3. Phenomenology of the impact events

In figure 2, we compare the behaviour of a typical silicone oil drop ($R = 1.0\ {\rm mm}$, $V = 0.35\ {\rm m}\ {\rm s}^{-1}$ and $\eta _d = 4.6\ {\rm mPa}\ {\rm s}$, i.e. $(\mathit {We}_{d}, \mathit {Oh}_{d}, \mathit {Bo}_{d}) = (4, 0.034, 0.5)$) impacting on films with fixed viscosity $\eta _f = 96\ {\rm mPa}\ {\rm s}$ ($\mathit {Oh}_{f} = 0.67$) but contrasting thicknesses: $h_{f} = 0.01$, $0.35$ and $0.85\ {\rm mm}$ (i.e. $\varGamma = 0.01$, $0.35$ and $0.85$, respectively). We show a one-to-one comparison between experimental and DNS snapshots, and display three key pieces of information: the position of the liquid–air interfaces (green lines) that can be compared directly with experiments, the rate of viscous dissipation per unit volume (left-hand part of each numerical snapshot), and the magnitude of the velocity field (right-hand part of each numerical snapshot).

Figure 2.

Effect of the film thickness on the drop impact process: comparison of the experimental and DNS snapshots of the impact process on films with differing $h_f$ values (a) $0.01\ {\rm mm}$, (b) $0.35\ {\rm mm}$, and (c) $0.85\ {\rm mm}$. In each panel, the top row contains the experimental images with (green) interface outline from DNS, and the bottom row contains numerical snapshots showing the dimensionless rate of viscous dissipation per unit volume ($\tilde {\xi }_\eta = 2\,\mathit {Oh}\,(\tilde {\boldsymbol {\mathcal {D}}}:\tilde {\boldsymbol {\mathcal {D}}})$) on the left and the magnitude of the dimensionless velocity field ($\tilde {\boldsymbol {v}}$) on the right. We show $\tilde {\xi }_\eta$ on a $\log _{{10}}$ scale to identify regions of maximum dissipation (marked with black for $\tilde {\xi }_\eta \geqslant 10$). For all cases in this figure, $R = 1\ {\rm mm}$, $V = 0.3\ {\rm m}\ {\rm s}^{-1}$, $\eta _{d} = 4.6\ {\rm mPa}\ {\rm s}$ and $\eta _{f} = 96\ {\rm mPa}\ {\rm s}$, corresponding to $(\mathit {We}_{d}, \mathit {Oh}_{d}, \mathit {Oh}_{f}) = (4, 0.034, 0.67)$. Supplementary movies 1–3 are available at https://doi.org/10.1017/jfm.2023.13.

For the thinnest film ($h_f = 0.01\ {\rm mm}$, figure 2(a) and supplementary movie 1), the drop deforms as it comes into apparent contact with the film mediated by the air layer, an instant that we choose as the origin of time $t = 0$. The drop spreads until it reaches its maximal lateral extent, recoils, and rebounds in an elongated shape after a time $t_c = 15.6 \pm 0.1\ {\rm ms}$, called the contact time. Throughout the impact process, viscous stresses inside the drop dissipate energy (see times $t = 1.5$ and 7.5 ms). Consequently, after take-off, the drop reaches a maximal centre of mass height $H = 2.0 \pm 0.1\ {\rm mm}$ relative to the undisturbed film surface, from which we deduce the restitution coefficient defined as $\varepsilon = \sqrt {2g(H-R)}/V$; here, $\varepsilon = 0.48\pm 0.05$. The liquid–air interface profiles obtained from experiments and numerics are in excellent agreement, and we measure the same values of the contact time and restitution coefficient in simulations, using the method described in Appendix E. This behaviour is in quantitative agreement with that reported for the impact of a viscous drop on a superhydrophobic surface by Jha et al. (Reference Jha, Chantelot, Clanet and Quéré2020), suggesting that the presence of both the air and liquid film has a negligible influence on the macroscopic dynamics of the rebound, and that viscous dissipation in the drop determines the rebound height.

For $h_f = 0.35\ {\rm mm}$ (figure 2(b) and supplementary movie 2), despite the noticeable deformation of the liquid film, the qualitative features of the bounce are similar. We further observe that as the drop takes off, the film free surface has not yet recovered its undisturbed position. We measure an increase of the contact time to $t_c = 17 \pm 0.1\ {\rm ms}$, and a decrease in the rebound elasticity, with $H = 1.6 \pm 0.1\ {\rm mm}$ implying $\varepsilon = 0.37 \pm 0.04$. The DNS snapshots show that in this case, viscous dissipation occurs in both the drop and the underlying liquid. Qualitatively, the instantaneous rate of viscous dissipation in the drop is similar for $h_f = 0.01\ {\rm mm}$ and $h_f = 0.35\ {\rm mm}$, suggesting that the decrease in rebound elasticity is linked primarily to the increased film dissipation.

Finally, for $h_f = 0.85\ {\rm mm}$ (figure 2(c) and supplementary movie 3), the film deformation increases and the substrate loses its repellent ability. The drop centre of mass does not take off above $H=R$; the drop floats on top of the liquid film, a situation that corresponds to the inhibition of bouncing for which $\varepsilon \approx 0$, and the contact time diverges. In this case, we notice that the experimental and numerical interface profiles differ at $t= 0\ {\rm ms}$. This initial discrepancy, caused by drop oscillations upon detachment from the needle, does not affect the subsequent impact dynamics and the impact outcome, as evidenced by the good agreement of the interface profiles at later instants.

We now vary systematically the film thickness $h_{f}$ while keeping the drop and film viscosities constant ($\eta _d = 4.6\ {\rm mPa}\ {\rm s}$ and $\eta _f = 96\ {\rm mPa} {\rm s}$), and plot in figures 3(a) and 3(b) the contact time $t_c$ and the coefficient of restitution $\varepsilon$ extracted from experiments (circles) and DNS (hexagrams). Experiments and simulations are in excellent agreement when varying the film thickness by two orders of magnitude, $h_{f} = 0.01\unicode{x2013}1\ {\rm mm}$. The existence of two regimes is readily apparent.

Figure 3.

Effect of the film thickness on the rebound characteristics for $R = 1\ {\rm mm}$, $V = 0.3\ {\rm m}\ {\rm s}^{-1}$, $\eta _{d} = 4.6\ {\rm mPa}\ {\rm s}$ and $\eta _{f} = 96\ {\rm mPa}\ {\rm s}$, i.e. $(\mathit {We}_{d}, \mathit {Oh}_{d}, \mathit {Oh}_{f}) = (4, 0.034, 0.67)$: (a) contact time $t_{c}$, and (b) restitution coefficient $\varepsilon$, as a function of film thickness $h_{f}$. Circles and hexagrams represent experiments and DNS, respectively. In (a,b), the horizontal black dashed lines represent the substrate-independent limits of contact time and restitution coefficient, respectively, while the solid black lines show the results from the phenomenological model (see § 4) with parameters $c_{k} = 2$, $c_{d} = 5.6$ and $c_{f} = 0.46$. The vertical grey dashed line marks the transition from the bouncing to the non-bouncing (floating) regime. The inset of (b) illustrates the variation of the restitution coefficient normalized by its substrate-independent value $\varepsilon ^*$ as a function of the film thickness. Here, the horizontal grey line represents $\varepsilon = 0.9\varepsilon ^*$, marking the transition from substrate-independent to substrate-dependent bouncing at $h_f = h_{f, 1}$. (c) Schematic diagram of the phenomenological model that describes the drop impact process on a liquid film. The parameters $\rho _d R^{3}$, $\eta _{d}R$ and $\gamma$ are associated with the drop properties, and $\eta _{f} \varGamma ^{-3} R$ is associated with the film properties.

First, for $h_{f} \lesssim 0.1\ {\rm mm}$, both $t_{c}$ and $\varepsilon$ are independent of $h_f$. The value of the contact time in this regime, $t_c = 15.6 \pm 0.5\ {\rm ms}$, corresponds to that expected from the inertio-capillary scaling (Wachters & Westerling Reference Wachters and Westerling1966; Richard et al. Reference Richard, Clanet and Quéré2002). The contact time is proportional to $\tau _\gamma = \sqrt {\rho _d R^3/ \gamma }$, with a prefactor $2.2 \pm 0.1$, in good agreement with that calculated by Rayleigh (Reference Rayleigh1879) for the fundamental mode of drop oscillation ${\rm \pi} /\sqrt 2$. Similarly, the plateau value of the coefficient of restitution $\varepsilon = 0.47 \pm 0.04$ is in reasonable agreement with that reported for the impact of water drops on superhydrophobic substrates for a similar drop Ohnesorge number $\mathit {Oh}_{d}$ and impact Weber number $\mathit {We}_{d}$ (Jha et al. Reference Jha, Chantelot, Clanet and Quéré2020). We therefore refer to this regime as substrate-independent rebound (see also Appendix B).

Second, for $h_{f} \gtrsim 0.1\ {\rm mm}$, the contact time and coefficient of restitution are influenced by the film thickness. We observe that $t_c$ increases (figure 3a) and $\varepsilon$ decreases (figure 3b) with increasing $h_f$ until $t_c$ diverges and bouncing ceases ($\varepsilon = 0$) for $h_f \approx 0.75$ mm. This critical thickness marks the threshold of the rebound behaviour and the transition to the non-bouncing (floating) regime. Here, the rebound characteristics vary significantly with $h_f$ and we therefore refer to this regime as substrate-dependent.

Finally, we characterize the transition from the substrate-independent to the substrate-dependent regime by introducing the thickness $h_{f,1}$ (in dimensionless form $\varGamma _1 = h_{f,1}/R$) that marks the decrease of $\varepsilon$ to $0.9$ times its plateau value $\varepsilon ^*$. Similarly, we define the critical thickness $h_{f,2}$ (respectively, $\varGamma _2 = h_{f,2}/R$) associated with the transition from the substrate-dependent to the non-bouncing (floating) regime as the smallest film thickness that results in $\varepsilon = 0$. The impact dynamics can be categorized into three distinct regimes: a substrate-independent regime for $\varGamma = h_{f}/R \leqslant \varGamma _1$, a substrate-dependent regime for $\varGamma _1 < \varGamma < \varGamma _2$, and a non-bouncing (floating) regime for $\varGamma \geqslant \varGamma _2$.

4. Phenomenological model

We now seek to rationalize the dependence of the rebound time and elasticity with the substrate and drop properties by constructing a minimal model, guided by our experimental and numerical observations. We build on the classical description of a drop as a liquid spring that reflects the balance of inertia and capillarity during a rebound (Richard et al. Reference Richard, Clanet and Quéré2002; Okumura et al. Reference Okumura, Chevy, Richard, Quéré and Clanet2003). Here, we consider viscous drops and further add a damping term to the liquid spring, an approach that has been shown to capture successfully the variation of contact time and coefficient of restitution across over two orders of magnitude variation in liquid viscosities (Jha et al. Reference Jha, Chantelot, Clanet and Quéré2020). Similarly, we interpret the film behaviour through the liquid spring analogy. The film motion contrasts with that of the drop: while the latter displays a full cycle of oscillation during a rebound, the former never returns to its undisturbed position (see figure 2 and supplementary movies 1–3). This observation leads us to consider that the damping component dominates the behaviour of the liquid film, and to neglect the contributions of inertia and surface tension. We further discuss this assumption and its validity in § 6.

In figure 3(c), we present a sketch of the model, where we assume that the droplet and the film are connected in series during apparent contact, and show the scaling forms of the drop and film components. The scaling relations for the drop mass, stiffness and damping are taken from the work of Jha et al. (Reference Jha, Chantelot, Clanet and Quéré2020) as proportional to $\rho _d R^3$, $\gamma$ and $\eta _d R$, respectively, with corresponding prefactors $1$, $c_k$ and $c_d$. We determine the values of $c_k$ and $c_d$ from results in the substrate-independent bouncing regime (see Appendix B). The scaling form of the film damping term is chosen as proportional to $\eta _{f} \varGamma ^{-3} R$, where $\varGamma = h_f/R$, with corresponding prefactor of $c_{f}$ (figure 3c). This is built on two key assumptions. First, we assume that the viscous lubrication approximation holds in the film as, for sufficiently high film Ohnesorge numbers ($\mathit {Oh}_{f} \gtrsim 0.1$), the slopes associated with the film deformations are small ($\varGamma \ll 1$, $\mathit {Oh}_{f} \sim {O}(1)$; see § 6 for limitations). And second, we choose to consider the drop as an impacting disk rather than a sphere, owing to the rapid drop spreading upon impact (Eggers et al. Reference Eggers, Fontelos, Josserand and Zaleski2010; Wildeman et al. Reference Wildeman, Visser, Sun and Lohse2016), which results in a damping term proportional to $\varGamma ^{-3}$ instead of $\varGamma ^{-1}$ (Leal Reference Leal2007). Finally, we fit the prefactor $c_f$ to our experiments and simulations.

With these assumptions, the equations of motion for the model system (figure 3c) read

(4.1)\begin{gather} {\rho_d} R^{3} \ddot{y} ={-}c_{k} \gamma \left( y - x \right) - c_{d} \eta_{d} R \left( \dot{y} - \dot{x} \right), \end{gather}

(4.2)\begin{gather}0 ={+}c_{k} \gamma \left( y - x \right) + c_{d} \eta_{d} R \left( \dot{y} - \dot{x} \right) - c_{f} \eta_{f} \varGamma^{{-}3} R \dot{x}, \end{gather}

where $y$ and $x$ are the displacements of the drop and the film relative to their initial positions in the reference frame of the laboratory, and the dots denote time derivatives. We point out that by setting $\dot {x} = x = 0$, we recover the model proposed by Jha et al. (Reference Jha, Chantelot, Clanet and Quéré2020), which extends the analogy between the drop impact process and a spring–mass system (Okumura et al. Reference Okumura, Chevy, Richard, Quéré and Clanet2003) by adding a damper to account for viscous dissipation in the drop. Here, additionally, we consider viscous dissipation in the liquid coating and model the film as a damper without inertia. We make this modelling assumption, guided by the overdamped dynamics of the film (figure 2), to keep the number of free parameters to as few as possible (namely $c_k$, $c_d$ and $c_f$). We stress here that $c_k$ and $c_d$ are fixed in this study, and that their values are in quantitative agreement with the corresponding prefactors derived by Jha et al. (Reference Jha, Chantelot, Clanet and Quéré2020).

Similarly as for the governing equations in DNS, we make (4.1) and (4.2) dimensionless using the length scale $R$ and the time scale $\tau _{\gamma }$, and use tildes to identify dimensionless variables. Next, we obtain an equation of motion for the drop deformation $\tilde {z} = \tilde {y} - \tilde {x}$, namely

(4.3)\begin{equation} \left( 1 + \frac{c_{d}\,\mathit{Oh}_{d}}{c_{f}\,\mathit{Oh}_{f}\,\varGamma^{{-}3}} \right) \ddot{\tilde{z}} + c_{d}\, \mathit{Oh}_{d} \left( 1 + \frac{c_{k}}{c_{d}\,\mathit{Oh}_{d} \times c_{f}\,\mathit{Oh}_{f}\,\varGamma^{{-}3}} \right) \dot{\tilde{z}} + c_{k} \tilde{z} = 0, \end{equation}

which admits oscillatory solutions (i.e. drop rebound) under the condition

(4.4)\begin{equation} \omega^2 = 4 c_{k} - \left( c_{d}\,\mathit{Oh}_{d} - \dfrac{c_{k}}{c_{f}\,\mathit{Oh}_{f}\,\varGamma^{{-}3}} \right)^{2} > 0. \end{equation}

We note that $\omega ^2$ decreases with increasing $\varGamma$ for fixed $\mathit {Oh}_{d}$ and $\mathit {Oh}_{f}$, in qualitative agreement with the existence of a critical film height above which bouncing stops (figure 3b). Equation (4.4) allows us to determine the bounds of the bouncing regime in terms of a critical drop Ohnesorge number $\mathit {Oh}_{d,c}$ and film thickness $\varGamma _2$. Discarding the two roots of the equation $\omega ^2 = 0$ that yield unphysical negative values of $\mathit {Oh}_{d,c}$ and $\varGamma _2$, we obtain

(4.5)\begin{equation} \mathit{Oh}_{d,c} = \frac{1}{c_d}\left(2\sqrt{c_k} + \frac{c_{k}}{c_{f}}\left(\varGamma_2/\mathit{Oh}_{f}^{1/3}\right)^3\right) \end{equation}

and

(4.6)\begin{equation} \varGamma_2/\mathit{Oh}_{f}^{1/3} = \left(\frac{c_{f}}{c_{k}}\left( c_{d}\,\mathit{Oh}_{d} + 2 \sqrt{c_{k}} \right)\right)^{1/3}. \end{equation}

Equations (4.4)–(4.6) evidence that the role of the film viscosity and height are intertwined as we find the combination $\varGamma /\mathit {Oh}_{f}^{1/3}$ that can be inferred as the effective film thickness or mobility. Furthermore, the substrate-independent bouncing threshold is recovered when this film mobility, $\varGamma /\mathit {Oh}_{f}^{1/3}$, tends to 0, that is, for very thin and/or very viscous films. Indeed, (4.5) and (4.6) become

(4.7)\begin{equation} \mathit{Oh}_{d,c} = \frac{2\sqrt{c_k} }{c_d} \end{equation}

and

(4.8)\begin{equation} \varGamma_2/\mathit{Oh}_{f}^{1/3} = \left(2\,\frac{c_{f}}{\sqrt{c_{k}}}\right)^{1/3}, \end{equation}

for the limiting cases of substrate-independent ($\varGamma /\mathit {Oh}_{f}^{1/3} \to 0$), and inviscid drop ($\mathit {Oh}_{d} \to 0$) asymptotes, respectively.

To go further, we solve (4.3) with the initial conditions $\tilde {z} = 0$ and $\dot {\tilde {z}} = \sqrt {\mathit {We}_{d}}$ at $\tilde {t} = 0$, yielding

(4.9)\begin{equation} \tilde{z}(\tilde{t}) = \frac{2 \sqrt{\mathit{We}_{d}}} {\varOmega}\exp\left(-\frac{\mathcal{\phi} \tilde{t}}{2}\right) \sin\left(\frac{\varOmega\tilde{t}}{2}\right), \end{equation}

where

(4.10)\begin{equation} \mathcal{\phi} = \frac{c_k + c_d\,\mathit{Oh}_{d}\,c_f\,\mathit{Oh}_{f}\,\varGamma^{{-}3}}{c_d\,\mathit{Oh}_{d}+c_f\,\mathit{Oh}_{f}\,\varGamma^{{-}3}} \end{equation}

and

(4.11)\begin{equation} \varOmega = \omega\left(1+\frac{c_d\,\mathit{Oh}_{d}}{c_f\,\mathit{Oh}_{f}\,\varGamma^{{-}3}}\right)^{{-}1} \end{equation}

can be interpreted as an effective damper and angular frequency, respectively, by comparing the above expression (4.9) to the one obtained by Jha et al. (Reference Jha, Chantelot, Clanet and Quéré2020) for $\varGamma /\mathit {Oh}_{f}^{1/3} \to 0$. We can deduce the expressions for both the contact time and the coefficient of restitution using these pieces of information. The contact time is taken as the instant at which the drop deformation $\tilde {z}$ comes back to zero, which occurs at $\varOmega \tilde {t} = 2{\rm \pi}$, giving

(4.12)\begin{equation} \frac{t_{c}}{\tau_{\gamma}} = \dfrac{2 {\rm \pi}}{\omega} \left( \dfrac{c_{d}\,\mathit{Oh}_{d}}{c_{f}\, \mathit{Oh}_{f}\,\varGamma^{{-}3}} + 1 \right). \end{equation}

Equation (4.12) is then used to compute the coefficient of restitution $\varepsilon$ as the ratio of the rebound velocity $\dot {\tilde {z}}(\tilde {t}_c)$ to the impact velocity $\sqrt {\mathit {We}_{d}}$. We notice immediately that this definition yields an expression for $\varepsilon$ that does not depend on $\mathit {We}_{d}$, in contrast with the experimentally observed decrease of $\varepsilon$ with $\mathit {We}_{d}$. We account for the Weber number dependence of $\varepsilon$, which is not captured by spring–mass models (Jha et al. Reference Jha, Chantelot, Clanet and Quéré2020), by scaling the coefficient of restitution by $\varepsilon _0(\mathit {We}_{d})$, its $\mathit {We}_{d}$-dependent value in the substrate-independent limit for inviscid drops:

(4.13)\begin{equation} \varepsilon(\mathit{We}_{d}, \mathit{Oh}_{d}, \mathit{Oh}_{f}, \varGamma) = \varepsilon_0(\mathit{We}_{d}) \exp\left( -\dfrac{\rm \pi}{\omega} \left( c_{d}\,\mathit{Oh}_{d} + \dfrac{c_{k}}{c_{f}\,\mathit{Oh}_{f}\,\varGamma^{{-}3}} \right) \right), \end{equation}

where the prefactor $\varepsilon _0(\mathit {We}_{d})$ is not a model prediction. We obtain the other prefactors, $c_k$ and $c_d$, by fitting the substrate-independent experiments following Jha et al. (Reference Jha, Chantelot, Clanet and Quéré2020). This simplification allows us to recover the expressions for $t_c$ and $\varepsilon$ for viscous drop impact on non-wetting substrates (Jha et al. Reference Jha, Chantelot, Clanet and Quéré2020), and thus to determine $c_k = 2$ and $c_d = 5.6$, which we keep fixed during this study. For details of this simplification and on the determination of the prefactors, see Appendix B.

We test the model predictions for the contact time and rebound elasticity in the substrate-dependent regime by comparing the data (symbols) presented in figures 3(a) and 3(b) to least squares fits of (4.12) and (4.13), with $c_f$ as a free parameter (solid lines), and taking $\varepsilon _0 = 0.58$ (see Appendix B). We find that the model predicts accurately the variation of $t_c$ and $\varepsilon$ with $\varGamma$ for $c_f = 0.46 \pm 0.1$. For the rest of this work, we fix $c_f = 0.46$ and assess the predictive ability of the simplified model.

5. Influence of drop and film parameters

We now test the model predictions and limits by varying experimentally and numerically the drop and film Ohnesorge numbers. We give particular attention to the value of the coefficient $c_f$ (fixed at $0.46$) necessary to fit the model to these data, and to the two asymptotes predicted by the model that bound the bouncing domain (4.7)–(4.8).

5.1. Influence of the film Ohnesorge number $\mathit {Oh}_{f}$

We first vary the film Ohnesorge number $\mathit {Oh}_{f}$ while keeping the drop and impact properties constant. In figure 4(a), we show the evolution of the coefficient of restitution $\varepsilon$ for drops with $\mathit {Oh}_{d} = 0.034$ as a function of the dimensionless film thickness $\varGamma$, while exploring two decades in film viscosity, $\mathit {Oh}_{f} = 0.01 \unicode{x2013} 2.0$. On the one hand, as expected, the values of the coefficient of restitution are not affected in the substrate-independent limit. On the other hand, the substrate-dependent behaviour shows the influence of $\mathit {Oh}_{f}$, and we identify two regimes. For $\mathit {Oh}_{f} < 0.1$, the evolution of $\varepsilon$ with $\varGamma$ does not depend on $\mathit {Oh}_{f}$, as illustrated by the data collapse in figure 4(a). However, for $\mathit {Oh}_{f} > 0.1$, increasing the film viscosity leads to a larger extent of the substrate-independent plateau and to an increase of the critical film thickness at which bouncing stops. This change in the $\mathit {Oh}_{f}$ dependence can be characterized by the two dimensionless critical film thicknesses $\varGamma _1 = h_{f,1}/R$ and $\varGamma _2 = h_{f,2}/R$, which increase from $0.17$ to $0.33$, and from $0.58$ to $1.1$, respectively, when $\mathit {Oh}_{f}$ is increased from $0.1$ to $2.0$.

Figure 4.

Influence of the film parameters on the impact characteristics: variation of the coefficient of restitution $\varepsilon$ as a function of (a) the film thickness $\varGamma$, and (b) the film mobility $\varGamma / \mathit {Oh}_{f}^{1/3}$. In (a,b), the circles and hexagrams correspond to the results from experiments and simulations, respectively. The coloured dashed lines in (a) and the solid black line in (b) illustrate the results from the phenomenological model (4.13) with parameters $c_{k} = 2$, $c_{d} = 5.6$ and $c_{f} = 0.46$. Black dashed lines in (a,b) mark the substrate-independent limit of the restitution coefficient $\varepsilon ^*$. For all cases in this figure, $\mathit {Oh}_{d} = 0.034$ and $\mathit {We}_{d} = 4$.

We interpret the two types of behaviour in the substrate-dependent regime in the light of our minimal model, which predicts that the film mobility $\varGamma /\mathit {Oh}_{f}^{1/3}$ controls the dissipation in the substrate. In figure 4(b), we plot the coefficient of restitution data presented in figure 4(a) after rescaling the horizontal axis by $\mathit {Oh}_{f}^{-1/3}$. The data now collapse for $\mathit {Oh}_{f} > 0.1$, indicating that the proposed approximations capture the large viscosity limit but break down for lower film Ohnesorge numbers. We further evidence the validity and failure of the minimal model by plotting the predictions of (4.13) with $c_f = 0.46$ (dashed coloured lines in figure 4(a) and solid black line in figure 4(b)). The minimal model predicts the restitution coefficient accurately for $\mathit {Oh}_{f} > 0.1$, suggesting that our modelling assumptions are valid in this regime: the liquid film dynamics is dominated by viscous dissipation, and the flow can be modelled successfully in the lubrication approximation by assimilating the impacting drop to a cylinder.

5.2. Influence of the drop Ohnesorge number $\mathit {Oh}_{d}$

In this subsection, we focus on the influence of the drop Ohnesorge number on the rebound elasticity. In figure 5(a), we plot the coefficient of restitution as a function of the dimensionless film thickness for a fixed $\mathit {Oh}_{f} = 0.667$ and for varying $\mathit {Oh}_{d}$ spanning the range 0.01–0.133. Increasing $\mathit {Oh}_{d}$ affects $\varepsilon$ across all film thicknesses. In the substrate-independent region, the coefficient of restitution decreases with increasing drop Ohnesorge number. In Appendix B, we show that the plateau values reported in figure 5(a) decay exponentially with increasing $\mathit {Oh}_{d}$ as predicted by Jha et al. (Reference Jha, Chantelot, Clanet and Quéré2020). To better illustrate the influence of $\mathit {Oh}_{d}$ in the substrate-dependent regime, we normalize the coefficient of restitution $\varepsilon$ by its substrate-independent value $\varepsilon ^*$ (figure 5b). With this normalization, we expect the data to follow the prediction of (4.13) (solid line). The data collapse only for small $\varGamma$, indicating that the phenomenological model predicts the influence of $\mathit {Oh}_{d}$ only in the substrate-independent limit. This suggests that the model fails to account for the interplay between the drop and the film properties that affects dissipation in both liquids, and ultimately the coefficient of restitution. Here as well, we monitor the $\mathit {Oh}_{d}$ dependence of the coefficient of restitution, and its deviation from the prediction of (4.13), through the evolution of $\varGamma _1$ and $\varGamma _2$, that both decrease with increasing Ohnesorge number.

Figure 5.

Influence of the drop parameters on the rebound elasticity: variation of (a) the coefficient of restitution $\varepsilon$, and (b) the coefficient of restitution normalized with its substrate-independent value $\varepsilon /\varepsilon ^*$, as a function of the normalized film thickness $\varGamma$. The circles and hexagrams correspond to the results from the experiments and simulations, respectively. In (a), the dashed lines denote the plateau values of the restitution coefficient $\varepsilon ^*(\mathit {Oh}_{d})$. In (a,b), the solid lines represent the results from the phenomenological model (4.13) with parameters $c_{k} = 2$, $c_{d} = 5.6$ and $c_{f} = 0.46$. For all cases in this figure, $\mathit {Oh}_{f} = 0.667$ and $\mathit {We}_{d} = 4$.

5.3. Influence of $\mathit {Oh}_{f}$ and $\mathit {Oh}_{d}$ on the critical film thicknesses

We now quantify the influence of the drop and film Ohnesorge numbers by reporting their effects on the critical thicknesses for substrate-independent to substrate-dependent ($\varGamma _1$) and bouncing to non-bouncing (floating) ($\varGamma _2$) transitions. Indeed, we have shown above that these two critical thicknesses are good proxies to characterize the continuous transition from substrate-independent bouncing to rebound inhibition. In figures 6(a,b), we show $\varGamma _1$ and $\varGamma _2$ as functions of the film Ohnesorge number for $\mathit {Oh}_{d}$ in the range 0.01–0.133. This representation reflects the existence of the two distinct regimes reported in figure 4.

Figure 6.

Critical film thickness marking the transition from (a) substrate-independent to substrate-dependent bouncing $\varGamma _1$, and (b) bouncing to non-bouncing (floating) $\varGamma _2$ as a function of $\mathit {Oh}_{f}$ at different $\mathit {Oh}_{d}$. Prefactors (c) $\alpha _{1}$ and $\beta _{1}$, and (d) $\alpha _{2}$ and $\beta _{2}$, as a function of $\mathit {Oh}_{d}$. The solid black line in (c) represents the model prediction for $\beta _1$, (5.1). The solid black lines in (d) represent the model predictions for $\beta _2$ using (4.5) and (4.6), and the black dashed lines show the two asymptotes, (4.7) and (4.8).

First, when $\mathit {Oh}_{f} < 0.1$, $\varGamma _1$ and $\varGamma _2$ are independent of $\mathit {Oh}_{f}$. We write $\varGamma _1 = \alpha _1(\mathit {Oh}_{d})$ and $\varGamma _2 = \alpha _2(\mathit {Oh}_{d})$, and report the values of $\alpha _1(\mathit {Oh}_{d})$ and $\alpha _2(\mathit {Oh}_{d})$ in figures 6(c,d). This observation is in contradiction with the expectations from our minimal model, which predicts that $\mathit {Oh}_{f}$ influences the values of $\varGamma _1$ and $\varGamma _2$. Surprisingly, this $\mathit {Oh}_{f}$ independence of the critical thicknesses and the collapse observed in figure 4(a) suggest that the energy transfer to the film (in the form of kinetic and surface energies) and the film viscous dissipation are independent of film viscosity for $\mathit {Oh}_{f} < 0.1$. We will elaborate further on this regime in § 6.

Second, for larger film Ohnesorge numbers, the dissipation in the film is captured by the lubrication approximation ansatz. As a result, both critical thicknesses follow the relations $\varGamma _1 = \beta _1(\mathit {Oh}_{d})\mathit {Oh}_{f}^{1/3}$ and $\varGamma _2 = \beta _2(\mathit {Oh}_{d})\mathit {Oh}_{f}^{1/3}$, as predicted by the model. Beyond this scaling relation, the accuracy of the minimal model is tied to its ability to predict the prefactors $\beta _1$ and $\beta _2$ when $\mathit {Oh}_{f} \gtrsim 0.1$. In figures 6(c,d), we plot $\beta _1$ and $\beta _2$ as functions of the drop Ohnesorge number. Both prefactors show a plateau for $\mathit {Oh}_{d} \lesssim 0.03$ before decreasing monotonically with the drop Ohnesorge number. We compare the measured prefactors to the model predictions, which we plot as solid lines in figures 6(c,d). Here, $\beta _1$ is obtained by solving $\varepsilon = 0.9\varepsilon ^*$, yielding

(5.1)\begin{equation} \beta_{1} = c_{f}^{1/3} \left[ \dfrac{-c_{d}\,\mathit{Oh}_{d}\,(1 - r^{2}) + 2 r \sqrt{c_{k} (1 + r^{2}) - c_d^{2}\,\mathit{Oh}_{d}^{2}}}{c_{k} (1 + r^{2})} \right]^{1/3}, \end{equation}

where

(5.2)\begin{equation} r = \dfrac{c_{d}\,\mathit{Oh}_{d}}{\sqrt{4 c_{k} - c_{d}^{2}\,\mathit{Oh}_{d}^{2}}} - \dfrac{\ln(0.9)}{\rm \pi}, \end{equation}

and $\beta _2$ is given by (4.6). The model fails to capture the decrease of both $\beta _1$ and $\beta _2$ with $\mathit {Oh}_{d}$. Yet we can interpret the evolution of these two prefactors along the inviscid and viscous drop limiting cases. Indeed, for inviscid drops (i.e. small $\mathit {Oh}_{d}$), the model predictions for $\beta _1$ and $\beta _2$ show a plateau whose value is in good agreement with that reported in experiments. Conversely, for viscous drops (i.e. large $\mathit {Oh}_{d}$), $\beta _2$ decreases with $\mathit {Oh}_{d}$ to match the asymptote associated with the substrate-independent bouncing inhibition occurring at $\mathit {Oh}_{d,c} \approx 0.5$ (see (4.5) and dashed line in figure 6d).

We stress that the model predictions shown in figures 6(c,d) consider a unique value $c_f = 0.46 \pm 0.1$, determined from a least squares fit in § 4. We attribute the failure of the model to predict the dependence on $\mathit {Oh}_{d}$ away from the two asymptotes to its simplified representation of the drop–film interactions. While remarkably, these oscillator-based models predict the global outcome of a rebound – that is, for example, the contact time, coefficient of restitution and bounds of bouncing – they fail to represent accurately the interactions, such as the drop or film deformations (4.9), and their dynamics. For example, the force associated with drop impact is maximal at early times, when the drop shape is spherical, while the force exerted by a spring is proportional to deformation.

Our phenomenological model captures successfully the behaviour of the liquid film for $\mathit {Oh}_{f} \geqslant 0.1$, giving support to the modelling assumptions. Within this regime, the model allows for a quantitative prediction of the contact time and coefficient of restitution for both inviscid (i.e. low $\mathit {Oh}_{d}$) and viscous (i.e. large $\mathit {Oh}_{d}$) drops, for a fixed set of constants $c_k$, $c_d$ and $c_f$. In between these two limits, the model fails to predict the $\mathit {Oh}_{d}$ dependence, a fact that we attribute to the simplicity of the representation of drop–film interactions. More intriguingly, the minimal model also breaks down for $\mathit {Oh}_{f} \lesssim 0.1$, where we observe that the coefficient of restitution does not depend on the film Ohnesorge number. We will demystify this behaviour in the next section.

6. Bouncing inhibition on low Ohnesorge number films

We now investigate the independence of the rebound elasticity with the film Ohnesorge number, illustrated by the data collapse of figure 4(a), for $\mathit {Oh}_{f} < 0.1$. Figure 7(a) shows two typical impact scenarios in this regime, with $\mathit {Oh}_{f} = 0.01$ (figure 7a-i) and $\mathit {Oh}_{f} = 0.1$ (figure 7a-ii), where bouncing is inhibited by the presence of the liquid film. Although these two representative cases differ by an order of magnitude in $\mathit {Oh}_{f}$, qualitatively, the drop shape and flow anatomy remain similar (figure 7(a), $t/\tau _\gamma = 0.2, 1$), suggesting an equal loading on the film. Nonetheless, the film response varies. We observe surface waves post-impact on the film–air interface for $\mathit {Oh}_{f} = 0.01$, which vanish for $\mathit {Oh}_{f} = 0.1$ owing to increased bulk viscous attenuation (figure 7(a), $t/\tau _\gamma = 0.2, 2.65$).

Figure 7.

$\mathit {Oh}_{f}$ independent inhibition of bouncing. (a) Typical drop impact dynamics on low-viscosity films. The snapshots show the dimensionless rate of viscous dissipation per unit volume on the left and the magnitude of dimensionless velocity field on the right. (b) Energy budgets for the two representative cases shown in (a), normalized by the available energy at the instant of impact. Here, the subscripts $g$, $k$, $\gamma$ and $\eta$ denote gravitational potential, kinetic, surface and viscous dissipation energies, respectively. The superscripts $d$, $f$ and $a$ represent drop, film and air, respectively. The grey dashed line in each plot marks the instant when the normal reaction force between the drop and the film is minimum, and represents the last time instant when the drop could have bounced off the film. For all cases, $(\mathit {We}_{d}, \mathit {Oh}_{d}, \varGamma ) = (4, 0.034, 1)$.

To further elucidate the drop–film interaction, we compute the energy budgets associated with the two representative cases with $\mathit {Oh}_{f} = 0.01$ (figure 7b-i) and $\mathit {Oh}_{f} = 0.1$ (figure 7b-ii). The overall energy budget reads

(6.1)\begin{equation} E_0 = \left(E_k^d + \Delta E_\gamma^d + \Delta E_g^d\right) + E_\eta^d + \left(E_k^f + \Delta E_\gamma^f + \Delta E_g^f\right) + E_\eta^f + E_t^a, \end{equation}

where $E_0$ is the energy at impact (i.e. the sum of the drop's kinetic and gravitational potential energies). The subscripts $g$, $k$, $\gamma$ and $\eta$ denote gravitational potential, kinetic, surface and viscous dissipation energies, respectively. Moreover, the superscripts $d$, $f$ and $a$ represent drop, film and air, respectively. Finally, reference values to calculate $\Delta E_g$ and $\Delta E_\gamma$ are at minimum $E_g$ and $E_\gamma$ at $t = 0$, respectively. Note that the contribution of the total energy associated with air ($E_t^a = E_k^a + E_\eta ^a$) is negligible compared to other energies ($E_t^a(t/\tau _\gamma = 4) \approx 0.01E_0$). Readers are referred to Landau & Lifshitz (Reference Landau and Lifshitz1987), Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016), Ramírez-Soto et al. (Reference Ramírez-Soto, Sanjay, Lohse, Pham and Vollmer2020) and Sanjay et al. (Reference Sanjay, Sen, Kant and Lohse2022) for details of energy budget calculations.

In both cases highlighted in figure 7, the magnitudes of the drop energy (the sum of the drop's kinetic, gravitational potential and surface energies) at the end of the rebound cycle – that is, for $t = 3.25\tau _\gamma$ when $\mathit {Oh}_{f} = 0.01$, and $t = 2.65\tau _\gamma$ when $\mathit {Oh}_{f} = 0.1$ (vertical grey dashed lines in figure 7) – are similar, as expected from the independence of $\varepsilon$ with $\mathit {Oh}_{f}$. Note that the end of the cycle has been determined from the instant at which the reaction force between the drop and the film is minimum (see Appendix E and Zhang et al. Reference Zhang, Sanjay, Shi, Zhao, Lv and Lohse2022). Moreover, the energy budget evidences that the viscous dissipation in the drop during the rebound is similar, indicating that the magnitude of the energy transferred from the impacting drop to the film (the sum of the film's kinetic, gravitational potential and surface energies, and viscous dissipation) is not affected by the one order of magnitude change in $\mathit {Oh}_{f}$. Yet the distribution of the film energy is dramatically different in the two cases that we consider. For $\mathit {Oh}_{f}$ = 0.1, the energy transferred to the film is mostly lost to viscous dissipation, while for $\mathit {Oh}_{f} = 0.01$, the energy stored in the film's kinetic, surface and potential components dominates. We stress here that the $\mathit {Oh}_{f}$-independent behaviour does not imply that dissipation is negligible. Indeed, the viscous dissipation in the film accounts for approximately $40\,\%$ and $85\,\%$ of the total energy transferred to the film for $\mathit {Oh}_{f} = 0.01$ and $0.1$, respectively. This difference in the film energy distribution hints at the failure of our assumptions to neglect the film's inertia and surface tension. The minimal model is relevant only when the energy transferred to the liquid film is lost predominantly to viscous dissipation.

Guided by the energy budget analysis in the above two extreme cases, we now evidence the minimal model breakdown as the film mobility $\varGamma /\mathit {Oh}_{f}^{1/3}$ fails to describe the film deflection $\delta _f$. In figure 8, we report the normalized maximum film deflection $\tilde {\delta }_f = \delta _f/R$ as a function of $\varGamma$ (figure 8a) for $\mathit {Oh}_{f}$ in the range $0.01$–$2$ while keeping $\mathit {Oh}_{d}$ constant. For $\mathit {Oh}_{f} > 0.1$, the deflection decreases with increasing $\mathit {Oh}_{f}$, and the data collapses once the horizontal axis is rescaled by $\mathit {Oh}_{f}^{-1/3}$ (figure 8b), confirming the relevance of the film mobility. However, for lower film Ohnesorge number $\tilde {\delta }_f$ scales with $\varGamma$ independent of $\mathit {Oh}_{f}$, illustrating the limits of our hypotheses. Here, one might be tempted to replace empirically the effective control parameter in our model $\propto \mathit {Oh}_{f}\,\varGamma ^{-3}$ with $\propto \varGamma$, in light of the $\tilde {\delta }_f$ collapse with ${\sim }\varGamma$ in the low $\mathit {Oh}_{f}$ regime. However, such a replacement still fails to account appropriately for the kinetic and surface energies of the film. Indeed, low $\mathit {Oh}_{f}$ films are associated with surface waves, and the maximum deflection $\delta _f$ might not be the correct length scale to mimic their behaviour in a simplified model. As future work, it would be interesting to couple a linearized quasi-potential fluid model (Lee & Kim Reference Lee and Kim2008; Galeano-Rios, Milewski & Vanden-Broeck Reference Galeano-Rios, Milewski and Vanden-Broeck2017; Galeano-Rios et al. Reference Galeano-Rios, Cimpeanu, Bauman, MacEwen, Milewski and Harris2021) for the liquid pool/film with a spring–mass–damper system for the liquid drop to investigate this regime further.

Figure 8.

Dimensionless film deflection $\tilde {\delta }_f = \delta _f/R$ measured from the initial film free surface (see the inset in a) as a function of (a) the film thickness $\varGamma$, and (b) the film mobility $\varGamma / \mathit {Oh}_{f}^{1/3}$ in the DNS. The solid black lines represent $\delta _f/R = \varGamma$ and $\delta _f/R = \varGamma /\mathit {Oh}_{f}^{1/3}$ in (a,b), respectively. For all cases in this figure, $\mathit {Oh}_{d} = 0.034$ and $\mathit {We}_{d} = 4$.