3.1. Pre-deformation

As the drop approaches the pool, the two liquid bodies eventually get close enough that they ‘feel’ the presence of each other through pressure build-up inside the intervening gas layer. A schematic of this stage is shown in figure 4(b). In the pre-deformation stage, the pressure peaks in the centre of the film. From the scaling arguments above, it is clear that this stage commences when $H\sim R{St}^{-1/2}$. The gas pressure at this stage is not large enough to deform any of the surfaces. As such, in this stage, the problem is similar to the problem of a solid cylinder approaching a wall.

3.2. Cusp formation

While the drop approaches the pool in the pre-deformation stage, the pressure in the centre of the gas film grows as $H$ decreases, according to (3.2). At some stage, the gas pressure becomes large enough to deform the drop. The peak pressure in the film centre eventually deforms the drop and pool locally until two cusps form. These cusps are the regions where the thickness of the air film is minimum while gas pressure and interface curvature are maximum. This stage is shown schematically in figure 4(c).

The requirement for deformation is $\partial {\boldsymbol {u}}_{l}/\partial t\sim \boldsymbol {\nabla }{P}_{l}/{\rho }_{l}\sim \boldsymbol {\nabla }{P}_{g}/{\rho }_{l}$. The relevant time scale here is $T\sim H/U$, and from (3.2) we arrive at the scaling relations for this stage given by

(3.4)\begin{gather} H\sim R{St}^{-2/3}, \end{gather}

(3.5)\begin{gather}L\sim R{St}^{-1/3}, \end{gather}

(3.6)\begin{gather}{P}_{g}\sim {\rho}_{l}{U}^{2}{St}^{1/3}. \end{gather}

Note that these scaling laws were also previously identified by Hicks & Purvis (Reference Hicks and Purvis2011) as well as Hendrix et al. (Reference Hendrix, Bouwhuis, Vandermeer, Lohse and Snoeijer2016). Now, using (3.4) to (3.6), we can non-dimensionalize and rescale the physical variables, keeping in mind that as opposed to the gas, the potential flow in the liquid results in the same length scales in the $x$ and $y$ directions. Thus, from ${x}_{l}^{*}=x/(R{St}^{-1/3})$, ${y}_{l}^{*}=y/(R{St}^{-1/3})$, ${\boldsymbol {u}_{l}}^{*}={\boldsymbol {u}_{l}}/U$, ${P}_{g,l}^{*}={P}_{g,l}/({\rho }_{l}{U}^{2}{St}^{1/3})$, $h^{*}=h/(R{St}^{-2/3})$ and ${t}^{*}=tU/(R{St}^{-2/3})$, we obtain the non-dimensional form of the BIM equations that were introduced in § 2,

(3.7)\begin{gather} \frac { \partial {\boldsymbol{ u }_{ l } } }{ \partial t } =-\boldsymbol{\nabla} { P }_{ l }-{ St }^{ -1/3 } \boldsymbol{\nabla}{ \left| { {\boldsymbol{ u }_{ l } } } \right| }^{ 2 }, \end{gather}

(3.8)\begin{gather}\frac { \partial }{ \partial { x }_{ t } } ({ h }^{ 3 }\frac { \partial { P }_{ g } }{ \partial { x }_{ t } } )=6{St}^{-1/3}\frac { \partial }{ \partial { x }_{ t } } (({ u }_{ t }^{ + }+{ u }_{ t }^{ - })h)+12({ u }_{ n }^{ + }-{ u }_{ n }^{ - }), \end{gather}

(3.9)\begin{gather}{P}_{g}\pm{We}^{-1}{St}^{-1/3}\frac { {\partial}^{2} {h}^{\pm}}{\partial {x}^{2} }={P}_{l}, \end{gather}

(3.10)\begin{gather}\frac { \partial {h}^{\pm} }{ \partial t } ={v}_{l}^{\pm}-{ St }^{ -1/3 }{ u }_{ l }^{\pm}\frac { \partial {h}^{\pm} }{ \partial x }, \end{gather}

where the ‘$*$’ symbols have been dropped for easier reading. Considering that we are interested in the inertial impact regime (Bouwhuis et al. Reference Bouwhuis, van der Veen, Roeland, Tran, Keij, Winkels, Peters, van der Meer, Sun, Snoeijer and Lohse2012) in which $St\sim {O}({10}^{5})$ and $We\sim {O}(10)$, the leading order equations are given by

(3.11)\begin{gather} \frac { \partial {\boldsymbol{ u }_{ l } } }{ \partial t } =-\boldsymbol{\nabla} { P }_{ l }, \end{gather}

(3.12)\begin{gather}\frac { \partial }{ \partial { x }_{ t } } \left({ h }^{ 3 }\frac { \partial { P }_{ g } }{ \partial { x }_{ t } } \right)=12{ \mu }_{ g }({ u }_{ n }^{ + }-{ u }_{ n }^{ - }), \end{gather}

(3.13)\begin{gather}{P}_{g}={P}_{l}, \end{gather}

(3.14)\begin{gather}\frac{ \partial {h}^{\pm} }{ \partial t } ={v}_{l}^{\pm}. \end{gather}

These leading order equations have been solved in the work of Hicks & Purvis (Reference Hicks and Purvis2011) for drop impact onto pools of various depths. They observe the formation of disk-type films at contact time. In this work we solve (2.1)–(2.4) along with (2.5). In other words, we retain the linear part of the momentum flux, surface tension forces, the gas mass flux due to Couette flow and the contribution of the tangential velocity in the kinematic boundary condition. Similar to drop-solid simulations of Mandre et al. (Reference Mandre, Mani and Brenner2012) and Duchemin & Josserand (Reference Duchemin and Josserand2011), this decision was made with the intention of capturing all the physical mechanisms possible within our numerical BIM framework. Moreover, converse to Hendrix et al. (Reference Hendrix, Bouwhuis, Vandermeer, Lohse and Snoeijer2016), we allow for the simulations to continue when the distance of the liquid bodies reach ${O}$(100 nm). Namely, we only stop the simulations and declare contact when the distance of the liquid bodies reaches 10 nm. We do not explicitly account for Van der Waals forces in our numerical scheme, but check for contact due to molecular forces in the post-processing stage by defining a film thickness at which these forces cause contact. As we will see, the combination of these two choices – including the non-leading order terms and not declaring contact when film thickness becomes ${O}$(100 nm) – allow us to numerically observe a capillary transition that prevents early contact for the first time. In the following, it is useful to first examine the results from BIM simulations with zero surface tension ($\sigma =0$).

3.2.1. BIM with no surface tension

Using the BIM method introduced in § 2, but with no surface tension, the dynamics are primarily governed by the leading order equations ((3.11) to (3.14)). In figure 5 one can observe how the profiles of the water drop and pool with $\sigma =0$ evolve in time for $R=1.7$ mm and $U=1$ m s$^{-1}$. In this work, $t=0$ is defined as the time that impact would happen in a vacuum. Similar to the simulations of Hicks & Purvis (Reference Hicks and Purvis2011), which solved the leading order equations ((3.11) to (3.14)), we see the entrapment of a disk-type film. Moreover, in figure 6(a), the disk-type film entrapped at contact time is shown for twelve different simulations performed with $(U,R)\in \{0.9,\ 1.1\ \text {m}\ \text {s}^{-1}\}\times \{0.25, 0.55, 0.85, 1.15, 1.45, 1.75\ \text {mm}\}$ pairs. We observe that from our simulations, contact time happens at $tU/(R{St}^{-2/3})\approx 10.4$ for all of these impacts, which is close to the value of $11$ predicted by Hicks & Purvis (Reference Hicks and Purvis2011). The thickness of these entrapped films as a function of $x$ is plotted in figure 6(b). It is then clear from figure 6(c) that if one plots the film thickness with the scaled coordinates $x/(R{St}^{-1/3})$ and $y/(R{St}^{-2/3})$, the profiles collapse. The film thickness and lateral extent predicted by these simulations are also in quantitative agreement with the solutions of the leading order equations by Hicks & Purvis (Reference Hicks and Purvis2011).

Figure 5. Evolution of the water–air interfaces for a drop impacting a deep pool obtained from BIM with no surface tension ($\sigma =0$). Drop diameter is $D=3.4\ \text {mm}$ and impact velocity is $U=1$ m s$^{-1}$. The final snapshot is at $t=8.51\ \mathrm {\mu }$s and ${\rm \Delta} t=8\ \mathrm {\mu }$s.

Figure 6. Boundary integral method with no surface tension ($\sigma =0$) results at contact time, which is $tU/(R{St}^{-2/3})\approx 10.4$, for $R\in \{0.25, 0.55, 0.85, 1.15, 1.45, 1.75\ \text {mm}\}$ plotted in {dark blue, light blue, cyan, green, yellow, orange} with $U\in \{0.9, 1.1\ \text {m}\ \text {s}^{-1}\}$, showing (a) disk-type films (b) disk-type film thicknesses (c) scaled disk-type film thicknesses.

The evolution of the air film thickness can provide quantitative insight into the dynamics, especially given the connection it provides to drop-solid impact simulations. In figure 7 the evolution of the film thickness for a simulation with no surface tension force is shown (for one side of the system due to symmetry around $x=0$). The formation of cusps is clear in this figure. As the minimum thickness of the film, which is at the cusp, becomes smaller, the length scale of the cusp also decreases. The pressure peaks at the cusp location and grows in time. It is worth noting that although both surfaces deform between the cusps, the film does not drain past the pressure maxima at the cusps. This is clear from the almost constant-in-time film thickness between the cusp location and $x=0$ in figure 7. Similar to drop-solid impacts, in § 3.2.2 we show for the first time that during this stage of the film evolution the solutions around the cusp are self-similar (Mandre et al. Reference Mandre, Mani and Brenner2009; Mani et al. Reference Mani, Mandre and Brenner2010; Duchemin & Josserand Reference Duchemin and Josserand2011) to the leading order. Additionally, we show that in the absence of regularizing effects such as liquid viscosity and surface tension, a finite-time singularity would occur for drop-pool impacts as the film thickness approaches zero at the cusps.

Figure 7. Evolution of the film thickness in time for a simulation of $(U,R)=(0.7\ \text {m}\ \text {s}^{-1}, 0.55\ \text {mm})$ with no surface tension ($\sigma =0$). The final time is $tU/(R{St}^{-2/3})= 10.34$, and the intervals between the snapshots are $({\rm \Delta} t)U/(R{St}^{-2/3})= 1.22$.

3.2.2. Self-similar solution

During the initial stages of the formation of cusps, despite solving the full set of equations given in § 2, the leading order equations ((3.11) to (3.14)) adequately describe the dynamics. In particular, surface tension forces initially play an insignificant role in cusp formation. In this section we show that a self-similar solution exists around the drop and pool cusps. This self-similar solution involves a finite-time singularity, whereby gas pressure and curvature values approach infinity as minimum film thickness approaches zero. However, because of the faster rate of growth of capillary pressures, we predict that the self-similar solution eventually breaks down due to capillary effects and a transition takes place, preventing the early contact and entrapment of a disk-type film. This theoretical prediction is confirmed with our numerical solutions in § 3.3, where the results from simulations with realistic surface tension value of $\sigma =0.072$ N m$^{-1}$ are presented. A similar transition has been observed in drop-solid impacts (Mandre et al. Reference Mandre, Mani and Brenner2009), where the capillary transition prevents the contact of the drop and solid wall. The drop was numerically found to ‘skate’ on a thin air film. Skating was later observed experimentally by Kolinski et al. (Reference Kolinski, Rubenstein, Mandre, Brenner, Weitz and Mahadevan2012); Kolinski, Mahadevan & Rubinstein (Reference Kolinski, Mahadevan and Rubinstein2014), de Ruiter, van den Ende & Mugele (Reference de Ruiter, van den Ende and Mugele2015) and Pack et al. (Reference Pack, Hu, Kim, Zheng, Stone and Sun2017). Other numerical studies including Mandre et al. (Reference Mandre, Mani and Brenner2012) and Duchemin & Josserand (Reference Duchemin and Josserand2011) furthered our understanding of drop skating on solid walls after the capillary transition.

The non-dimensionalized leading order equations governing the evolution of the cusp ((3.11) to (3.14)) can be written in terms of the velocity potential values, ${\phi }^{+}, {\phi }^{-}$, and $y$ coordinate values, ${h}^{+}, {h}^{-}$, on the drop and pool surfaces, respectively,

(3.15)\begin{gather} \frac{\partial{h}^{\pm}}{\partial t}=\frac{\partial{\phi}^{\pm}}{\partial y}, \end{gather}

(3.16)\begin{gather}\frac{\partial{\phi}^{\pm}}{\partial t}=-{P}_{g}, \end{gather}

(3.17)\begin{gather}\frac { \partial }{ \partial { x }} \left({ h }^{ 3 }\frac { \partial { P }_{ g } }{ \partial { x } } \right)=12\frac{\partial h}{\partial t}, \end{gather}

where $h={h}^{+}-{h}^{-}$. An important relation we observe from the numerical results is that to the leading order during the cusp formation stage, the drop and pool surfaces deform equally. In non-dimensional units, deformation of the drop is given by ${{\rm \Delta} h}^{+}={h}^{+}+t$, where $t=0$ corresponds to impact time in a vacuum. The pool deformation is ${{\rm \Delta} h}^{-}=-{h}^{-}$. From figure 8 we see that as the cusps are forming for an impact with $R=0.85$ mm and $U=0.5$ m s$^{-1}$ (with realistic surface tension value of $\sigma =0.072\,{\rm N}\,{\rm m}^{-1}$), we have ${{\rm \Delta} h}^{-}\approx -{{\rm \Delta} h}^{+}$. This symmetry between the drop and pool deformation during the cusp formation stage was also observed numerically in Hendrix et al. (Reference Hendrix, Bouwhuis, Vandermeer, Lohse and Snoeijer2016). We can also introduce an auxiliary velocity potential, ${\tilde {\phi }}^{+}$, for the drop by removing the initial downward velocity ($-Uy$ in dimensional form). Then, from ${{\rm \Delta} h}^{-}=-{{\rm \Delta} h}^{+}$ and inspection of $y$-velocity values, we have

(3.18)\begin{equation} \frac{\partial{\tilde{\phi}}^{-}}{\partial y}=-\frac{\partial{\tilde{\phi}}^{+}}{\partial y}. \end{equation}

This reduces the variables in (3.15)–(3.17), and we have the following representative set of equations which resembles the equations derived by Duchemin & Josserand (Reference Duchemin and Josserand2011) for drop-solid impacts:

(3.19)\begin{gather} \frac{\partial{h}}{\partial t}=2\frac{\partial{\tilde{\phi}}^{+}}{\partial y}, \end{gather}

(3.20)\begin{gather}\frac{\partial{\tilde\phi}^{+}}{\partial t}=-{P}_{g}, \end{gather}

(3.21)\begin{gather}\frac { \partial }{ \partial { x }} \left({ h }^{ 3 }\frac { \partial { P }_{ g } }{ \partial { x } } \right)=12\frac{\partial h}{\partial t}. \end{gather}

Now, we can introduce the ansatz for the film thickness $h(x,t)={h}_{min}(t)H(\eta )$, gas pressure ${P}_{g}(x,t)={P}_{max}(t)\Pi (\eta )$ and potential ${\tilde {\phi }}^{+}(x,y,t)={\tilde {\phi }}_{0}^{+}(t)\Phi (\eta ,\xi )$, defined in coordinate systems around the cusp,

(3.22)\begin{gather} \eta=\frac{x-{x}_{0}}{l(t)}, \end{gather}

(3.23)\begin{gather}\xi=\frac{y}{l(t)}, \end{gather}

where $x_{0}$ is the cusp location, $l(t)$ is an unknown length scale for the cusp and ${h}_{min}(t)$, ${P}_{max}(t)$ and ${\tilde {\phi }}_{0}^{+}(t)$ are all defined locally at the cusp. The cusp (on the drop side) is shown schematically in figure 9.

Figure 8. Deformation of the drop and pool for a simulation with $R=0.85$ mm, $U=0.5$ m s$^{-1}$ and $\sigma =0.072$ N m$^{-1}$ plotted at times (a) $t=-25.95\ \mathrm {\mu }$s (b) $t=-17.22\ \mathrm {\mu }$s (c) $t=3.52\ \mathrm {\mu }$s and (d) $t=17.75\ \mathrm {\mu }$s, where $t=0$ corresponds to impact time in a vacuum. During the cusp formation step, the drop and pool deformation have approximately equal magnitude but opposite signs.

Figure 9. Schematic of the cusp (on the drop surface) and its length scale, $l(t)$.

Similar to drop-solid impact, we find that the behaviour of the solution around the cusp is wave-like to the leading order. As such, temporal derivatives can be approximated using appropriate spatial derivatives. For instance, $\partial h/\partial t \approx -( {\dot{x}_{0}}{h}_{min}/l)\partial H/\partial \eta$, where $\dot {x_{0}}$ is the velocity of the cusp and can be assumed constant during this stage (Mani et al. Reference Mani, Mandre and Brenner2010; Duchemin & Josserand Reference Duchemin and Josserand2011). Substituting the ansatz into (3.19) to (3.21) and replacing the temporal derivatives, we obtain the following scaling rules:

(3.24)\begin{gather} -\frac{\dot{x_{0}}{h}_{min}}{l}\frac{\partial H}{\partial \eta}\sim \frac{{\tilde{\phi}}_{0}^{+}}{l}\frac{\partial \Phi}{\partial \xi}, \end{gather}

(3.25)\begin{gather}-\frac{\dot{x_{0}}{\tilde{\phi}}_{0}^{+}}{l}\frac{\partial \Phi}{\partial \eta}\sim -{P}_{max}\Pi, \end{gather}

(3.26)\begin{gather}\frac{1}{{l}^{2}}\frac{\partial({h}_{min}^{3}{H}^{3}{P}_{max}(\partial \Pi/\partial \eta))}{\partial \eta}\sim -\frac{\dot{x_{0}}{h}_{min}}{l}\frac{\partial H}{\partial \eta}. \end{gather}

From (3.24) to (3.26), we can obtain the scaling laws giving

(3.27)\begin{gather} l\sim {h}_{min}^{3/2}, \end{gather}

(3.28)\begin{gather}{P}_{max}\sim {h}_{min}^{-1/2}. \end{gather}

Using these scaling laws coupled to the wave-like nature of the solution around the cusp, it can be shown that a finite-time singularity exists. If we denote the time at which ${h}_{min}=0$ with ${t}_{0}$, then we have ${h}_{min}\sim ({t}_{0}-t)^{2/3}$. As a result, in a finite time, the maximum gas pressure goes to infinity. However, the maximum curvature which also happens at the cusp scales as

(3.29)\begin{equation} {\kappa}_{max}\sim {h}_{min}^{-2}. \end{equation}

Crucially, the rate of growth of maximum capillary pressure ($\sigma {\kappa }_{max}$) is therefore faster than the maximum gas pressure. Eventually, capillary pressure cannot be neglected and the self-similar solution, which does not include capillary effects, breaks down. Surface tension regularizes the cusp, preventing early contact and entrapment of a disk-type film as we confirm using the numerical results in the following section. Additionally, in the following section we confirm the above scaling laws via our numerical simulations.

3.3. Boundary integral method with surface tension: transition to spreading

As predicted theoretically above, while becoming sharper in a self-similar manner, the cusp is at some point regularized by capillary intervention. The self-similar solution thus breaks down and a transition to another type of behaviour takes place. This transition is similar to the transition to skating of the drop on the solid wall observed in drop-solid impacts (Mandre et al. Reference Mandre, Mani and Brenner2009; Kolinski et al. Reference Kolinski, Rubenstein, Mandre, Brenner, Weitz and Mahadevan2012; de Ruiter et al. Reference de Ruiter, Oh, van den Ende and Mugele2012; Pack et al. Reference Pack, Hu, Kim, Zheng, Stone and Sun2017). For the case of drop-pool impact, the discovered transition allows for the drop to spread on the pool surface. As such, we denote it as transition to spreading behaviour. A schematic of the transition stage is also shown in figure 4(d).

Snapshots of the interface profiles before and after transition are shown in figure 10(a) for a $R=0.55$ mm, $U=0.7$ m s$^{-1}$ water impact ($\sigma =0.072$ N m$^{-1}$). Capillary waves are evident after transition occurs. The evolution of the film thickness is depicted in figure 10(b), showing that similar to the cusp formation stage, air does not drain from the central region of the film as the drop spreads after transition. This is due to the fact that the air cannot flow past the pressure peaks that exist around the spreading front. We elaborate further on this behaviour in § 3.4. In figure 10(c) the disk-type film entrapped when surface tension is zero is included for comparison. It is clear that while in the initial stages surface tension has a negligible effect, eventually it prevents the drop and pool getting close enough to contact (i.e. the disk-type film to rupture). A similar observation can be made from the film thickness comparison in figure 10(d).

Figure 10. Results from BIM for a $R=0.55$ mm and $U=0.7$ m s$^{-1}$ water impact ($\sigma =0.072$ N m$^{-1}$) showing (a) evolution of the drop and pool interfaces from pre-deformation to after transition, (b) evolution of the film thickness from pre-deformation to after transition, (c) disk mode film from simulations with no surface tension depicted in black on top of profiles from part (a), (d) disk mode film thickness from simulations with no surface tension depicted in black on top of film thickness plots from part (b). The $\sigma =0.072$ N m$^{-1}$ profiles start at $tU/(R{St}^{-2/3})=-32$ and are spaced by ${\rm \Delta} tU/(R{St}^{-2/3})=8.8$, while the disk film from $\sigma =0$ is at $tU/(R{St}^{-2/3})=10.34$.

To quantitatively illustrate the prevention of contact via capillary forces, figure 11(a) compares the time evolution of the minimum air film thickness between these simulations ($\sigma =0$ plotted in red versus $\sigma =0.072$ N m$^{-1}$ plotted in blue for $R=0.55$ mm, $U=0.7$ m s$^{-1}$). Note that $t=0$ corresponds to impact time in a vacuum. It is evident from figure 11(a) that for both simulations, after an initial linear portion corresponding to the pre-deformation stage (§ 3.1), the impacts are cushioned during the cusp formation stage (§ 3.2) before $t=0$. Without surface tension, the minimum film thickness decreases towards zero in finite time, resulting in contact at approximately $t=10\ \mathrm {\mu }$s (i.e. impact is delayed about $10\ \mathrm {\mu }$s). In the presence of surface tension forces, the minimum film thickness reaches an infimum around transition time (approximately at $t=10\ \mathrm {\mu }$s), after which it continuously grows. Note that there is a discontinuity in the derivative of ${h}_{min}$ with respect to time around $t=8\times {10}^{-5}$ s. In § 3.4.1 we show that this corresponds to a new location for the minimum film thickness between $x=0$ and the spreading front. A zoom in on figure 11(a) around the transition time is shown in figure 11(b). The value of the minimum film thickness at transition, denoted by ${h}_{tran}$, is shown to be about $230$ nm for these impact parameters. If we had followed Hendrix et al. (Reference Hendrix, Bouwhuis, Vandermeer, Lohse and Snoeijer2016) to declare contact and stop the simulation once the film thickness reached $400$ nm, we would have not been able to observe this transition. Here ${h}_{tran}$ is of interest in terms of determining the fate of an impact in the real world where Van der Waals forces are responsible for contact of the drop and pool. More specifically, as we will elaborate later on, if ${h}_{tran}$ is too small, Van der Waals forces precede the transition and early contact takes place. Alternatively, if $h_{tran}$ is large enough that Van der Waals forces cannot cause contact, the transition takes place, allowing for the drop to spread onto the pool surface and an elongated film to start to form. Lastly, ${h}_{tran}$ gives an estimate of the required mesh size if one aims to capture the transition with traditional interface-capturing methods (Mirjalili, Jain & Dodd Reference Mirjalili, Jain and Dodd2017).

Figure 11. (a) Minimum air film thickness plotted against time for a $R=0.55$ mm, $U=0.7\,{\rm m}\,{\rm s}^{-1}$ impact with $\sigma =0.072$ N m$^{-1}$ in blue and $\sigma =0$ in red, (b) a zoom in around time of $t=0$ when capillary transition takes place.

Finding the transition film thickness for all relevant $R$ and $U$ values by performing BIM simulations is a very expensive task due to the very high curvature values that need to be resolved. We take an alternative, theory-based approach similar to the work of Mandre et al. (Reference Mandre, Mani and Brenner2009) and Duchemin & Josserand (Reference Duchemin and Josserand2011) on drop-solid impacts. In figure 12 the maximum gas pressure is plotted along with the maximum capillary pressure ($\sigma {\kappa }_{max}$) on the pool surface against the minimum film thickness for a $R=0.55$ mm, $U=0.7$ m s$^{-1}$ water impact. The plot is cut off just before transition occurs for better visualization. The film thickness decreases in time, so time is progressing from right to left in the plot. It can be seen that initially in the pre-deformation stage, gas pressure scales with ${h}_{min}^{-2}$, which is in agreement with the scaling law in (3.2). At some point, when deformation starts, the slope of the profile changes such that the gas pressure scales with ${h}_{min}^{-1/2}$. This is also in agreement with the predictions from the self-similar solution (3.28) in the cusp formation stage. The maximum pool capillary pressure starts from zero, and asymptotes to scaling with ${h}_{min}^{-2}$, as predicted in (3.29). We observe transition to occur when the maximum capillary pressure catches up to the maximum gas pressure (figure 12). It is important to clarify that maximum curvature and maximum gas pressure both happen at the cusp location. When these two pressures become of the same order, the self-similar solution breaks and the minimum film thickness stops decreasing. Thus, we have not plotted the pressure values beyond transition in figure 12 to avoid a crowded plot. Moreover, due to the fact that prior to transition we have ${{\rm \Delta} h}^{-}\approx -{{\rm \Delta} h}^{+}$, the maximum capillary pressure in the drop and pool are very close at various ${h}_{min}$ values. Hence, we could have reached the same conclusions using maximum drop capillary pressure but have omitted that plot in figure 12 for better visualization. For the same range of minimum air film thicknesses, the maximum gas pressure from the simulation with $\sigma =0$ is also plotted on top of these results. Of course, capillary pressures are zero for the $\sigma =0$ simulation. From comparing these two simulations, it seems that as transition is approached in time, surface tension forces slightly reduce the peak gas pressure values, probably by regularizing the sharp cusps.

Figure 12. The maximum gas pressure and pool capillary pressure prior to transition plotted against minimum film thickness for a $R=0.55$ mm, $U=0.7$ m s$^{-1}$ water impact in blue. The maximum gas pressure from simulation with $\sigma =0$ is plotted in red.

Now similar to Mandre et al. (Reference Mandre, Mani and Brenner2009), but for drop-pool impacts instead of drop-solid impacts, we can analytically approximate the crossover between the maximum capillary pressure in the drop/pool and maximum gas pressure to find ${h}_{tran}$. At the start of deformation, which is also the start of the self-similar solution presented in § 3.2, the drop capillary pressure is known to be $\sigma /R$. The maximum gas pressure can also be found in a straightforward manner. Figure 13(a) shows the maximum gas pressure as deformation starts for numerous impacts given by $(R,U)\in \{0.55, 0.85, 1.15, 1.45, 1.75\ \text {mm}\}\times \{0.25, 0.35, 0.4, 0.5, 0.6, 0.7\ \text {m}\ \text {s}^{-1}\}$ pairs versus minimum film thickness. Rescaling ${h}_{min}$ and $max{({P}_{g})}$ using (3.4) and (3.6), respectively, we see that the plots collapse in figure 13(b), allowing us to extract an analytic value for maximum gas pressure as deformation starts.Extrapolating the maximum gas pressure and maximum drop capillary pressure using the scaling laws analytically derived and numerically verified before, we obtain the film thickness at transition to be

(3.30)\begin{equation} {h}_{tran}\approx 13R{St}^{-8/9}{We}^{-2/3}, \end{equation}

in non-dimensional form and

(3.31)\begin{equation} {h}_{tran}\approx 3\times{10}^{-9}{(R{U}^{4})}^{-5/9}, \end{equation}

in dimensional form where everything is presented in SI units. If the transition film thickness is too small then Van der Waals forces can become active prior to occurrence of the transition. In that case, these molecular forces cause premature contact of the drop and pool, leading to disk-type film entrapment and one or few central bubbles after retraction. If we assume that at a critical height denoted by ${h}_{VDW}$, Van der Waals attractive molecular forces cause contact of the drop and pool, then we have the following criterion for the early time dynamics of impact:

(3.32)\begin{equation} \text{outcome(}{h}_{tran}) =\begin{cases} \text{transition to spreading} & \text{if } { h }_{ tran }\ge { h }_{ VDW }, \\ {{\rm early}\ {\rm contact}} & \text{if } { h }_{ tran }<{ h }_{ VDW }. \end{cases} \end{equation}

In figure 14 contour lines of ${h}_{tran}$ in nanometres are plotted on top of Pumphrey's diagram (labelled in figure 1). It is typically reported that at film thicknesses of $\sim 100$ nm and below, Van der Waals forces cause contact (Couder et al. Reference Couder, Fort, Gautier and Boudaoud2005; Dorbolo et al. Reference Dorbolo, Reyssat, Vandewalle and Quéré2005; Thoroddsen et al. Reference Thoroddsen, Thoraval, Takehara and Etoh2012; Tran et al. Reference Tran, de Maleprade, Sun and Lohse2013). The critical film thickness depends on the working liquid. In this work we use ${h}_{VDW}=100$ nm as the critical thickness for spontaneous contact due to Van der Waals forces. With this assumption (${h}_{VDW}=100$ nm in (3.32)), all $U$ and $R$ values below the ${h}_{tran}=100$ nm contour line in figure 14 should allow for transition to spreading behaviour. On the other hand, above the $100$ nm contour line, one expects a disk-type film to be entrapped in the short-time dynamics of drop-pool impact. From experiments (Thoroddsen et al. Reference Thoroddsen, Etoh and Takehara2003; Saylor & Bounds Reference Saylor and Bounds2012; Thoroddsen et al. Reference Thoroddsen, Thoraval, Takehara and Etoh2012; Tran et al. Reference Tran, de Maleprade, Sun and Lohse2013), we know that Mesler entrainment requires elongated hemispherical-type films, and further that early contact (disk-type films) results in one or few central bubbles. As such, we propose the ${h}_{tran}=100$ nm contour line in figure 14 as an upper boundary for Mesler entrainment. It must be clarified that this upper boundary is not tight in the sense that an impact under this line, despite going through transition, does not necessarily lead to micro-bubbles. It is encouraging that the regular entrainment region of Pumphrey's diagram (solid black lines in figure 14) lies above this upper boundary, as regular entrainment experiments are associated with early contact.

Figure 13. Maximum gas pressure shown for $R\in \{0.55, 0.85, 1.15, 1.45, 1.75\ \text {mm}\}$ plotted in {blue, cyan, green, yellow, orange} with $U\in \{0.25, 0.35, 0.4, 0.5, 0.6, 0.7\ \text {m}\ \text {s}^{-1}\}$ plotted using different markers for drop-pool impact cases in (a) unscaled form versus unscaled minimum film thickness, (b) scaled form versus scaled minimum film thickness.

Figure 14. Contour lines of ${h}_{tran}$ in nanometres given by (3.31) plotted on top of Pumphrey's diagram (labelled in figure 1).

To explore the accuracy of this prediction in more detail, in figure 15 we keep the contour lines while including experimental results from water drop-pool impacts. We see that no experimental observation of Mesler entrainment (solid black symbols) lies above the ${h}_{tran}=100$ nm contour line, validating that this line is in fact an upper boundary for Mesler entrainment. The previous estimate for an upper boundary for Mesler entrainment in water was suggested by Liow & Cole (Reference Liow and Cole2007) to be ${We}_{D}={ { \rho }_{ l }{ U }^{ 2 }(2R) }/{ \sigma }=20$. This suggested upper boundary is also included as a red dashed line in figure 15. It is clear that converse to our suggested boundary, this line does not bound all of the observations of Mesler entrainment.

Figure 15. Experimental observations of Mesler entrainment (solid black symbols) and non-Mesler entrainment (open magenta symbols) in water drop-pool impacts plotted on top of contour lines of ${h}_{tran}$ in nanometres (3.31). The red dashed line ($We_{D}=20$) is the suggested boundary for Mesler entrainment from Liow & Cole (Reference Liow and Cole2007).

There are many instances of non-Mesler entrainment below the $100$ nm contour lines in figure 15. While we must point out that most of these instances are results from the experiments of Salerno et al. (Reference Salerno, Levoni and Barozzi2015), which did not succeed in capturing Mesler entrainment in any of their attempts, other authors have also reported the issue of repeatability of Mesler entrainment for water (Sigler & Mesler Reference Sigler and Mesler1990; Thoroddsen et al. Reference Thoroddsen, Etoh and Takehara2003; Mills et al. Reference Mills, Saylor and Testik2012). Since the occurrence of transition and spreading of a drop on the pool surface do not guarantee Mesler entrainment, these experimental data do not invalidate our suggested upper boundary. Nonetheless, the issue of unrepeatability of Mesler entrainment for water impacts deserves a discussion which is provided in § 4, where we examine the transition dynamics for liquids other than water.

As mentioned before, the transition film thickness given by (3.31) and depicted with the contour lines in figure 15 also informs on the necessary resolution requirements for successful simulations of drop-pool impacts in this regime using interface-capturing schemes such as VoF, level set or phase field methods (Mirjalili et al. Reference Mirjalili, Jain and Dodd2017). First of all, to capture curvature accurately, sufficient resolution must be provided in the $x$ direction. Additionally, in contrast to the BIM we used here, such methods must resolve the gas film thickness at transition time. Despite including all physical terms in the formulation, with grids coarser than these requirements, such high-fidelity simulations erroneously predict the early contact of the drop and pool, leading to disk-type film entrapment and one or few central bubbles after retraction. From figure 15, it can be seen that a film thickness of ${O}$(100 nm) has to be resolved to simulate any of the experimental observations of Mesler entrainment. This is an impractically high resolution requirement, even in two dimensions as one would need to have mesh sizes smaller than ${\sim }100$ nm in a simulation domain of size ${\sim }1~\text {cm}\times {\sim }1~\text {cm}$. In fact, using a recently developed conservative diffuse interface method (Mirjalili, Ivey & Mani Reference Mirjalili, Ivey and Mani2020), we have also tackled this problem in two-dimensional high-fidelity simulations. In our simulations we used non-uniform fixed Cartesian grids. We observed that the simulations were predictive and convergent up until the cusp formation stage. However, to prevent premature numerical rupture of the air film, prohibitively small mesh cells would be required ($<$100 nm). Adaptive mesh refinement (AMR) could potentially provide the capability to resolve the capillary transition by temporarily refining the mesh around the cusps when transition is taking place. Indeed, researchers have utilized AMR successfully in other regimes and for other types of drop impact problems (Thoraval et al. Reference Thoraval, Takehara, Etoh, Popinet, Ray, Josserand, Zaleski and Thoroddsen2012; Shetabivash, Ommi & Heidarinejad Reference Shetabivash, Ommi and Heidarinejad2014; Agbaglah et al. Reference Agbaglah, Thoraval, Thoroddsen, Zhang, Fezzaa and Deegan2015; Josserand et al. Reference Josserand, Ray and Zaleski2016).

3.4. Post-transition spreading

After capillary forces regularize the sharp cusps, the drop penetrates the pool while also spreading outward without violating the integrity of the air film. A schematic of this stage of the process is shown in figure 4(e). Figure 10(c) showed the drop and pool profiles around transition time for a $R=0.55$ mm and $U=0.7$ m s$^{-1}$ water impact in comparison to the disk-type film that is entrapped if $\sigma =0$. Figure 16(a) is a continuation of that figure in time, depicting the evolution of the profiles well after transition. It is clear that the transition allows for penetration of the drop into the pool while the spreading front of the drop moves outward on the pool free surface. In figure 16(b), for the same impact parameters but in dimensional units, we compare the disk-type film that is entrapped at contact time ($t=1.06\times {10}^{-5}$ s) for the surface-tension free simulation ($\sigma =0$, plotted in red), against the post-transition geometry at $t=10.25\times {10}^{-5}$ s with $\sigma =0.072$ N m$^{-1}$ (plotted in blue). We have already seen from figure 11 that once the system passes through the ‘bottleneck’ of transition (${h}_{tran}>{h}_{VDW}$), the minimum film thickness slowly but steadily increases in time. As such, in the presence of Van der Waals forces, the film remains intact and the geometry shown in figure 16(b) can be reached at time $t=10.25\times {10}^{-5}$ s. It is important to remember that the lubrication zone, in which the gas pressure is computed has a fixed horizontal length of $0.9D$. This can prohibit time integration to continue to a stage when rupture (${h}_{min}\le {h}_{VDW}$), or at lower impact velocities, bouncing, takes place. For instance, the elongated film shown in figure 16(b) is approximately the final geometry this framework can robustly capture, as the boundary conditions at the edge of the lubrication zone are invalidated once the effects of the spreading front of the drop reach their vicinity. Nonetheless, although we may not be able to predict the final outcome of the elongation process with our simulations, studying the numerical results after transition provides understanding of the dynamics of contact-free spreading of a drop on a pool surface, in addition to asymptotic scaling laws and features that had not been identified prior to this work.

Figure 16. For a water drop-pool impact with $(U,R)=(0.7\ \text {m}\ \text {s}^{-1}, 0.55\ \text {mm})$, (a) the evolution of the drop and pool surfaces from $\sigma =0.072$ N m$^{-1}$ simulations are shown at equal time intervals from $t=-3.28\times {10}^{-5}$ s to $t=9.42\times {10}^{-5}$ s on top of the disk-type film entrapped from $\sigma =0$ simulations at time $t=1.06\times {10}^{-5}$ s (plotted in black), (b) the disk-type film entrapped at contact time from a $\sigma =0$ simulation plotted in red ($t=1.06\times {10}^{-5}$), compared to the elongated film at $t=1.025\times {10}^{-4}$ s from the $\sigma =0.072$ N m$^{-1}$ simulation, plotted in blue.

We utilize simulations of three water drop-pool impact cases below the ${h}_{tran}=100$ nm contour line in figure 14 to study the post-transition spreading dynamics. These impact cases are $(U,R)\in \{(0.5\ \text {m}\ \text {s}^{-1}, 0.85\ \text {mm}), (0.4\ \text {m}\ \text {s}^{-1}, 1.15\ \text {mm}), (0.7\ \text {m}\ \text {s}^{-1}, 0.55\ \text {mm})\}$. By means of the capillary transition, the pool accommodates for the initial difference in geometry between the impacting surfaces while avoiding contact. To demonstrate how the drop spreads on the pool surface after transition, figure 17(a) shows the time evolution of the position of maximum gas pressure (${x}_{max{(P_{g})}}$), maximum pool capillary pressure (${x}_{max{(\kappa )}}$, pool), maximum drop capillary pressure (${x}_{max{(\kappa )}}$, drop) and minimum air film thickness (${x}_{min{(h)}}$) for the $(U,R)=(0.7\ \text {m}\ \text {s}^{-1}, 0.55\ \text {mm})$ simulation. The position of maximum gas pressure for the $\sigma =0$ simulation is also plotted for reference (${x}_{max{(P_{g})}}$, $\sigma =0$). Just prior to $t=0$, the pool and drop start to deform, resulting in non-zero values for ${x}_{min{(h)}}$ and meaningful values for ${x}_{max{(\kappa )}}$ on the drop and pool surfaces. The value for ${x}_{max{(P_{g})}}$ follows the $\sigma =0$ values until transition takes place at $tU/R\approx 0.014$ and steadily grows afterwards. In general, we observe that the position of all these extrema move outwards along with the spreading front of the drop. Note that the discontinuity in ${x}_{min{(h)}}$ at $tU/R\approx 0.11$ is because at that time the film thickness minimum switches from the spreading front to an inner kink, as we will explain in § 3.4.1. The largest value belongs to ${x}_{max{(\kappa )}}$ for the pool surface which lies to the outside of the spreading front of the drop (see the pool capillary waves in figures 10 and 16). We observe linear profiles when we plot the maxima locations to the power of two in figure 17(b). This $\sqrt {t}$ behaviour is similar to the growth rate of the spreading edge at early times for drop impacts when gas cushioning is neglected (Wagner Reference Wagner1932; Josserand & Zaleski Reference Josserand and Zaleski2003; Purvis & Smith Reference Purvis and Smith2005; Josserand et al. Reference Josserand, Ray and Zaleski2016). To explain, we reiterate that the air film stops draining past the pressure maxima at the cusps prior to transition. After transition, the pressure maxima continue to move outward along with the spreading front, thereby effectively confining the air mass between the spreading fronts (see figure 10b). Thus, to the leading order, the film is carried below passively by the drop as it penetrates into the pool. As a result, the large-scale dynamics follow the early dynamics of drop-pool impacts in the absence of gas cushioning, studied by Wagner (Reference Wagner1932) and Purvis & Smith (Reference Purvis and Smith2005), whereby the spreading front of the drop moves outward with a $\sqrt {t}$ dependence. The penetration velocity, examined below, also follows the cushioning-free classical solutions of Wagner theory (Wagner Reference Wagner1932). Note that the fact that the film does not drain past the gas pressure maxima does not imply that the air mass encapsulated between the spreading fronts is constant in time. As the drop spreads on the pool surface, the air between the liquid bodies at the spreading front is continuously deposited into the film as we will show in more detail in § 3.4.1.

Figure 17. For simulations of $(U,R)=(0.7\ \text {m}\ \text {s}^{-1}, 0.55\ \text {mm})$ water drop-pool impact, with $\sigma =0$ and $\sigma =0.072$ N m$^{-1}$, the temporal evolution is shown for (a) gas and capillary pressure maxima positions, in addition to minimum air film position, (b) gas and capillary pressure maxima positions and minimum air film position, all squared. For $(U,R)\in \{(0.5\ \text {m}\ \text {s}^{-1}, 0.85\ \text {mm}),\ (0.4\ \text {m}\ \text {s}^{-1}, 1.15\ \text {mm}),\ (0.7\ \text {m}\ \text {s}^{-1}, 0.55\ \text {mm})\}$ and $\sigma =0.072$ N m$^{-1}$, the evolution of the positions of maximum gas pressure, representing the location of the spreading front, are plotted (c) in dimensional units, (d) scaled by $R$.

Following Wagner theory, the location of the spreading front at time $t$ scales with $\sqrt {URt}$. In figure 17(c) we plot the time evolution of ${x}_{max{(P_{g})}}$ as a proxy for spreading front location for the three different case studies. Using the non-dimensionalization of $tU/R$ for time, it follows that ${x}_{max{(P_{g})}}(tU/R)\sim R\sqrt {tU/R}$. Hence, at the same non-dimensional time ($tU/R$), the spreading front scales with $R$, as is evident from figure 17(d). From figure 17, it is clear that the axial length scale of the film after transition is no longer governed by $L\sim \sqrt {RH}$ but grows in time as the drop spreads within the pool. Moreover, figures 16(b) and 17 show that at a time of $tU/R\sim {O}(0.1)$, the film's axial length scale is already $L\sim R$ and barring contact due to Van der Waals forces, no length scale other than $R$ exists to prohibit further spreading.

The drop and pool surface velocities ($\partial {h}^{+}/\partial t$ and $\partial {h}^{-}/\partial t$) are also of interest. In figure 18 the time evolution of the $y$ coordinate of the drop and pool surfaces at $x=0$ are shown for the same three cases introduced above, all depicting an approximate penetration velocity of $-U/2$. Recall that to the leading order, there is no drainage from the central regions of the air film, and the air film is carried passively below by the drop as it penetrates within the pool. Since the air layer no longer affects the liquid dynamics in this central region of the air film, the dynamics are to the leading order given by the solutions in the absence of gas effects due to Wagner theory (Wagner Reference Wagner1932; Smith et al. Reference Smith, Li and Wu2003; Josserand et al. Reference Josserand, Ray and Zaleski2016), resulting in a penetration velocity of $-U/2$ for the drop and pool. This is also in agreement with measurements of the penetration velocity by Thoroddsen et al. (Reference Thoroddsen, Thoraval, Takehara and Etoh2012) and Tran et al. (Reference Tran, de Maleprade, Sun and Lohse2013) for Mesler entrainment experiments using silicone oils. Since Tran et al. (Reference Tran, de Maleprade, Sun and Lohse2013) found that this property is weakly dependent on viscosity, one expects it to hold for water drop-pool impacts that avoid early contact as well, in agreement with our finding here.

Figure 18. The position of the drop and pool interfaces at $x=0$ as a function of time, showing the constant penetration velocity of $-U/2$ after transition.

The time scale we have used to non-dimensionalize time so far in this section, $T\sim R/U$, is consistent with the found length ($R$) and velocity scales ($U$). Note the contrast between the length and time scales governing the spreading stage, and those of the cusp formation stage (§ 3.2).

3.4.1. Anatomy of contact-free drop-pool spreading

So far, we have shown that while an unruptured gas film separates the two liquid bodies, the post-transition kinematics exhibit general features of drop spreading in drop-liquid impacts (Wagner Reference Wagner1932; Yarin Reference Yarin2006; Josserand et al. Reference Josserand, Ray and Zaleski2016). Here, we study the solutions of our three water impact cases during this stage to reveal the complex details of the dynamics. Note that we do not have any $\sigma =0$ data for this stage; therefore, $\sigma =0.072\,{\rm N}\,{\rm m}^{-1}$ for the remainder of this section. In figure 19, for four time instances after transition of the $(U,R)=(0.7\ \text {m}\ \text {s}^{-1}, 0.55\ \text {mm})$ impact, the gas and liquid surface pressures (in blue), surface $y$-velocities (in red) and film thickness (in black), scaled by $max{({P}_{g})}$, $U$ and ${h}_{0}=h(x=0)$, respectively, are plotted against $\tilde {x}=(x-{x}_{max{({P}_{g})}})/{x}_{max{({P}_{g})}}$ for $x\ge 0$. With this coordinate transform, the centre of the air film is located at $\tilde {x}=-1$, and the peak gas pressure happens at $\tilde {x}=0$. We have chosen ${x}_{max{({P}_{g})}}$, which monotonically increases as the drop spreads (figure 17), to represent the spreading front position.

Figure 19. For a water drop-pool impact at $(U,R)=(0.7\ \text {m}\ \text {s}^{-1}, 0.55\ \text {mm})$, the normalized post-transition values of gas pressure, pool surface pressure, drop surface pressure, pool $y$-velocity, drop $y$-velocity and film thickness plotted against $\tilde {x}=(x-{x}_{max{({P}_{g})}})/{x}_{max{({P}_{g})}}$ at times (a) $tU/R=0.037$ (b) $tU/R=0.065$, (c) $tU/R=0.093$ and (d) $tU/R=0.121$.

As indicated by figure 19, between the spreading fronts of the drop, $\tilde {x}<0$, the drop and pool surface both have a downward velocity of $\approx -U/2$ at all times. Another time-invariant feature of the solutions is the presence of capillary waves to the right of the spreading front ($\tilde {x}>0$). Note how these waves do not enter the zone between the spreading fronts. In agreement with figure 17(a), the minimum film thickness is situated to the right of peak gas pressure, followed by the peak capillary pressures on the capillary waves. This spatial ordering is temporary, as at $Ut/R\approx 0.11$ the minimum film thickness is no longer at the spreading front of the drop, but rather at a second kink in between the maximum gas pressure position and $x=0$. This event appears as discontinuities in the evolution of $h_{min}$ in figure 11(a) and $x_{min(h)}$ in figure 17. A similar double kink structure has been also observed in low $We$ drop-solid impacts (de Ruiter et al. Reference de Ruiter, Oh, van den Ende and Mugele2012, Reference de Ruiter, van den Ende and Mugele2015). The double kink structure we observe here has not yet been reported for drop-liquid impacts, perhaps due to the limited radial extent when studying these characteristically curved air films with interferometry (Tran et al. Reference Tran, de Maleprade, Sun and Lohse2013; Tang et al. Reference Tang, Saha, Law and Sun2019). While the appearance of the double kink structure is similar to low $We$ drop-solid impacts, there is an important distinction. In drop-solid impacts transition is accompanied by lift-off of the spreading front of the drop, after which the minimum film thickness location is immobile. Later, a sharper secondary cusp forms at the spreading front (de Ruiter et al. Reference de Ruiter, Oh, van den Ende and Mugele2012; Kolinski et al. Reference Kolinski, Mahadevan and Rubinstein2014; de Ruiter et al. Reference de Ruiter, van den Ende and Mugele2015). As can be observed from the temporal evolution of the air film in figure 20, we do not observe that sort of lift-off during transition, but rather the minimum film thickness, while growing, moves outward along with the spreading front. The inner kink starts to form later on, after the formation of a local gas pressure minimum between $x=0$ and the spreading front (figure 19b). On the left side of this local pressure minimum, the central region of the air film, also known as the dimple (Tran et al. Reference Tran, de Maleprade, Sun and Lohse2013), depletes and becomes flatter in time. On the other side, the region between the local gas pressure minimum and gas pressure maximum is continually fed from the air under the spreading front. Eventually, the film thickness at the spreading front becomes larger than the film thickness at the inner kink (at $tU/R\approx 0.11$). All along this process, as the drop spreads on the pool, the air volume between the spreading fronts increases and the film regularizes, gradually becoming more uniform in space as is also evident from figure 20. Note that the temporal evolution described above is not unique to this case, and similar post-transition dynamics are demonstrated for the other drop-pool impact cases in appendix B.

Figure 20. Evolution of the air film thickness for a $(U,R)=(0.7\ \text {m}\ \text {s}^{-1},\ 0.55\ \text {mm})$ water drop-pool impact from $tU/R=-0.0235$ to $tU/R=0.1245$ with equal time intervals between the profiles (${{\rm \Delta} }tU/R=0.0074$).

We know from figure 17(d) that $x_{max{(P_{g})}}$, representing the spreading front, grows in time and scales with $R$. From figure 19 we see that the kink position has an almost constant $\tilde {x}$ value in time, suggesting that all lateral features in the film grow in time and scale with $R$. This is further illustrated by the very similar kink positions at the same $tU/R$ among the different impact cases shown in figures 19, 27 and 28. At a common $tU/R$, $x_{max{(P_{g})}}\sim R$, hence, the kink position also scales with $R$. This is later utilized and confirmed numerically in § 3.4.2.

Recall that prior to capillary transition, the gas and liquid capillary pressures all rise as the minimum film thickness decreases in time (see figure 12). For the $(U,R)=(0.7\ \text {m}\ \text {s}^{-1},\ 0.55\ \text {mm})$ case, we show in figure 21(a) that after transition takes place around non-dimensional time of $tU/R=0.014$, the pressure values relax and drop monotonically as the minimum film thickness slowly increases. As the drop spreads on the pool surface while also penetrating into it, the maximum pressure magnitude in the system is a pool capillary pressure, as can be also observed from figure 19. This pressure is in fact a large negative value, corresponding to a local minimum in ${h^{-}}$ and $\partial {h}^{-}/\partial t\approx 0$, situated to the right of the minimum film thickness, gas pressure maximum and drop pressure maximum. Capillary waves are the result of the interplay between liquid inertia and capillary forces, with wavelengths of $\lambda \sim {We}^{-1}R$. The magnitudes of these pressure peaks thus all scale with $\rho _{l}{U}^2$. This is verified in figures 21(b) and 21(c), where $P_{g,max}$ and $P_{g,max}/({\rho }{U}^2)$ are plotted as a function of time for the three impact cases.

Figure 21. Evolution of the (a) pressure maxima for a $(U,R)=(0.7\ \text {m}\ \text {s}^{-1},\ 0.55\ \text {mm})$ water drop-pool impact, (b) maximum gas pressure in dimensional form for three water drop-pool impact cases and (c) maximum gas pressure scaled by ${\rho }_{l}{U}^{2}$ for the same three impact cases, plotted after the onset of deformation.

The air mass in the film does not drain past the pressure peaks that move outward with the spreading front of the drop, and is instead carried with the drop downward into the pool. However, between the gas pressure peaks, pressure gradients exist and the air film redistributes its mass as spreading takes place, while also being continuously fed from the air under the spreading front (figures 19, 20). In figure 22(a) we inspect the temporal evolution of the film thickness at the film centre ($x=0$), denoted by ${h}_{0}$, which is also known as the dimple height. It is clear that in the pre-deformation stage prior to $t=0$, the linear profiles collapse with the simple scaling of ${h}_{0}/R$. In figure 22(b) we zoom in on the times after the onset of deformation (slightly before $t=0$) to show that for all three cases, the dimple height grows during the cusp formation stage until transition takes place around $tU/R=0.014$, after which the dimple slowly depletes. It is clear that the ${h}_{0}/R$ scaling does not hold at any of the stages after pre-deformation. We explained in § 3.2 that during the cusp formation stage, the dimple height and time scales are given by $R{St}^{-2/3}$ and $R{St}^{-2/3}/U$, respectively, as was also previously reported by other authors (Hicks & Purvis Reference Hicks and Purvis2011; Hendrix et al. Reference Hendrix, Bouwhuis, Vandermeer, Lohse and Snoeijer2016). However, the scaling laws governing the dimple height, and in general the film thickness between the spreading fronts of the drop after capillary transition, have not yet been identified. In the following, we tackle this problem by taking advantage of the post-transition data available from the simulations.

Figure 22. Film thickness at film centre, ${h}_{0}$, plotted as a function of time, (a) scaled by $R$, (b) scaled by $R$, zoomed in on times after deformation starts, (c) scaled by $R{Ca}^{1/3}$, given in (3.33) and (d) scaled by $R{Ca}^{1/2}$, which holds for the dimple height in $We<1$ drop-solid impacts (Bouwhuis et al. Reference Bouwhuis, van der Veen, Roeland, Tran, Keij, Winkels, Peters, van der Meer, Sun, Snoeijer and Lohse2012; Klaseboer, Manica & Chan Reference Klaseboer, Manica and Chan2014).

3.4.2. Film thickness scaling

The air film between the spreading fronts consists of two parts: (1) a central dimple zone between the local gas pressure minima, (2) an outer zone between the local minima and global maxima of gas pressure. These two zones reside on the two sides of the kink, after it forms. While we explained in § 3.4.1 that the length scales of these two zones both scale with $R$, ${L}_{in}\sim {L}_{out}\sim R$, and the velocity scales are both $U$, the pressure gradients in these two zones are different in nature, resulting in different scaling laws for the film thickness in the inner dimple zone, denoted by $H_{in}$, and the outer zone denoted by $H_{out}$. The pressure gradients in the inner zones are due to the variations in the liquid surface capillary pressures, giving ${(\partial {P}_{g}/\partial x)}_{in}\sim \sigma /{R}^2$. From conservation of mass in the film, we have the asymptotic balance of $(\partial /\partial x)({h^3(\partial {P}_{g}/\partial x)}_{in})\sim \sigma {H}_{in}^{3}/{R}^{3}\sim \mu _{g}U$, and, therefore,

(3.33)\begin{equation} H_{in}\sim R{Ca}^{1/3}, \end{equation}

where $Ca=We{St}^{-1}=({\mu }_{g}U)/{\sigma }$ is the capillary number based on the gas viscosity. If we apply this scaling law to the evolution of $h_0$ post-transition, presented in figure 22(b), we obtain figure 22(c). Clearly, after an initial transient period, the post-transition evolution of the dimple height collapses for the three cases.

While having similar form, (3.33) is in contrast to the scaling law governing the dimple thickness for drop-solid impacts in capillary dominated regimes ($We<1$). Bouwhuis et al. (Reference Bouwhuis, van der Veen, Roeland, Tran, Keij, Winkels, Peters, van der Meer, Sun, Snoeijer and Lohse2012) and Klaseboer et al. (Reference Klaseboer, Manica and Chan2014) examined such low velocity drop-solid impacts to show that capillary cushioning results in ${H}\sim R{Ca}^{1/2}$. The regime of interest in our work corresponds to $We\sim {O}(10)$, equivalent to the inertial regime of Bouwhuis et al. (Reference Bouwhuis, van der Veen, Roeland, Tran, Keij, Winkels, Peters, van der Meer, Sun, Snoeijer and Lohse2012) and Klaseboer et al. (Reference Klaseboer, Manica and Chan2014). For the drop-pool problem in this regime, we have found that capillary effects only start to become important as the sharp cusp is regularized to cause transition, after which they affect the evolution of the dimple. The dimple height during spreading is thus governed by a capillary scaling for pressure which conceptually links our analysis to the capillary dominated cushioning regime of $We<1$ drop-solid impacts. The contrast between the scaling law for post-transition dimple height suggested here (3.33) and that of $We<1$ drop-solid impacts (Bouwhuis et al. Reference Bouwhuis, van der Veen, Roeland, Tran, Keij, Winkels, Peters, van der Meer, Sun, Snoeijer and Lohse2012; Klaseboer et al. Reference Klaseboer, Manica and Chan2014), is due to the different scales for the lateral length and gas pressure gradients. Namely, in the early dynamics of these drop-solid impacts, $L\sim \sqrt {RH}$, as the drop is geometrically constrained to flatten its spherical shape upon impact on the solid, yielding capillary pressure gradients to scale with $\sigma {H}/{L}^3\sim \sigma /(R\sqrt {RH})$ (Bouwhuis et al. Reference Bouwhuis, van der Veen, Roeland, Tran, Keij, Winkels, Peters, van der Meer, Sun, Snoeijer and Lohse2012; Klaseboer et al. Reference Klaseboer, Manica and Chan2014). In contrast, as was shown in detail in this section, in the drop-pool problem, after the cusps are regularized during transition, as the drop penetrates into the pool, carrying the non-draining air film with itself, the air film length scale for both zones is $L\sim R$ and the pressure gradients in the inner region scale as $\sigma /{R}^2$. These differences in scales thus explain the difference in the exponent of $Ca$ in the scaling laws governing the dimple height in these two problems. Indeed, it is clear from figure 22(d) that the dimple height does not scale with $R{Ca}^{1/2}$ for the problem of interest in this work.

The dimple height ($h_{0}$) is an indicator of the film thickness in the inner region. However, to further test the asymptotic scaling law presented in (3.33), the full film thickness profile should be examined. In figures 23(a), 23(b) and 23(c), for the three water impact cases, we show the air film profile in dimensional form against $x/R$ for non-dimensional times of $tU/R=0.05, 0.065, 0.093$, respectively. These times respectively correspond to when the kink has not yet formed, the kink has formed, and late times when the film thickness at the kink is comparable to the minimum film thickness at the spreading front. Note that only results from two of the simulations can be seen for the latest time, as the simulations were not all stable until the same non-dimensional time. It can be seen from figures 23(d), 23(e) and 23(f) that if we scale the film thickness with $R{Ca}^{1/3}$, the profiles collapse very well in the inner region where the dimple exists, confirming (3.33). However, (3.33) does not collapse the outer region between the kink and spreading front. This was forecast above, as we alluded to the different nature between the pressure gradients in the two zones. Before deriving the scaling law for the outer zone, we note the invalidity of the ${H}\sim R{Ca}^{1/2}$ scaling law in the inner and outer regions from figures 23(g), 23(h) and 23(i).

Figure 23. Post-transition film thickness for various impact cases at non-dimensional times of $tU/R=0.05, 0.065, 0.093$, corresponding to the left, middle and right panels, plotted against $x/R$, (a–c) in SI units (d–f) scaled by $R{Ca}^{1/3}$ from (3.33), (g–i) scaled by $R{Ca}^{1/2}$, which holds for the dimple height in $We<1$ drop solid impacts (Bouwhuis et al. Reference Bouwhuis, van der Veen, Roeland, Tran, Keij, Winkels, Peters, van der Meer, Sun, Snoeijer and Lohse2012; Klaseboer et al. Reference Klaseboer, Manica and Chan2014) and (j–l) scaled by $R{St}^{-1/3}$ from (3.34).

As explained before, similar to the inner zone, the length scale (${L}_{out}$) and velocity scale for the outer zone scale with $R$ and $U$, respectively. However, the pressure gradients in these two regions scale differently, resulting in different scaling laws for their film thicknesses. The pressure in the outer region has an inertial scaling, $P_{g,max}\sim \rho {U}^2$, as was shown in figure 21(c). As a result, conservation of mass in the air film asymptotically gives $\rho {U}^{2}{H}_{out}^{3}/{R}^{2}\sim \mu _{g}U$, yielding

(3.34)\begin{equation} H_{out}\sim R{St}^{-1/3}. \end{equation}

Applying this asymptotic scaling to the simulation profiles gives very good collapses of the outer zones shown in figures 23(j), 23(k) and 23(l). Of course, this scaling fails at collapsing the inner zone. Moreover, the collapse of the instantaneous profiles in the inner (figures 23d, 23e and 23f) and outer (figures 23j, 23k and 23l) zones are both achieved by using $x/R$ for the lateral direction of the film, verifying that both zones have a lateral length that scales with $R$. Interestingly, based on (3.33) and (3.34), we have the following scaling relation for the ratio between the thickness of the outer zone and the inner zone (dimple height):

(3.35)\begin{equation} \frac{{H}_{out}}{{H}_{in}}\sim {We}^{-1/3}. \end{equation}

Equation (3.35) is in agreement with the intuition that capillary effects, which are stronger at lower $We$, result in more regularized and uniform films.

From a practical viewpoint, the scaling laws proposed in (3.33) and (3.34) can be utilized along with (3.30) for subgrid-scale modelling of air entrainment due to liquid-liquid impacts in turbulent breaking waves or other large-scale two-phase flows (Chan, Urzay & Moin Reference Chan, Urzay and Moin2018; Mirjalili, Chan & Mani Reference Mirjalili, Chan and Mani2018; Chan et al. Reference Chan, Mirjalili, Jain, Urzay, Mani and Moin2019). Before discussing viscous effects and drop-pool impacts for materials other than water, it is necessary to make some comments regarding the outcome of drop-pool impacts in the low $We$ regime studied here. In this section we observed from our simulation results that capillary transition is a ‘bottleneck’ for an impact in the sense that if the film's integrity survives attractive Van der Waals forces at transition time, it attains a regularized geometry with a continuously growing minimum film thickness afterwards. As indicated before, our lubrication zone, in which the gas pressure is computed has a fixed horizontal length of $0.9D$. This limits time integration to the final times shown in the results of this section, during which the film thicknesses never drop below ${h}_{VDW}$. As such, the final fate of these regularized unruptured films cannot be predicted via the specific numerical framework adopted in this work. In general, for low $We$ impacts that do not result in bouncing, experiments using viscous silicone oils have shown that rupture dynamics involve three-dimensional instabilities driven by Van der Waals forces (Thoroddsen et al. Reference Thoroddsen, Thoraval, Takehara and Etoh2012; Tran et al. Reference Tran, de Maleprade, Sun and Lohse2013). Such effects cannot be studied via two-dimensional BIM simulations. Furthermore, these experiments found that the rupture position exhibits a strong dependence on the viscosity of the liquid phase. As we will show in § 4, for the viscous oils used in the aforementioned experiments, liquid viscous forces dominate capillary forces during the cusp formation stage and cannot be neglected. As such, we do not expect to be able to capture the transition and post-transition evolution for such liquids within our framework. Quite possibly in fact, the observations about such viscous liquids during spreading and rupture (which have been found to be a function of liquid viscosity) cannot be safely extended to water systems, which are capillary dominated. In summary, for systems that are capillary dominated, our findings shed light on the transition to contact-free drop spreading on the pool surface and the early post-transition dynamics.