3.1. Review of previous results

The predictions in Mandre & Brenner (Reference Mandre and Brenner2012), later extended by Chantelot & Lohse (Reference Chantelot and Lohse2021, Reference Chantelot and Lohse2023) to the case of superheated substrates, are based on the following idea: the minimum gas film thickness is attained when a self-similar solution describing the pre-impact stage at the region where the distance to the wall is minimum, fails to predict the flow for $\tau \geq \tau ^*$ because the capillary and convective terms in the momentum equation, initially neglected, become of the order of the dominant terms in the approximate solution. Hence, as far as we understand, the results in Mandre et al. (Reference Mandre, Mani and Brenner2009), Mandre & Brenner (Reference Mandre and Brenner2012) and Chantelot & Lohse (Reference Chantelot and Lohse2021, Reference Chantelot and Lohse2023) have been deduced using an argument that neither describes nor identifies the physical mechanism that prevents the contact between the liquid and the wall.

In contrast, the physical model in Gordillo & Riboux (Reference Gordillo and Riboux2022) describes the lubricated impact of a drop over a wall for instants of time $\tau \geq \tau ^*$. The physical idea behind the predictions in Gordillo & Riboux (Reference Gordillo and Riboux2022) relies on the well-known lubrication mechanism depicted in figure 5, where the gas velocity field calculated numerically along the region where the gas pressure is maximum, is represented in a frame of reference moving with the wetting velocity at which the local maximum pressure propagates radially outwards, namely (see figures 2 and 3)

(3.1)\begin{equation} V_m(s)=\frac{{\rm d}a}{{\rm d}t}=\frac{U}{2}\sqrt{\frac{3}{s}} , \end{equation}

where the subscript $m$ is used to denote the values of quantities particularized at $r=a(s)$; see (2.7). Figure 5 shows that in the moving frame of reference, the gas velocity field can be expressed as the superposition of the Poiseuille (parabolic) velocity profile induced by the favourable pressure gradient pointing radially outwards, towards the atmosphere (see figure 2b), plus the Couette (linear) velocity profile caused by the relative motion between the point of maximum pressure and the wall, which is directed towards the axis of symmetry. Notice that the point of maximum pressure is attained at the radial position where the liquid interfacial velocity in the laboratory frame of reference equals the velocity of the moving frame of reference; see e.g. Wagner (Reference Wagner1932) and Gordillo & Riboux (Reference Gordillo and Riboux2022).

Figure 5.

Radial component of the dimensionless gas velocity field, $u_r/U$, represented in the frame of reference moving with the wetting velocity $V_m$ given in (3.3) for three different values of $(r-\bar {a})/\bar {h}_a$ and two different instants of time: (a) $\tau =20$ and (b) $\tau =40$. Here, $\bar {a}\approx R \sqrt {3 s}$ indicates the radial position where the maximum gas pressure is attained (see figure 3) and $\bar {h}_a=2/(9{\rm \pi} )\,\bar {a}^3\approx h_{a,m}$, with $h_{a,m}$ given in (3.4). Here, $We=12$, $St=2.60\times 10^4$, whereas $z(1)$ indicates the vertical coordinate of the interface at the minimum value of $(r-\bar {a})/\bar {h}_a$ represented in the figure.

Hence, for the case of isothermal substrates, the distance between the drop and the wall at $r=a(s)$ during the instants close to the one for which the minimum film thickness is attained, can be quantified through the equation (Gordillo & Riboux Reference Gordillo and Riboux2022)

(3.2)\begin{equation} {-}\frac{h_m^3}{12\mu_a}\,\frac{\partial p}{\partial r}-\frac{V_m\,h_m}{2}\approx 0 .\end{equation}

Non-continuum effects, which could have been retained in (3.2) by considering values for the slip lengths at the interface and at the wall different from zero (see e.g. Duchemin & Josserand (Reference Duchemin and Josserand2012) and Riboux & Gordillo (Reference Riboux and Gordillo2014)), will be considered in § 4 using the approach detailed in Li (Reference Li2016) in his numerical study of the head-on collision of drops. This approximation consists in modifying the value of the actual viscosity by using the equation for the effective viscosity deduced in Zhang & Law (Reference Zhang and Law2011), which depends explicitly on the Knudsen number defined in terms of the gas film thickness. While gas kinetic effects need to be retained in order to compare our predictions with experiments, van der Waals effects can be neglected safely in the modelling because these forces become relevant only when $h_m\lesssim 20$ nm (Sprittles Reference Sprittles2024), namely, for values of the gas film thicknesses that are well below those measured experimentally by de Ruiter et al. (Reference de Ruiter, Oh, van den Ende and Mugele2012) for the case of isothermal impacts, and by Chantelot & Lohse (Reference Chantelot and Lohse2021, Reference Chantelot and Lohse2023) for the case of drops impacting the wall in the dynamic Leidenfrost regime.

Notice that (3.2) expresses that the value of the Poiseuille flow rate per unit length induced by the large pressure gradient generated around $r=a(s)$ (see figure 3) needs to be balanced by the Couette flow because, otherwise, the gas beneath the region where the pressure is maximum would flow radially outwards, emptying this volume, and consequently the liquid would make contact with the wall.

In order to deduce an equation for $h_m$, the next step that we followed in Gordillo & Riboux (Reference Gordillo and Riboux2022) was to make use of the fact that the vertical interfacial velocities during $\tau \geq \tau ^*$ are much smaller than the impact velocity, i.e. $(1/U) \partial h/\partial t\ll 1$, and also that the gas film is slender, $\partial h/\partial r\ll 1$. In this way, since the Reynolds number verifies $Re=St\, \mu _a/\mu \gg 1$, and hence the production of vorticity at the gas–liquid interface is confined within small boundary layers, the liquid velocity and pressure fields can be approximated by the irrotational values calculated using Wagner's theoretical framework Wagner (Reference Wagner1932). At this point, notice that the condition $(1/U) \partial h/\partial t\ll 1$ also implies that $\ell \,{\rm d}h_m/ {\rm d}t\ll V_m h_m$ – with $\ell$ indicating the characteristic length along which $h_m$ varies, of the order of $\sim h_m$ – due to the fact that $\ell >h_m$ and also because $V_m\propto U\,St^{1/3}\gg U$, this being the reason why the term $\ell d h_m/ dt$ has been neglected in the mass balance (3.2).

Therefore, Wagner's potential flow theory (Wagner Reference Wagner1932) predicts that the component of the liquid velocity parallel to the wall at $r\approx a(\tau )$ (see also Appendix C, where we discuss the role played by the gas shear stresses) can be expressed as

(3.3)\begin{equation} V_m(\tau)= U\,\frac{\sqrt{3}}{2}\,\tau^{-1/2}\,St^{1/3} ,\end{equation}

where use of (2.6a,b) and (3.1) has been made, and Wagner's theory also predicts that the length along which the liquid interfacial velocity changes of the order of $V_m$ is

(3.4)\begin{equation} h_{a,m}(\tau) = R\,\frac{\sqrt{12\tau^3}}{3{\rm \pi}}\,St^{-1};\end{equation}

see Gordillo & Riboux (Reference Gordillo and Riboux2022) for details. Moreover, Wagner's theoretical framework (Wagner Reference Wagner1932) also reveals that the tangential liquid velocity changes from $V_m$ to $2 V_m$ in a region of width $h_{a,m}$ located around $r=a(s)$, and therefore in the moving frame of reference where (3.2) applies, there exists a stagnation point of the flow. Then, by virtue of the Euler–Bernoulli equation and of fact that the local flow is quasi-steady in the moving frame of reference, the relative pressure with respect to that of the surrounding atmosphere at the stagnation point reads

(3.5)\begin{equation} \Delta p_m(\tau)=\frac{1}{2}\,\rho V^2_m=\frac{3}{8\tau}\,\rho U^2\,St^{2/3} .\end{equation}

Consequently, the expression for $h_m(\tau )$ follows from (3.2) once we approximate the pressure gradient as (see figures 2 and 3)

(3.6)\begin{equation} {-}\frac{\partial p}{\partial r}\sim \frac{\Delta p_m(\tau)}{\ell(\tau)}=\frac{1}{2}\,\rho\,\frac{V^2_m}{\ell(\tau)}, \end{equation}

with $V_m(\tau )$ and $\Delta p_m(\tau )$ given in (3.3) and (3.5), respectively. In (3.6), $\ell (\tau )$ refers to the characteristic length where the gas pressure varies of the order of $\sim O(\Delta p_m)$, and hence (Gordillo & Riboux Reference Gordillo and Riboux2022)

(3.7)\begin{equation} \left. \begin{aligned} & \mathrm{if}\ \ell_c< h_{a,m},\text{ then }\ell=h_{a,m};\\ & \ \mathrm{if}\ \ell_c>h_{a,m},\text{ then }\ell=\ell_c . \end{aligned} \right\} \end{equation}

Indeed, in (3.7), $\ell _c$ indicates the characteristic capillary length where capillary forces balance the pressure in (3.5), namely (Gordillo & Riboux Reference Gordillo and Riboux2022)

(3.8)\begin{equation} \frac{\sigma h_m}{\ell^2_c}\propto \Delta p_m\Rightarrow \ell_c(\tau)\propto\sqrt{\frac{\sigma h_m}{\Delta p_m}}=R\,We^{-1/2}\sqrt{\frac{h_{m}}{R}} \left(\frac{V_m}{U}\right)^{-1} .\end{equation}

The substitution of (3.3)–(3.6) into (3.2) yields the following equation for $h_m/R$:

(3.9)\begin{equation} y\left(-1+\frac{1}{6}\,y\right)=0,\end{equation}

with

(3.10)\begin{equation} y=\frac{St}{2(\ell/R)}\,\frac{V_m}{U} \left(\frac{h_m}{R}\right)^2, \end{equation}

and therefore,

(3.11)\begin{equation} y=6\quad\mathrm{and} \quad\frac{h_m}{R}=\left(12\,St^{-1}\left(\frac{V_m}{U}\right)^{-1}\frac{\ell}{R}\right)^{1/2} .\end{equation}

Then, by virtue of (3.7), if $\ell _c>h_{a,m}$, then $\ell /R=\ell _c/R$, hence (3.8) and (3.11) yield the following expression for $h_m/R$ under the so-called capillary regime:

(3.12)\begin{equation} \frac{h_{m,c}}{R}\propto y^{2/3} \tau^{2/3}\,We^{-1/3}\,St^{-10/9}.\end{equation}

From this, using (3.8), we deduce the following expression for $\ell _c/R$:

(3.13)\begin{equation} \frac{\ell_c}{R}\propto y^{1/3} \tau^{5/6}\,We^{-2/3}\,St^{-8/9} .\end{equation}

In contrast, if $h_{a,m}>\ell _c$, then $\ell /R=h_{a,m}/R$ (see (3.4) and (3.7)), and the expression for $h_m/R$ in the so-called inertial regime follows from the definition of $y$ in (3.10):

(3.14)\begin{equation} \frac{h_{m,i}}{R}\propto y^{1/2} \tau\,St^{-7/6} .\end{equation}

Then the parameter controlling whether the minimum film thickness is given by either the capillary equation (3.12) or the inertial equation (3.14) is the ratio (see (3.4), (3.7) and (3.13))

(3.15)\begin{equation} \frac{h_{a,m}}{\ell_c}= We^{2/3}\,y^{-1/3}\,St^{-1/9}\,\tau^{2/3}=\bar{\xi}^{2/3} ,\end{equation}

where

(3.16)\begin{equation} \bar{\xi}= \tau \xi,\quad\mathrm{with}\ \xi=We\,St^{-1/6}\,y^{-1/2} \end{equation}

and, consequently, all the results deduced above can be summarized, making use of (3.6), as follows.

(3.17)\begin{align} \left. \begin{aligned} & \mathrm{If}\ \xi\lesssim \xi^*,\text{ then }\frac{h_{m,c}}{R}=A y^{2/3}\tau^{2/3}\,We^{-1/3}\,St^{-10/9}\text{ and } {-\frac{\partial p}{\partial r}}\propto\frac{\rho U^2}{R}\,\tau^{-11/6}\,St^{14/9}\,We^{2/3}\,y^{-1/3}. \\ & \mathrm{If}\ \xi\gtrsim \xi^*,\text{ then }\frac{h_{m,i}}{R}=B y^{1/2} \tau\,St^{-7/6}\text{ and }{-\frac{\partial p}{\partial r}}\propto \frac{\rho U^2}{R}\,\tau^{-5/2}\,St^{5/3}. \end{aligned} \right\} \end{align}

Here, $A$ and $B$ are order unity prefactors to be determined in § 3.2, and the corresponding expressions for the pressure gradients have been obtained making use of (3.6)–(3.7). Indeed, the substitution of the equations for $V_m/U$ and $\ell /R$ given in (3.3) and (3.7) into (3.11) would give $A\approx 6$ and $B\approx 2$, but these are only approximate values, which will be quantified accurately in § 3.2 using the results of the numerical simulations. Let us anticipate here that the analysis in § 3.2 will also reveal that the correct value of $\xi ^*$ – namely, of the threshold value of $\xi$ separating the capillary and inertial regimes – is $\xi ^*\approx 3.5$, which is substantially larger than the ad hoc threshold value $\xi ^*=6^{-1/2}$ that we used in Gordillo & Riboux (Reference Gordillo and Riboux2022).

Notice that the result corresponding to the capillary-dominated regime in (3.17) was already obtained by Duchemin & Josserand (Reference Duchemin and Josserand2011), but using a type of approach different to the one followed here. Indeed, among other things, the lubrication equations in Duchemin & Josserand (Reference Duchemin and Josserand2011) do not satisfy the continuity of the tangential velocities at the interface, therefore the term representing the Couette contribution to the flow rate is missing in their approximate description, this being a key ingredient of the physical model presented in Gordillo & Riboux (Reference Gordillo and Riboux2022), based on the classical lubrication mechanism illustrated in figure 5, which we improve here by retaining gas kinetic effects, as will be detailed below. In addition, here we extend the analysis corresponding to the capillary and inertial regimes in Gordillo & Riboux (Reference Gordillo and Riboux2022) with the purpose of describing the so-called lift-off mechanism described by Kolinski et al. (Reference Kolinski, Mahadevan and Rubinstein2014b); hence here we include the dependence with time in the equations for the minimum film thickness. Moreover, the values of the prefactors $A$ and $B$ in (3.17) corresponding to the case of low viscous liquids will also be provided here; indeed, in view of the results in Mishra et al. (Reference Mishra, Rubinstein and Rycroft2022), these two values could also depend on the viscosity ratio.

3.2. Comparison between predictions and the numerical results

The two expressions for the minimum film thickness in (3.17) depend on the value of the parameter $\bar {\xi }$ defined in (3.15)–(3.16), which expresses a measure of the relative importance between the capillary and the dynamic pressure given in (3.5). In this way, values of the parameter $\xi$ such that $\xi >\xi ^*$ indicate that capillary effects are subdominant and the impact can be considered as inertial, whereas in the opposite limit, capillary effects can no longer be neglected, and the drop impacts the wall in the so-called capillary regime. Likewise, Mandre & Brenner (Reference Mandre and Brenner2012) deduced the following equations, analogous to those given in (3.17) depending on whether or not capillary effects can be neglected.

(3.18)\begin{equation} \left. \begin{aligned} & \text{Capillary regime:}\quad \frac{h_{mc,MB}}{R}\propto We^{-2/3}\,St^{-8/9}.\\ & \text{Inertial regime:}\quad \frac{h_{mi,MB}}{R}\propto St^{-4/3} . \end{aligned} \right\} \end{equation}

In order to decipher which of the two different predictions is in closer agreement with the numerical results – namely, either that deduced by Mandre & Brenner (Reference Mandre and Brenner2012) (and later used by de Ruiter et al. Reference de Ruiter, Oh, van den Ende and Mugele2012; Chantelot & Lohse Reference Chantelot and Lohse2023) or the more recent ones in Gordillo & Riboux (Reference Gordillo and Riboux2022) – we compare in figure 6 the ratios between the numerical values of the minimum gas film thickness calculated at $\tau ^*=12$, $h_{min}(\tau =12)$, and the predictions given in (3.17) and (3.18). The results depicted in figure 6, where the proportionality constants in (3.17) and (3.18) have been selected in order to maximize the agreement with the numerical values, reveal that our predictions in (3.17) reproduce the numerical results better using, in addition, prefactors of order unity, namely $A=3.5$ and $B=1$, which are similar to the ones obtained by substituting the values of $V_m/U$ and $\ell /R$ given in (3.3) and (3.7) into (3.11), i.e. $A\approx 6$ and $B\approx 2$.

Figure 6.

Comparison between the numerical values of the minimum gas layer thickness calculated as $h_{min}(\tau ^*=12)$ and the predicted values $h_{th}$, given either in (3.18) (Mandre & Brenner Reference Mandre and Brenner2012; Chantelot & Lohse Reference Chantelot and Lohse2023) or in (3.17) (Gordillo & Riboux Reference Gordillo and Riboux2022) for the case of isothermal substrates, $y=6$. The predicted values in (a,b) correspond, respectively, to the capillary and inertial regimes in (3.17) and (3.18). The values predicted by Mandre & Brenner (Reference Mandre and Brenner2012) (M&B) and Chantelot & Lohse (Reference Chantelot and Lohse2023) (C&L) are represented using open symbols ($\circ$): in (a), $h_{th}=17.0 R\,We^{-2/3}\,St^{-8/9}$, whereas in (b), $h_{th}=180.0 R\,St^{-4/3}$. The values corresponding to the predictions in Gordillo & Riboux (Reference Gordillo and Riboux2022) are represented using full symbols ($\bullet$): in (a), $h_{th}=3.5 R\tau ^{*2/3}y^{2/3}\,St^{-10/9}\,We^{-1/3}$, whereas in (b), $h_{th}=1.0 R\tau ^*\,St^{-7/6}\,y^{1/2}$. Dashed lines indicate variations $h_m/h_{th}=1.0\pm 0.2$. (c) Comparison between the numerical results and the theoretical predictions in (3.17) as a function of $\xi =We\,St^{-1/6}\,y^{-1/2}$, with $h_{th}$ calculated as $h_{th}=3.5 R\tau ^{*2/3}y^{2/3}\,St^{-10/9}\,We^{-1/3}$ for the case of the capillary regime, or as $h_{th}=1.0 R\tau ^*\,St^{-7/6}\,y^{1/2}$ for the case of the inertial regime. In (c), dashed lines indicate variations $h_m/h_{th}=1.0\pm 0.1$.

As pointed out already, figure 7 shows that the values of $h_{min}$ and $h_m$ are very similar to each other up to $\tau =\tau ^*$. Beyond that instant of time, the values of $h_{min}$ keep on decreasing but very smoothly, so that the curve corresponding to $h_m$ lies above the one for $h_{min}$ for $\tau >\tau ^*$. Figure 7 also compares the results of numerical simulations with the time-dependent predictions given in (3.17), once the origin of times is fixed appropriately. Indeed, notice that the virtual origin of times is set in figure 7 at a dimensionless instant $\tau '=-3$, a value that is consistent with the time taken to entrap the central bubble. This result, which could be interpreted as if the impact began before the drop reaches the wall, shares similarities with previous findings on the subject; see e.g. Peters et al. (Reference Peters, van der Meer and Gordillo2013), where both numerical and experimental results reveal that the gas cushioning effects can be quantified using a virtual origin of times, or the numerical results in Ross & Hicks (Reference Ross and Hicks2019), which show that a two-dimensional impacting solid deforms the interface before contacting the liquid.

Figure 7.

Comparison between the numerical values of $h_{min}(\tau )$ (empty symbols) and $h_m=h(r/R=\sqrt {3 s},\tau )$ (full symbols), with the values predicted in (3.17) for the case of isothermal substrates, $y=6$, using the following values for the prefactors: $h_m/R=2.42\,y^{2/3}(\tau -\tau ')^{2/3}\,We^{-1/3}\,St^{-10/9}$ (solid lines) or $h_m/R=0.48\,y^{1/2}(\tau -\tau ')\,St^{-7/6}$ (dashed lines), with $\tau '=-3$. In each of the plots, the thin vertical dashed line shows the value $\tau =12$. Values are: (a) $We=4$, $St=3.0\times 10^4$; (b) $We=12$, $St=2.6\times 10^4$; (c) $We=12$, $St=1.2\times 10^4$; (d) $We=22$, $St=2.1\times 10^4$; (e) $We=36$, $St=2.6\times 10^4$; (f) $We=48$, $St=4.5\times 10^4$; (g) $We=48$, $St=3.6\times 10^4$; (h) $We=48$, $St=1.8\times 10^4$; (i) $We=48$, $St=1.2\times 10^4$; (j) $We=60$, $St=1.8\times 10^4$.

Interestingly, the results in figure 7 are the manifestation of the so-called ‘lift-off instability’, described empirically by Kolinski et al. (Reference Kolinski, Mahadevan and Rubinstein2014b) and analysed very recently using two-dimensional numerical simulations by Mishra et al. (Reference Mishra, Rubinstein and Rycroft2022). Indeed, figure 7 shows that, consistently with our predictions, the growth in time of the minimum film thickness can be quantified using either the capillary or the inertial limits of (3.17), depending on whether the value of the parameter $\xi$ is larger or smaller than $\xi ^*\approx 3.5$. The fair agreement between predictions and numerical results in figure 7 indicates that the lift-off instability corresponding to low values of the liquid to gas viscosity ratio can be quantified using the equations below.

(3.19)\begin{equation} \left. \begin{aligned} & \mathrm{If}\ \xi\lesssim 3.5,\text{ then }\frac{h_{m,c}}{R}=2.42 y^{2/3}(\tau+3)^{2/3}\, We^{-1/3}\,St^{-10/9}.\\ & \mathrm{If}\ \xi\gtrsim 3.5,\text{ then }\frac{h_{m,i}}{R}=0.48 y^{1/2} (\tau+3)\, St^{-7/6}. \end{aligned} \right\} \end{equation}

Notice that the values of the prefactors in (3.19), $A=2.42$ and $B=0.48$, are not the same as those deduced from figure 6, $A=3.5$ and $B=1$; the reason for these slight differences lies in the fact that the numerical values in figure 6 correspond to $h_{min}(\tau =12)$, namely, to an instant of time around which the transition between the pre-impact and post-impact stages depicted in figure 4 takes place, whereas the values predicted in (3.19) correspond to $h_{m}(\tau >12)$, namely, within the post-impact stage described by our model; see figure 7. Then the small differences in the values of the proportionality constants are caused by the fact that the results in figure 6 correspond to an instant of time that is slightly smaller than that from which our modelling starts being valid, as it is evidenced by the fact that the actual function $h_{m}(\tau )$ depicted in figure 7 possesses a minimum at $\tau \approx 12$, whereas our (3.19) predict that $h_m$ increases monotonically with $\tau$. In fact, the results in figure 6 indicate that the expressions for $h_m$ corresponding to the post-impact stage deduced here can be used to describe the minimum film thickness at the instant of time $\tau =12$, at which the analysis is not strictly valid, by simply introducing slight changes in the values of the constants $A$ and $B$ in (3.19). Let us point out here that we could not include in figure 7 the analogous predictions by Mandre & Brenner (Reference Mandre and Brenner2012) and Chantelot & Lohse (Reference Chantelot and Lohse2023) because these authors limited their analysis to the instant when the minimum film thickness is attained, therefore their results do not depend on time.

Appendix B compares the numerical values of the pressure and the pressure gradient with the predictions given in § 3.1, providing further support to our physical description.

4.1. Modelling the vapour production and the inclusion of gas kinetic effects

A previous step before comparing our predictions with the experimental measurements is to add to (3.2) the term representing the flow rate per unit length $q_v$ produced by the evaporation of the liquid. Indeed, as explained in the Introduction, we will consider here not only the case of isothermal impacts already analysed in §§ 2 and 3, but also the case of drops impacting over superheated substrates, i.e. substrates with temperature $T_s>T_b$, with $T_b$ indicating the boiling temperature of the liquid. Then, in order to predict the experimental measurements for the minimum film thickness reported in Chantelot & Lohse (Reference Chantelot and Lohse2021, Reference Chantelot and Lohse2023), where $\Delta T=T_s-T_b$ is varied within an ample range of values, we make use here of the result in Sobac et al. (Reference Sobac, Rednikov, Dorbolo and Colinet2014), which provided the following expression for the flow rate per unit length of vapour produced in a region of length $\ell$, where the distance between the liquid and the wall is $h_m$:

(4.1)\begin{equation} q_v\approx \frac{k_v\,\Delta T}{\rho_v\mathcal{L}}\,\frac{\ell}{h_m} . \end{equation}

In (4.1), $\mathcal {L}$ refers to the latent heat of vaporization of the liquid, and the viscosity, density and thermal conductivity of the vapour will be denoted in what follows as $\mu _{v}$, $\rho _{v}$ and $k_{v}$. Notice that (4.1) expresses the balance between the conductive heat flux from the substrate into the drop and the evaporation rate of the liquid at the interface; indeed, we justified in Gordillo & Riboux (Reference Gordillo and Riboux2022) that the heat flux across the liquid thermal boundary layer can be neglected for the particular case of the experiments reported in Chantelot & Lohse (Reference Chantelot and Lohse2021, Reference Chantelot and Lohse2023).

Therefore, the equation analogous to (3.2) for the case of drops impacting over superheated substrates reads

(4.2)\begin{equation} {-}\frac{h_m^3}{12\mu_v}\,\frac{\partial p}{\partial r}-\frac{V_m\,h_m}{2}\approx \frac{k_v\,\Delta T}{\rho_v \mathcal{L}}\,\frac{\ell}{h_m},\end{equation}

where we have made use of (4.1). The substitution of (3.3)–(3.6) into (4.2) yields the following equation for $h_m/R$:

(4.3)\begin{equation} y\left(-1+\frac{1}{6}\,\frac{\mu_a}{\mu_v}\,y\right)=\beta^*, \end{equation}

with

(4.4)\begin{equation} y=\frac{St}{2(\ell/R)}\,\frac{V_m}{U} \left(\frac{h_m}{R}\right)^2\quad\mathrm{and}\quad \beta^* =\beta\left(\frac{\rho}{\rho_v}\right)\left(\frac{\mu_v}{\mu_a}\right),\quad\mathrm{where}\ \beta=\frac{k_v\,\Delta T}{\mu_v\mathcal{L}} . \end{equation}

Consequently,

(4.5)\begin{equation} \frac{h_m}{R}=\left(2y\,St^{-1}\left(\frac{V_m}{U}\right)^{-1}\frac{\ell}{R}\right)^{1/2} ,\end{equation}

with

(4.6)\begin{equation} y=3\left(\frac{\mu_v}{\mu_a}\right)\left[1+\sqrt{1+\frac{2\beta^*}{3}\left(\frac{\mu_a}{\mu_v}\right)}\right] .\end{equation}

The substitution of (4.6) and of either (3.4) or (3.8) into (4.5) provides the same expressions as those given in (3.17), which are then valid to describe the skating of drops over either a gas or a vapour layer, the only difference being that $y=6$ for the case of isothermal impacts, whereas $y$ is given by (4.4) and (4.6) for the cases of drops impacting a substrate in the dynamic Leidenfrost regime. Hence our physical description differs substantially from that of Chantelot & Lohse (Reference Chantelot and Lohse2021, Reference Chantelot and Lohse2023), who deduce different equations depending on whether the substrate is superheated or not, therefore the isothermal case cannot be recovered using their results corresponding to superheated substrates in the limit in which the production of vapour tends to zero.

The recent review on the subject by Sprittles (Reference Sprittles2024) states clearly that gas kinetic effects cannot be neglected in the description of the head-on collision of drops (Li Reference Li2016) or in the impact of drops on a substrate (Riboux & Gordillo Reference Riboux and Gordillo2014; Chubynsky et al. Reference Chubynsky, Belousov, Lockerby and Sprittles2020) if the value of the Knudsen number, defined as

(4.7)\begin{equation} Kn=\frac{\lambda}{h_m},\end{equation}

with $\lambda$ denoting the mean free path of the gas, becomes $Kn\gtrsim 0.1$, which is the case of interest here. Indeed, the typical values of the gas film thickness in the experiments reported by de Ruiter et al. (Reference de Ruiter, Oh, van den Ende and Mugele2012) and Chantelot & Lohse (Reference Chantelot and Lohse2021, Reference Chantelot and Lohse2023) vary between $10^3$ and $10^2$ nanometres, whereas the values of the mean free path of air and ethanol at normal conditions are, respectively,

(4.8a,b)\begin{equation} \lambda_{a}\approx 69\, \mathrm{nm}\quad \mathrm{and}\quad \lambda_{eth}\approx 50\ \mathrm{nm}. \end{equation}

Then, in order to account for gas kinetic effects, we define an effective gas viscosity $\mu ^*(Kn)$ (Sprittles Reference Sprittles2024) and make use of the expression deduced by Zhang & Law (Reference Zhang and Law2011), employed successfully by Li (Reference Li2016) in his study of the head-on collision of drops:

(4.9)\begin{equation} \mu^*=\frac{\mu}{1 + 6.0966\,Kn + 0.9650\,Kn^2 + 0.6967\,Kn^3},\end{equation}

where $\mu$ refers to the value of the actual gas viscosity. Notice that for the case of drops impacting a non-heated substrate, $\mu _v$ in (4.6) refers to the effective air viscosity, namely, $\mu _v=\mu ^*_a$ (see (4.9)), whereas the Knudsen number defined in (4.7) is calculated using the value of the mean free path given by

(4.10)\begin{equation} \lambda=\lambda_a\,\frac{P_a}{P_a+\Delta p_m} , \end{equation}

with $\lambda _a$ and $\Delta p_m$ given, respectively, in (4.8a,b) and (3.5). For the case of drops impacting a superheated substrate, $\mu _v=\mu ^*_{eth}$ and

(4.11)\begin{equation} \lambda=\lambda_{eth}\,\frac{P_a}{P_a+\Delta p_m}\,\frac{273+0.5\left(T_b+T_s\right)}{T_a}, \end{equation}

with $T_a=298$ K, and $T_b=78$ and $T_s$ refer to the values in Celsius of the boiling temperature of ethanol and of the substrate temperature, respectively. For the case of drops impacting a superheated substrate, here we also include gas kinetic effects in the heat transfer from the wall into the liquid, making use of the effective heat conductivity for the vapour given by (Sharipov et al. Reference Sharipov, Cumin and Kalempa2007)

(4.12)\begin{equation} k^*_v=\frac{k_v}{1+3.91\,Kn} .\end{equation}

As pointed out in the Introduction, van der Waals effects are not retained in the analysis because in all the experimental results reported by de Ruiter et al. (Reference de Ruiter, Oh, van den Ende and Mugele2012) and Chantelot & Lohse (Reference Chantelot and Lohse2021, Reference Chantelot and Lohse2023), the gas film thickness is well above 20 nm, which is the length scale below which these forces become relevant (Sprittles Reference Sprittles2024).

4.2. Comparison with experiments

The comparison with the numerical results in § 3 confirms our physical description, and it is now our purpose in this subsection to check whether our results, once gas kinetic effects are taken into consideration, can also be used to predict the experimental data reported by de Ruiter et al. (Reference de Ruiter, Oh, van den Ende and Mugele2012) and Chantelot & Lohse (Reference Chantelot and Lohse2021, Reference Chantelot and Lohse2023) for drops impacting over either isothermal or superheated substrates.

The equations given in (4.6) and (3.17) can be used to predict the minimum gas film thickness for arbitrary values of $T_s$. However, the unified description for the skating of a drop over a gas or vapour film provided by (4.6) and (3.17) also requires us to introduce the effect of the vapour produced at the dimple on the instant of time $\tau ^*$ at which the central bubble is formed, i.e. at the instant when the minimum film thickness is attained at $r\approx R\,St^{-1/3}\sqrt {3 \tau ^*}$.

Then, for the case of superheated substrates, the term on the left-hand side of the mass conservation equation (2.2) needs to be modified in order to take into account the evaporation of the liquid. Consequently, in this case, the flow rate that needs to be evacuated radially outwards from the axis of symmetry is ${\rm \pi} (Rh_d)(U+k_v\,\Delta T/(\rho _v\,h_d\mathcal {L}))$. Therefore, expressing $h_d/R=\tau ^*\,St^{-2/3}$ and following the same steps as those detailed in § 3, the mass balance (2.2) provides the following equation for $\tau ^*$ (Gordillo & Riboux Reference Gordillo and Riboux2022):

(4.13)\begin{equation} \tau^{*5/2}=12.4^{3/2}\left(\tau^*+\beta^*\,St^{-1/3}\right) ,\end{equation}

with $\beta ^*$ defined in (4.4). Notice that (4.13) recovers the value of $\tau ^*\approx 12$ corresponding to the case of isothermal substrates, for which $\beta ^*=0$.

Before comparing our predictions with the experimental values provided in de Ruiter et al. (Reference de Ruiter, Oh, van den Ende and Mugele2012) and Chantelot & Lohse (Reference Chantelot and Lohse2021, Reference Chantelot and Lohse2023), it is first necessary to express the real temperature-dependent material properties of the gas and of the vapour as functions of the substrate temperature. This is done here using the Python routines provided as supplementary material available at https://doi.org/10.1017/jfm.2024.20, which implement (3.17), (4.6) and (4.7)–(4.12) and make use of the values $\sigma =22\times 10^{-3}\ \textrm {N}\ \textrm {m}^{-1}$ or $\sigma =17\times 10^{-3}\ \textrm {N}\ \textrm {m}^{-1}$ for the interfacial tension coefficient of ethanol at either room or boiling temperature, with these values taken from www.ddbst.com. Notice that all the material properties are quantified at the mean temperature $(T_b+T_s)/2$ as detailed in Gordillo & Riboux (Reference Gordillo and Riboux2022), and this is reflected in the Python routines provided as supplementary material. Let us point out here that we have not considered the effect of Marangoni stresses in our physical model because the liquid located at a distance $h_m$ from the wall is evaporating along a region of length $\ell$, therefore the interfacial temperature remains constant and equal to the boiling temperature of the liquid at the region of interest here, namely, where the minimum film thickness is attained.

Figure 8 reveals that the minimum gas film thickness can be predicted using the equations corresponding to the capillary or inertial limits in (3.17) and (4.4)–(4.12) with relative errors ${\sim }30\,\%$. For the case of isothermal substrates, the predicted value for the minimum film height has been calculated in figure 8 retaining gas kinetic effects and using the capillary limit in (3.17) with $A=3.5$, namely, the value deduced from figure 6 for the case of drops impacting a wall at room temperature in the ideal case $Kn=0$. Hence for the case of substrates at room temperature, we conclude that the experimental data can be approximated using the equation (see (3.17))

(4.14)\begin{equation} \frac{h_m}{R}\approx 3.5\times \left(6\,\frac{\mu^*_a}{\mu_a}\right)^{2/3}12^{2/3}\,We^{-1/3}\, St^{-10/9}, \end{equation}

with $\mu ^*_a=\mu ^*$ defined in (4.9).

Figure 8.

(a) Comparison between the predictions given in (4.14)–(4.16) and the values for the minimum thickness of the air or vapour layers measured by de Ruiter et al. (Reference de Ruiter, Oh, van den Ende and Mugele2012) (DeR) and Chantelot & Lohse (Reference Chantelot and Lohse2023) (C&L) for different values of the impact velocity $U$ and for different values of the substrate temperature $T_s$. Here, the values of the effective gas viscosity and the effective gas conductivity have been modified taking into account kinetic effects through (4.7)–(4.12). Solid lines represent the predictions for $h_m$ corresponding to the capillary regime, whereas the predictions for $h_m$ in the inertial regime are represented using dashed lines.(b) Comparison between the predicted and measured values of $h_m$ in (a) as a function of $\xi$. (c) The experimental data are compared here only with the predictions given by (4.16). In this case, the value of the prefactor is 1.15 instead of 1.25. (d) Comparison between the predicted and measured values of $h_m$ in (c) as a function of $\xi$. The dashed horizontal lines in (b,d) are placed at $1\pm 0.3$.

Figure 8 also compares the experimental measurements in Chantelot & Lohse (Reference Chantelot and Lohse2021, Reference Chantelot and Lohse2023) with our predictions. In this case, since the vapour layer prevents contact between the liquid and the wall, a transition between the capillary and inertial regimes in (3.17) is observed when the impact velocity increases. In this case, the equations for the minimum film thickness used in the comparisons of figure 8 are (see (3.17))

(4.15)\begin{equation} \frac{h_m}{R}\approx 2.3\,y^{2/3}\,\tau^{*2/3}\,We^{-1/3}\,St^{-10/9} \end{equation}

and

(4.16)\begin{equation} \frac{h_m}{R}\approx 1.25\tau^*\,St^{-7/6}\,y^{1/2}\end{equation}

for the capillary and inertial regimes, respectively. In (4.15)–(4.16), $y$ is given by (4.6), $\tau ^*$ has been calculated using (4.13), and gas kinetic effects have been quantified through (4.7)–(4.12). The value of the prefactor in (4.15), corresponding to the capillary limit in (3.17), differs from that in (4.14). This could be due to the differences in the local geometry of the interface for $r\approx a$; indeed, the experiments in figure 3 of Chantelot & Lohse (Reference Chantelot and Lohse2021) show that the curvature of the interface near $r\approx a$ for the case of Leidenfrost drops increases with $T_s$ (see also Kolinski et al. Reference Kolinski, Mahadevan and Rubinstein2014b), a fact implying that the prefactor affecting $\ell$ in (4.5) for the case of superheated substrates should be smaller than for the case of isothermal impacts. Notice also that since the local curvature is very much dependent on whether the substrate is superheated or not, it could also be the case that de Ruiter et al. (Reference de Ruiter, Oh, van den Ende and Mugele2012) and Chantelot & Lohse (Reference Chantelot and Lohse2023) provide the experimental values of $h_{min}(\tau =12)$ for the case of isothermal substrates, whereas Chantelot & Lohse (Reference Chantelot and Lohse2021, Reference Chantelot and Lohse2023) provide $h_m(\tau \approx 12)$ for the case of superheated substrates; in that case, the value of the constant in (4.15) would be very similar to the one in (3.19). Figures 8(c,d) also show a comparison between the predicted and measured minimum film thicknesses under the approximation followed in Gordillo & Riboux (Reference Gordillo and Riboux2022), where we considered that the minimum film thickness could be predicted using the inertial approximation given in (4.16). In this case, figures 8(c,d) show that the experimental measurements by Chantelot & Lohse (Reference Chantelot and Lohse2021, Reference Chantelot and Lohse2023) can be predicted reasonably well, with relative errors $\pm 30\,\%$, using a value 1.15 for the prefactor in (4.16), which is very similar to that deduced from figure 6. Notice also that figure 8 also includes the predicted minimum film thickness corresponding to the largest temperature, $T_s=295\,^\circ \textrm {C}$, when the value of the prefactor in (4.16) is varied from 1.25 to 1.7. A possible reason why an increase in the value of the prefactor improves the comparison with the experimental data for the case of the highest substrate temperature could be the fact that the slender approximation under which (3.17) are deduced breaks for sufficiently large values of $T_s$. Indeed, the parameter expressing the ratio between the minimum film thickness and the length along which the pressure gradients in the liquid take place, namely,

(4.17)\begin{equation} \frac{h_m}{h_{a,m}}\approx y^{1/2}\,St^{-1/6} ,\end{equation}

where we have made use of (3.4) and (4.16), could become larger than unity for substrate temperatures $T_s$ exceeding the threshold value given by the condition (see (4.17))

(4.18)\begin{equation} y^2\gtrsim St^{2/3}\Rightarrow \frac{k_v(T_s-T_b)}{\mu_v\mathcal{L}}\left(\frac{\rho}{\rho_v}\right)\gtrsim St^{2/3} , \end{equation}

where use of (4.6) has been made. The results depicted in figure 9 reveal that, indeed, the spatial region where the maximum pressure gradient is attained is clearly not slender for the larger value of $T_s$ because the film thickness is larger than the length along which the liquid pressure gradient takes place. In these cases, since $h_m\approx h_{a,m}$, the capillary pressure is $\sim \sigma h_m/h^2_{a,m}\sim \sigma /h_{a,m}$, with this value being similar to the liquid overpressure $\Delta p_m$ given in (3.5) because

(4.19)\begin{equation} \frac{\Delta p_m}{\sigma/h_{a,m}}\sim We\,St^{-1/3}, \end{equation}

which happens to be of order unity for the experiments corresponding to the largest temperatures reported by Chantelot & Lohse (Reference Chantelot and Lohse2023). Then the loss of slenderness for the largest value of $T_s$ implies larger capillary pressures because $h_m$ is not much smaller than $h_{a,m}$, a fact also implying that the values of the gas pressure gradient are smaller than those corresponding to the slender limit in which (3.17) have been deduced. A reduction in the pressure gradient implies larger values of $h_m$ (see (4.2)), and this fact could be behind the result depicted in figure 8 for the case $T_s=295\,^\circ \textrm {C}$. Let us also point out that the discrepancies between the predictions and experimental measurements for the case $T_s=295\,^\circ \textrm {C}$ could also be originated as a consequence of limitations in the temporal resolution of the experiments carried out by Chantelot & Lohse (Reference Chantelot and Lohse2023) for the highest impact velocities.

Figure 9.

Ratio between the measured minimum film thickness and the length $h_{a,m}$ given in (3.4) along which pressure gradients take place. The meanings of the symbols are the same as in figure 8.