}$ , the ratio of $F_\nu$ and $F_a$ can be approximated using $\rho = 998\,\rm kg\,m^-{^3}$ , $\rho _a = 1.2\,\rm kg\,m^-{^3}$ , $c_a = 340\,\rm m\,s^-{^1}$ , $\nu = 1.0 \times 10^{-6}\,\rm m{^2}\,s^-{^1}$ , $h/e \approx 10$ and $t \approx 5$ ms as $0.6\alpha /\beta$ . This corroborates that $F_\nu$ and $F_a$ are of the same order and determined by the numerical values found for $\alpha$ and $\beta$ .

Figure 5. (a) Maximum height $h_{max}$ reached by the liquid film plotted against the corresponding time $t_{ max}$ at which the maximum was reached for different values of the plate thickness $D_{o}$ , namely $5\,\rm mm$ (squares), $10\,\rm mm$ (circles) and $20\,\rm mm$ (diamonds). Each colour denotes the same $U_{o}$ for all $D_{o}$ (see legend). (b) Same data as in (a), but now $h_{ max}$ and $t_{ max}$ are normalised with the viscous length $l_{\nu }$ and time $\tau _{\nu }$ scales defined in (3.9) and (3.8), respectively.

To independently corroborate the dominance of viscous forces in the dynamics of the liquid film, we measured the maximum liquid film height $h_{ max}$ and the corresponding time $t_{ max}$ at which the maximum is reached for different $D_{o}$ and $U_{o}$ . We observe that the values of $h_{ max}$ and $t_{ max}$ tend to increase linearly with both $U_{o}$ and $D_{o}$ (not shown) and in addition, when plotting $h_{ max}$ against $t_{ max}$ , we also find a linear relation (figure 5 a). Inspired by the importance of viscosity for the motion of the liquid film that follows from the simplified model for the film height dynamics introduced above, we define relevant viscous time and length scales as follows:

(3.8) \begin{eqnarray} \tau _{\nu } &=& e_{o}^{2}/\nu , \end{eqnarray}

(3.9) \begin{eqnarray} l_{\nu } &=& U_{o} \tau _{\nu } = U_{o}e_{o}^{2}/\nu . \end{eqnarray}

As mentioned before, the initial thickness $e_{o}$ is observed to be $0.32$ , $0.5$ and $0.85\,\rm mm$ for $D_{o} = 5$ , $10$ and $20\,\rm mm$ , respectively. Since it is difficult to observe any differences of $e_{o}$ between small and large $U_{o}$ at early times in our experimental set-up, we use the same value of $e_{o}$ for each impact velocity $U_{o}$ . In figure 5(b) we normalise $h_{ max}$ and $t_{ max}$ with the viscous length and time scales $l_{\nu }$ and $\tau _{\nu }$ defined above, and find that the maximum film height data all collapse to $h_{ max} \approx 0.03 l_{\nu }$ , independent of $D_{o}$ and $U_{o}$ . Also $t_{ max}/\tau _{\nu }$ have similar values, although for large $U_{o}$ this value slightly increases, which may be due to the fact that at high $U_{o}$ , $e_{o}$ becomes much thinner than for small $U_{o}$ , which, however, we could not capture precisely within the spatial resolution of our set-up.

Finally, it is worth noting that the observation that the viscosity of the liquid film is the dominant parameter for the maximum film height and the corresponding time stands in sharp contrast to droplet bouncing, where the contact time between the droplet and the solid substrate is determined solely by surface tension (Richard et al. Reference Richard, Clanet and Quéré2002; Okumura et al. Reference Okumura, Chevy, Richard, Quéré and Clanet2003; Bird et al. Reference Bird, Dhiman, Kwon and Varanasi2013).