The skating of drops impacting over gas or vapour layers

J. Fluid Mech. (2024), vol.980, A35, doi:10.1017/jfm.2024.20 The skating of drops impacting over gas or vapour layers P. García- Geijo1,G.Riboux 1and J.M. Gordillo1,† 1Área de Mecánica de Fluidos, Departamento de Ingeniería Aeroespacial y Mecánica de Fluidos, Universidad de Sevilla, Avenida de los Descubrimientos s/n, 41092 Sevilla, Spain (Received 12 June 2023; revised 22 December 2023; accepted 30 December 2023) We report numerical simulations confirming the predictions in Gordillo & Riboux (J. Fluid Mech., vol. 941, 2022, A10), where we elucidated the lubrication mechanism by which a drop of a low-viscosity liquid impacting over a smooth solid substrate skates over a thin gas film that prevents contact with the wall. Moreover, with the purpose of explaining the so-called lift-off mechanism reported in Kolinski et al. (Phys.Rev.Lett., vol. 112, issue 13, 2014, 134501), we extend our previous findings and derive expressions for the time-varying thickness of the gas layer at the region where the distance to the wall is minimum, finding good agreement with the numerical results. In addition, we report that our predictions for the minimum thickness of the gas film separating a falling drop from a wall at room temperature follow closely the experimental values when gas kinetic effects are retained in the analysis, and also report that the analogous equation for the minimum thickness of the vapour layer formed after a drop impacts a superheated wall predicts well the experimental measurements. Key words: boiling, drops 1. Introduction The impact of a drop over a substrate has been the subject of intense research efforts during the past decades as a consequence of its relevance in a myriad of natural and technological processes;see e.g. Josserand & Thoroddsen (2016). One of the many open questions that remains to be solved and has received substantial attention in recent years refers to the description and quantification of the conditions under which a drop impacting a superheated substrate skates over a vapour film, a phenomenon referred to as the dynamic †Email address for correspondence: [email protected] © The Author(s), 2024. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/ licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited. 980 A35-1 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press P. García-Geijo, G. Riboux and J.M. Gordillo Leidenfrost effect, which limits the cooling rate of solids in heat transfer technological applications (Tran et al. 2012,2013;Shirotaet al. 2016). Indeed, as a consequence of the smallness of the thermal conductivity of gases, the existence of a stable vapour film beneath the drop reduces the heat flux and therefore the cooling capacity of the liquid (van Limbeek et al. 2017). The theoretical and numerical studies aimed at describing the effect of a gas or vapour layer on the impact of a drop over a solid substrate were pioneered by Smith, Li & Wu (2003)and Korobkin, Ellis & Smith (2008), who coupled the equations for the inviscid motion of a two-dimensional drop impacting either a wall or a thin liquid layer with the lubrication equations for the gas. Later, Mandre, Mani & Brenner (2009), Mani, Mandre & Brenner (2010)and Mandre & Brenner (2012) made use of the theoretical and numerical framework already introduced by Smith et al. (2003)and Korobkin et al. (2008)withthe purpose of deriving scaling relationships for the dimple height – namely, the thickness of the pancake-shaped bubble entrapped at the centre of the impacting drop – and also for the minimum distance between the drop and the wall, which is attained not at the axis of symmetry, but at an off-centre position. The scaling for the dimple height deduced by Mandre et al. (2009) in the two-dimensional case and by Hicks & Purvis (2010)for the more realistic case of spherical drops was confirmed via experiments and also via numerical simulations by Bouwhuis et al. (2012) who, in addition, discovered a new regime dominated by capillarity that describes the entrapment of bubbles at low impact velocities. Let us point out here that the analysis of the entrapment of a gas pocket beneath a falling liquid mass is analogous to the cushioning effect of the air entrapped after a solid impacts a free surface, a physical situation that has been analysed by e.g. Wilson (1989), Ross & Hicks (2019)and Moore (2021), using Wagner’s original ideas (Wagner 1932). Again, back in the context of drops falling over a wall, Duchemin & Josserand (2011) developed a boundary integral method coupled with simplified lubrication equations for the gas flow, and reported numerical results for the minimum film thickness that did not follow the predictions in Mandre et al. (2009), Mani et al. (2010)and Mandre & Brenner (2012). From the purely experimental point of view, and making use of high-speed imaging techniques, Chandra & Avedisian (1991)and Thoroddsen et al. (2005)werethefirst to report the shape and thickness of the bubble entrapped at the centre of the drop, whereas Kolinski, Mahadevan & Rubinstein (2014b) reported experimental data on the time-varying minimum film thickness, and described what they called the lift-off mechanism, which takes place when a drop skates over a nanometric gas film, finding that this effect depends on the gas to liquid viscosity ratio. Very recently, Chantelot & Lohse (2021,2023) extended the previous experimental studies to the case of superheated substrates, reporting measurements of the minimum film thickness as a function of the impact velocity and of the substrate temperature, and they scaled and interpreted their own data using the ideas in Mandre et al. (2009)and Mandre & Brenner (2012). Recently, Gordillo & Riboux (2022) presented a physical model based on the idea that both the liquid pressure gradient and the component of the liquid velocity along the tangent direction to the wall drive the gas flow within the spatio-temporal region where the distance between the liquid and the solid is minimum. One of the main results of Gordillo & Riboux (2022), who made extensive use of Wagner’s theoretical framework (Wagner 1932), is that the classical lubrication mechanism, by which the Couette and Poiseuille flow rates are in balance in the slightly converging geometry formed between the drop and the wall, prevents the liquid contacting the solid provided that the minimum gas layer 980 A35-2 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press The skating of drops impacting over gas or vapour layers thickness calculated in this way is larger than the height of the substrate asperities or of the interfacial corrugations (Kim et al. 2011; Kolinski, Mahadevan & Rubinstein 2014a; Chantelot & Lohse 2021). The study by Gordillo & Riboux (2022) present results that differ from those in Mandre et al. (2009), Mandre & Brenner (2012)and Chantelot & Lohse (2021,2023), hence one of the main purposes in this contribution will be to decipher which of the two predictions for the minimum film thickness corresponding to the case of non-heated substrates (i.e. either the one given in Mandre et al. (2009), Mani et al. (2010)and Mandre & Brenner (2012), or the one deduced in Gordillo & Riboux (2022)) is in better agreement with numerical simulations carried out using Basilisk (Popinet 2015) in the limit in which the Knudsen number Kn defined in terms of the gas film thickness is zero, namely, when gas kinetic effects are absent (Sprittles 2024). It could appear that the main contribution here is nothing but a confirmation of already published results, but this is not the case. Indeed, here we will also provide a quantitative explanation of the so-called lift-off mechanism first described by Kolinski et al. (2014b), by which the minimum thickness of the gas film increases in time after the distance to the wall has reached a minimum. In spite of several contributions in the literature on the subject, some of which are very recent (see e.g. Mishra, Rubinstein & Rycroft 2022), we have not found any physical description or theory aimed at explaining and quantifying the original observations made by Kolinski et al. (2014b). Then here we deduce equations for the time-dependent width of the gas layer where the liquid pressure is maximum, finding that our predictions agree with the numerical results,which –as has been pointed out in the paragraph above – have been obtained in the ideal limit in which the Knudsen number characterizing the flow in the thin gas film is zero. This contribution also contains a comparison between our predictions and the experiments reported by de Ruiter et al. (2012)and Chantelot & Lohse (2023) for the case of a non-heated substrate, with such a comparison containing two fundamental differences from the analogous analysis reported in Gordillo & Riboux (2022). Indeed, the minimum film thickness is calculated here using an equation that has been validated previously by means of numerical simulations carried out in the ideal limit Kn =0. But, in addition, in order to compare with experiments, we retain in the algebraic expression for the minimum film thickness the effect of Kn by replacing the actual gas viscosity with the expression for the effective gas viscosity deduced by Zhang & Law (2011). The good agreement between our predictions –which do not include any kind of adjusting parameter since the only free constant will be determined using idealized numerical simulations – and experiments indicate that, as was pointed out already by Li (2016)and Chubynsky et al. (2020), gas kinetic effects are essential to predict the dynamics of impacting drops that skate over a gas layer; in addition, these results provide further support to our physical description which, as it was pointed out above, differ from the one given in Mandre et al. (2009), Mandre & Brenner (2012)and Chantelot & Lohse (2021,2023). Moreover, the predictions in Gordillo & Riboux (2022) for the cases of drops impacting a superheated substrate – namely, a substrate with a temperature larger than the boiling temperature of the liquid – will be compared with the experimental data reported by Chantelot & Lohse (2021,2023) once gas kinetic effects are taken into account through the expressions for the effective gas viscosity and the effective thermal conductivity reported by Zhang & Law (2011) and Sharipov, Cumin & Kalempa (2007), respectively. Let us point out clearly here that it is not the purpose of this contribution to describe the contact between the liquid and the solid, i.e. the so-called touchdown problem, which 980 A35-3 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press P. García-Geijo, G. Riboux and J.M. Gordillo Domain 2.1R Pa, Ta Ts 2.1R z0 ρ, μ R U z r Figure 1. Sketch showing the axisymmetric domain where the numerical simulations have been carried out using Basilisk (Popinet 2015); see Appendix A for details. Here, rand z, respectively, indicate the distance to the axis of symmetry and the distance to the wall in a cylindrical coordinate system. The numerical box is a square of length 2.1R. remains an open question due to the fact that either the contact or the rebound of a drop impacting a solid depends on a number of factors, such as the substrate roughness, the presence of contaminants at the free interface, asymmetries, van der Waals forces,or even electrostatic effects (Kim et al. 2011; Kolinski et al. 2014a; Sprittles 2024), which lead to differences on the instant at which the gas film destabilizes under very similar experimental conditions;see e.g. Kolinski et al. (2014a)and de Goede et al. (2019). The explanation for these differences is outside the scope of this contribution, which is then focused in the description of those regimes in which a drop impacting a solid substrate, which might be heated above the boiling point of the liquid or not, skates over a thin gas film, this being a subject of recent interest in the literature (Sprittles 2024). The manuscript is structured as follows.Section 2is devoted to presenting the results of numerical simulations carried out using Basilisk (Popinet 2015). In § 3,wereviewthe physical model presented in Gordillo & Riboux (2022), and compare our own predictions and those in Mandre & Brenner (2012)and Chantelot & Lohse (2023) with the numerical results. Taking into account gas kinetic effects, in § 4we compare our predictions for the minimum gas film thicknesses with the experimental values given in de Ruiter et al. (2012)and Chantelot & Lohse (2023). Finally, §5summarizes the main results in this contribution. 2. Numerical results corresponding to the case of isothermal substrates This section is devoted to presenting the results of simulations carried out using Basilisk (Popinet 2015) in the numerical domain depicted in figure 1, which shows a drop of radius Rof a liquid with density ρ, viscosity μ, and interfacial tension coefficient σ,falling with uniform velocity Uagainst a wall whose temperature Tsis equal to that of the gas, Ts=Ta (isothermal substrate);see Appendix A for details on the numerical implementation. Using R,R/Uand ρU2as the characteristic values of length, time and pressure, the numerical results in this section will be expressed in terms of the Stokes and Weber numbers defined as St =ρUR μa ,We =ρU2R σ,(2.1a,b) 980 A35-4 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press The skating of drops impacting over gas or vapour layers (a)(b) h/R p/(ρU2) 1.0 3(×10–3) 2 1 0 0 0.1 τ = 5 τ = 8 τ = 12 τ = 20 τ = 40 r = aˉ 0.2 0.3 0.4 0 0.2 0.4 r/R 0.6 0.8 1.0 0 0.05 0.10 0.15 r/R 0.20 0.25 0.30 0.35 0.40 15 10 5 0.8 0.6 0.4 0.2 r = 3s R τ = 5 τ = 8 τ = 12 τ = 20 τ = 40 r = aˉ Figure 2. (a) Time-evolving shapes of a drop impacting a wall with We =12 and St =2.6×104.Theinset shows the geometry of both the gas pocket and the region, located at r/R=√3s(solid line), where the drop skates over a thin gas layer. Notice that the radial position where the maximum gas pressure is attained, r=¯a (dashed line), verifies ¯a≈R√3s.(b) Spatio-temporal evolution of the gas pressure at the wall corresponding to the values of We and St of (a). The values of ∂p/∂rat the spatio-temporal region where the gas pressure is maximum are calculated as the slopes of the solid lines in the figure. and varied within the ranges 9 ×103≤St ≤4.5×104,4≤We ≤60, while keeping fixed the values of the ratios ρa/ρ =10−3and μa/μ =1.8×10−2,withρaand μaindicating the gas density and viscosity, respectively. For simplicity, neither compressibility nor gas kinetic effects have been retained in the numerical simulations,which, as has been pointed out above, have been carried out for the case of isothermal substrates. The numerical results depicted in figure 2(a) reveal that as the drop approaches the wall, a dimple is formed at the axis of symmetry, entrapping a nearly cylindrical gas pocket with radius ∝√Rhdand thickness hd.Figure 2 also shows that the liquid does not touch the solid but, instead, skates over a thin gas film whose minimum thickness hmin hdis attained at r∝√Rhd. The thickness of the entrapped bubble, hd, is deduced from the mass balance (Mandre et al. 2009; Bouwhuis et al. 2012) πU(Rhd)∼2πRhd h3 d 12μa pd √Rhd ,(2.2) where it has been taken into account that the gas flow rate per unit length induced by the pressure jump pd=p(r=z=0,t)−Pais pd∝(h3 d/μa)(pd/√Rhd).Since in a first approximation the liquid velocity field within the drop is irrotational, the Euler–Bernoulli equation particularized at r=0, z=hdyields (Bouwhuis et al. 2012) pd∼ρ∂φ ∂t,(2.3) where φ∝U√Rhdis the value of the velocity potential at r=0 created by a disk of radius √Rhdmoving into the liquid with a velocity U;see e.g. Peters, van der Meer & Gordillo (2013). Then since the dimple is formed in a characteristic time hd/U, the pressure jump deduced from (2.3) reads pd∝ρU2√R/hd, from which, using (2.2), we obtain hd∝RSt −2/3;(2.4) 980 A35-5 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press P. García-Geijo, G. Riboux and J.M. Gordillo (a)(b) (c)(d) p/(ρU2) h/R 0 0 3(×10–3) 15 12 9 6 3 2 1 0.05 0.10 0.15 0.20 0.25 0 p/(ρU2) 15 12 9 6 3 0 p/(ρU2) 15 12 9 6 3 0.05 0.10 0.15 0.20 0.25 0.05 0.10 r/R 0.15 0.20 0.25 00.05 0.10 r/R 0.15 0.20 0.300.25 τ = 20 r = aˉ r(hmin)/R r = 3s R Figure 3. (a) Gas layer thickness h(r)/Rat τ=20 for We =12 and St =2.6×104. The radial position where the minimum air film thickness is attained, r(hmin)/R(dotted vertical line), the radial position where the maximum pressure is reached, r/R=¯a/R(dashed vertical line), and the wetting radius, r/R=√3s(solid vertical line), are indicated. Values of the gas pressure jump at the wall for the following instants of time: (b) τ=12, (c)τ=15, and (d)τ=20. then the dimple is formed in a characteristic time given by td∝hd U→tdU R∝hd R∝St−2/3.(2.5) Hence, taking the origin of times at the instant when the drop would contact the substrate if the gas were not present, and defining the dimensionless times sand τas s=tU R,τ=St2/3s=St2/3tU R,(2.6a,b) the result in (2.5) indicates that the dimple is formed at the instant of time τ=τ∗,with τ∗≈12 (Gordillo & Riboux 2022). Figure 2(b) shows the spatio-temporal evolution of the gas pressure at the wall i.e. at z=0. For a fixed value of τ, the results in figure 2(b) reveal that the gas pressure increases radially, reaching a maximum at an off-centre position that moves towards larger values of ras time progresses. In addition, figure 2 shows that the maximum gas pressure at the wall, and also the values of the local pressure gradient at the radial position where the maximum pressure is located, increase with τ, reaching a maximum at τ=τ∗≈12. Moreover, the results in figure 3 reveal that the maximum gas pressure and the maximum pressure gradient, calculated as the slope of the lines depicted in figure 2(b), are reached at the radial position r=a(s)=R√3s,(2.7) namely, at the radius of the circular region that is wetted by a drop impacting with velocity Uover a wall, a result that was checked carefully against experiments and was deduced in Riboux & Gordillo (2014) in the context of drop impact using Wagner’s 980 A35-6 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press The skating of drops impacting over gas or vapour layers 0 0.5 hmin/R 1.0 1.5 10 20 30 40 τ 0 0.5 hm/R 1.0 1.5 10 20 30 40 Pre-impact We = 12, St = 1.2 × 104 We = 4, St = 3 × 104 We = 12, St = 2.6 × 104 We = 22, St = 2.1 × 104 We = 36, St = 2.6 × 104 We = 48, St = 1.8 × 104 We = 48, St = 3.6 × 104 We = 60, St = 1.8 × 104 τ (×10–3) (×10–3) (a)(b) Figure 4. Time evolution of (a) the minimum gas layer thickness hmin,and(b) the thickness of the gas layer measured at r/R=√3s,hm=h(r=R√3s), corresponding to the values of We and St indicated in the legend. The pre-impact stage τ<12 is highlighted in blue. theoretical framework (Wagner 1932). Indeed, Wagner’s theory, which has been known for nearly a century in the context of water entry problems, permits us to deduce theresultin(2.7) in a quite straightforward manner; see also Wilson (1989), where (2.7) was deduced in the context of ship slamming. Readers interested in the rigorous application of Wagner’s theory (Wagner 1932) to different physical phenomena using matched asymptotic techniques are directed to Korobkin & Pukhnachov (1988), Wilson (1989), Howison, Ockendon & Wilson (1991), Scolan & Korobkin (2001), Oliver (2002), Korobkin & Scolan (2006)and Moore (2014). The results in figure 3 also suggest that in spite of the minimum gas film thickness hmin(τ) is not attained at the radial position where the maximum gas pressure is reached, namely, hm(τ) /=hmin(τ ),with hm(τ) =h(r=a(s)), (2.8) hm(τ) ≈hmin(τ);seefigure 2(a). This result is appreciated more clearly in figure 4, which provides the values of hm(τ) and hmin(τ) for different values of the Weber and Stokes numbers explored in this study. Indeed, figure 4 reveals that hmin(τ) =hm(τ ) for the instants of time (highlighted in blue in figure 4)τ≤τ∗, and also that hm(τ) > hmin(τ ) for τ>τ ∗. Due to the fact that the values of the maximum gas pressure, the maximum pressure gradient and also the minimum of hm(τ) are attained at τ=τ∗,the instants of time τ<τ ∗will be termed, in what follows, as pre-impact stage, whereas those corresponding to τ>τ ∗will be referred here as post-impact stage; see also the time-evolving dimple shapes included as an inset of figure 2(a). Hence here we will consider that the ‘impact’ takes place when the maximum pressure is attained at τ=τ∗,so, using the result in (2.6a,b)–(2.7), this event is localized at the radial position r=R√3τ∗St−2/3≈6RSt −1/3;therefore the minimum thickness of the gas film is attained at the dimensionless instant τ=τ∗≈12 and at the dimensionless radial position r/R≈6St−1/3. The division of the impact process into two well-defined stages is crucial to understanding the differences between the present analysis and the predictions in Mandre & Brenner (2012)and Chantelot & Lohse (2021,2023), who describe the instants previous to the ‘impact’, namely, the pre-impact stage taking place for τ<τ ∗. In contrast, the predictions in Gordillo & Riboux (2022) were deduced with the purpose of describing both the impact and the post-impact stages, and therefore should be applicable to quantify the different events taking place for τ≥τ∗. 980 A35-7 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press P. García-Geijo, G. Riboux and J.M. Gordillo 3.0 0 1 2 ||ur/U|| = 5 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 (r – a¯)/h ¯a z/z(1) 3.0 0 1 2 3.5 4.0 4.5 5.0 5.5 6.02.0 2.5 z/z(1) (a) (b) Figure 5. Radial component of the dimensionless gas velocity field, ur/U, represented in the frame of reference moving with the wetting velocity Vmgivenin(3.3) for three different values of (r−¯a)/¯ haand two different instants of time: (a)τ=20 and (b)τ=40. Here, ¯a≈R√3sindicates the radial position where the maximum gas pressure is attained (see figure 3)and¯ ha=2/(9π)¯a3≈ha,m,withha,mgiven in (3.4). Here, We =12, St =2.60 ×104, whereas z(1)indicates the vertical coordinate of the interface at the minimum value of (r−¯a)/¯ harepresented in the figure. The next section is devoted to checking which of the predictions for the minimum film thickness – either the ones deduced in Mandre & Brenner (2012) or those in Gordillo & Riboux (2022)– are in better agreement with the numerical results. 3. Modelling the impact of drops over isothermal substrates 3.1. Review of previous results The predictions in Mandre & Brenner (2012), later extended by Chantelot & Lohse (2021,2023) 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 τ≥τ∗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. (2009), Mandre & Brenner (2012)and Chantelot & Lohse (2021,2023) 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 (2022) describes the lubricated impact of a drop over a wall for instants of time τ≥τ∗. The physical idea behind the predictions in Gordillo & Riboux (2022)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, 980 A35-8 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press The skating of drops impacting over gas or vapour layers namely (see figures 2 and 3) Vm(s)=da dt=U 23 s,(3.1) where the subscript mis 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 (1932)and Gordillo & Riboux (2022). 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 2022) −h3 m 12μa ∂p ∂r−Vmhm 2≈0.(3.2) 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 (2012)and Riboux & Gordillo (2014)), will be considered in § 4using the approach detailed in Li (2016) 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 (2011), 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 hm20 nm (Sprittles 2024), namely, for values of the gas film thicknesses that are well below those measured experimentally by de Ruiter et al. (2012) for the case of isothermal impacts, and by Chantelot & Lohse (2021,2023)forthe 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 hm, the next step that we followed in Gordillo & Riboux (2022) was to make use of the fact that the vertical interfacial velocities during τ≥τ∗are much smaller than the impact velocity,i.e.(1/U)∂h/∂t1, and also that the gas film is slender, ∂h/∂r1. In this way, since the Reynolds number verifies Re =St μa/μ 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 (1932). At this point, notice that the condition (1/U)∂h/∂t1 also implies that dhm/dtVmhm–with indicating the characteristic length along which hmvaries,of the order of ∼hm– due to the fact that >hmand also because Vm∝USt 1/3U,this being the reason why the term dhm/dt has been neglected in the mass balance (3.2). 980 A35-9 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press P. García-Geijo, G. Riboux and J.M. Gordillo in Gordillo & Riboux (2022) that the heat flux across the liquid thermal boundary layer can be neglected for the particular case of the experiments reported in Chantelot & Lohse (2021,2023). Therefore, the equation analogous to (3.2) for the case of drops impacting over superheated substrates reads −h3 m 12μv ∂p ∂r−Vmhm 2≈kvT ρvL  hm ,(4.2) where we have made use of (4.1). The substitution of (3.3)–(3.6)into(4.2) yields the following equation for hm/R: y−1+1 6 μa μv y=β∗,(4.3) with y=St 2(/R) Vm Uhm R2 and β∗=βρ ρvμv μa,where β=kvT μvL.(4.4) Consequently, hm R=2ySt −1Vm U−1 R1/2 ,(4.5) with y=3μv μa1+1+2β∗ 3μa μv.(4.6) 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 yis 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 (2021,2023), 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 (2024)states clearly that gas kinetic effects cannot be neglected in the description of the head-on collision of drops (Li 2016)orinthe impact of drops on a substrate (Riboux & Gordillo 2014; Chubynsky et al. 2020)ifthe value of the Knudsen number, defined as Kn =λ hm ,(4.7) with λdenoting the mean free path of the gas, becomes Kn 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. (2012) and Chantelot & Lohse (2021,2023)vary between 103 980 A35-16 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press The skating of drops impacting over gas or vapour layers and 102nanometres, whereas the values of the mean free path of air and ethanol at normal conditions are, respectively, λa≈69 nm and λeth ≈50 nm.(4.8a,b) Then, in order to account for gas kinetic effects, we define an effective gas viscosity μ∗(Kn)(Sprittles 2024) and make use of the expression deduced by Zhang & Law (2011), employed successfully by Li (2016) in his study of the head-on collision of drops: μ∗=μ 1+6.0966 Kn +0.9650 Kn2+0.6967 Kn3,(4.9) where μrefers to the value of the actual gas viscosity. Notice that forthecaseofdrops impacting a non-heated substrate, μvin (4.6) refers to the effective air viscosity, namely, μv=μ∗ a(see (4.9)), whereas the Knudsen number defined in (4.7) is calculated using the value of the mean free path given by λ=λa Pa Pa+pm ,(4.10) with λaand pmgiven, respectively, in (4.8a,b)and(3.5). For the case of drops impacting a superheated substrate, μv=μ∗ eth and λ=λeth Pa Pa+pm 273 +0.5(Tb+Ts) Ta ,(4.11) with Ta=298 K, and Tb=78 and Tsrefer 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. 2007) k∗ v=kv 1+3.91 Kn.(4.12) 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. (2012)and Chantelot & Lohse (2021,2023),the gas film thickness is well above 20 nm, which is the length scale below which these forces become relevant (Sprittles 2024). 4.2. Comparison with experiments The comparison with the numerical results in §3confirms our physical description,andit 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. (2012) and Chantelot & Lohse (2021,2023) 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 Ts. 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 τ∗at which the central bubble is formed, i.e. at the instant when the minimum film thickness is attained at r≈RSt −1/3√3τ∗. 980 A35-17 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press P. García-Geijo, G. Riboux and J.M. Gordillo 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 π(Rhd)(U+kvT/(ρvhdL)). Therefore, expressing hd/R=τ∗St−2/3and following the same steps as those detailed in §3, the mass balance (2.2)provides the following equation for τ∗(Gordillo & Riboux 2022): τ∗5/2=12.43/2τ∗+β∗St−1/3,(4.13) with β∗defined in (4.4). Notice that (4.13) recovers the value of τ∗≈12 corresponding to the case of isothermal substrates, for which β∗=0. Before comparing our predictions with the experimental values provided in de Ruiter et al. (2012) and Chantelot & Lohse (2021,2023), 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 σ=22 ×10−3Nm −1or σ=17 ×10−3Nm −1for 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 (Tb+Ts)/2as detailed in Gordillo & Riboux (2022),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 hmfrom the wall is evaporating along a region of length ,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 ∼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)withA=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)) hm R≈3.5×6μ∗ a μa2/3 122/3We−1/3St−10/9,(4.14) with μ∗ a=μ∗defined in (4.9). Figure 8 also compares the experimental measurements in Chantelot & Lohse (2021,2023) 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)) hm R≈2.3y2/3τ∗2/3We−1/3St−10/9(4.15) 980 A35-18 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press The skating of drops impacting over gas or vapour layers 0.2 0.3 0.4 0.6 U (m s–1) 1.0 0 0.5 1.0 1.5 2.0 2.5 3.0 123456 0.1 0.3 0.4 0.6 1.0 0.2 0.3 0.4 0.6 hm (µm)hm (µm) hm/hth hm/hth 1.0 0 0.5 1.0 1.5 2.0 2.5 3.0 123456 0.1 0.3 0.4 0.6 1.0 ξ (τ∗/12.4) Ts = 105 °C Ts = Ta, DeR Ts = 295 °C, B = 1.25 Ts = 295 °C, B = 1.70 Ts = 295 °C, B = 1.15 Ts = 295 °C, B = 1.70 Ts = Ta, C&L Ts = 144 °C Ts = 178 °C Ts = 230 °C Ts = 270 °C Ts = 295 °C (a)(b) (c)(d) 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. (2012) (DeR) and Chantelot & Lohse (2023) (C&L) for different values of the impact velocity Uand for different values of the substrate temperature Ts. 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 hmcorresponding to the capillary regime, whereas the predictions for hmin the inertial regime are represented using dashed lines. (b) Comparison between the predicted and measured values of hmin (a) as a function of ξ.(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 hmin (c) as a function of ξ.The dashed horizontal lines in (b,d) are placed at 1 ±0.3. and hm R≈1.25τ∗St−7/6y1/2(4.16) for the capillary and inertial regimes, respectively. In (4.15)–(4.16), yis given by (4.6), τ∗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≈a; indeed, the experiments in figure 3 of Chantelot & Lohse (2021) show that the curvature of the interface near r≈afor the case of Leidenfrost drops increases with Ts(see also Kolinski et al. 2014b), a fact implying that the prefactor affecting 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 casethatdeRuiteret al. (2012)and Chantelot & Lohse (2023)provide the experimental values of hmin(τ =12)for the case of isothermal substrates, whereas Chantelot & Lohse (2021,2023)provide hm(τ ≈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) 980 A35-19 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press P. García-Geijo, G. Riboux and J.M. Gordillo also show a comparison between the predicted and measured minimum film thicknesses under the approximation followed in Gordillo & Riboux (2022), 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 (2021,2023) can be predicted reasonably well, with relative errors ±30 %, using avalue1.15 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, Ts=295 ◦C, when the value of the prefactor in (4.16)isvaried 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)arededuced breaks for sufficiently large values of Ts. 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, hm ha,m≈y1/2St−1/6,(4.17) where we have made use of (3.4)and(4.16), could become larger than unity for substrate temperatures Tsexceeding the threshold value given by the condition (see (4.17)) y2St2/3⇒kv(Ts−Tb) μvLρ ρvSt2/3,(4.18) 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 Tsbecause the film thickness is larger than the length along which the liquid pressure gradient takes place. In these cases, since hm≈ha,m, the capillary pressure is ∼σhm/h2 a,m∼σ/ha,m, with this value being similar to the liquid overpressure pmgiven in (3.5) because pm σ/ha,m∼We St−1/3,(4.19) which happens to be of order unity for the experiments corresponding to the largest temperatures reported by Chantelot & Lohse (2023). Then the loss of slenderness for the largest value of Tsimplies larger capillary pressures because hmis not much smaller than ha,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 hm(see (4.2)), and this fact could be behind the result depicted in figure 8 for the case Ts=295 ◦C. Let us also point out that the discrepancies between the predictions and experimental measurements for the case Ts=295 ◦C could also be originated as a consequence of limitations in the temporal resolution of the experiments carried out by Chantelot & Lohse (2023) for the highest impact velocities. 5. Conclusions In this contribution, we have presented numerical simulations validating the predictions in Gordillo & Riboux (2022), where we described the lubrication mechanism by which a drop falling over a substrate skates over a gas or vapour layer. With the purpose of explaining and quantifying the lift-off mechanism reported empirically by Kolinski et al. (2014b)and 980 A35-20 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press The skating of drops impacting over gas or vapour layers 0.2 0 0.5 1.0 1.5 2.0 0.3 0.4 0.6 U (m s –1 ) 1.0 hm/ha,m Figure 9. Ratio between the measured minimum film thickness and the length ha,mgivenin(3.4) along which pressure gradients take place. The meanings of the symbols are the same as in figure 8. investigated numerically by Mishra et al. (2022), we have extended our previous results and have provided equations, including the values of the prefactors, that reproduce closely the time evolution of the minimum film thickness calculated numerically. Moreover, with the purpose of comparing our results with the measured values of the gas or vapour film thicknesses, we have included gas kinetic effects in the algebraic equations for the minimum thickness of the gas layer, finding that our predictions, up to prefactors, are in good agreement with the results reported by de Ruiter et al. (2012)and Chantelot & Lohse (2021,2023). Supplementary material. Supplementary material is available at https://doi.org/10.1017/jfm.2024.20. Acknowledgements. J.M.G. thanks P. Chantelot and D. Lohse for fruitful discussions. Funding. This work has been supported by the grant PID2020-115655GB-C21, financed by the Spanish MCIN/AEI/10.13039/501100011033, and all the simulations were run on the High Perfomance Computing cluster provided by the Centro Informático Científico de Andalucía (CICA). Declaration of interests. The authors report no conflict of interest. Author ORCIDs. P. García-Geijo https://orcid.org/0000-0001-8608-4804; G. Riboux https://orcid.org/0000-0003-2395-1653; J.M. Gordillo https://orcid.org/0000-0003-1431-3780. Author contributions. P.G.-G. carried out the numerical simulations and analysed the data, G.R. analysed the data and partly wrote the paper, and J.M.G. designed the research and the theory, and wrote the paper. All authors reviewed the results and approved the final version of the manuscript. Appendix A. Numerical simulations The numerical results in § 2, which have been calculated using the free software Basilisk (Popinet 2015), extending the ideas in the codes by Sanjay (2022)and Zhang et al. (2022), simulate the axisymmetric impact of a drop of radius Rinitially centred at z=1.006R,falling with a uniform velocity Uover an impermeable horizontal substrate. The numerical results have been obtained by imposing symmetry conditions at the axis, the impenetrability and no-slip boundary conditions at the wall limiting the computational domain depicted in figure 1,and outflow boundary conditions at the two remaining 980 A35-21 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press P. García-Geijo, G. Riboux and J.M. Gordillo 0.360 0.25 0 0.05 0.10 0.15 0.20 0.30 0.35 0.40 0.361 r/R r/R r/R h/R h/R 0.362 0 0 0.5 1.0 1.5 2.0 12 1 2 3(×10–3) (a) (b)(c) Figure 10. Each black dot represents the vertex of a numerical cell, whereas the interface is plotted using a blue solid line. (a) The numerical domain is divided into regions with different discretization levels: red box, h/R<0.1andr/R<1.0, with cell sizes Δ≥R/1950; yellow box, h/R<0.01 and r/R<√3s+0.15, with cell sizes Δ≥R/31208. For the rest of the numerical domain, Δ≥R/975. (b) A closer view of the region around the advancing front. (c) Detailed view of the spatial region around the wetting radius r=√3s(red dashed line). Here, We =12, St =2.60 ×104and τ=0.0435 ≈38 St−2/3. boundaries, where both the pressure and the gradient of normal velocities are set to zero. We have checked that the numerical results are unaffected by the dimensions of the computational domain. We have made use of an adaptive Cartesian mesh, which refines the solution both at the interface and at the regions with the largest velocity gradients. The tolerances for the volume fraction field, for the velocity field and for the curvature are set to 10−3,10 −2and 10−6,respectively, and the maximum level of grid refinement is defined in each of the different regions into which the computational domain has been divided; indeed, the grid size is Δ≥R/975 for z>0.1, but Δ≥R/31208 near the wall, where the velocity and pressure gradients reach the largest values;see the figure 10 caption for details. Moreover, we have carried out a sensitivity analysis in order to verify that the results reported are independent of the grid size; indeed, figure 11 shows that the values of the pressure, the 980 A35-22 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press The skating of drops impacting over gas or vapour layers τ 0 1000 2000 (∂p/∂r)/(ρU2/R)p/(ρU2) 3000 4000 5000 6000 10 20 30 40 R/Δ 0 1000 2000 3000 4000 5000 6000 0.5 1.00 1.5 2.52.0 3.53.0 0 5 10 15 20 25 5 10 15 20 10 20 30 40 00.5 1.00 1.5 2.52.0 3.53.0 (×104) (×104) hmin/R 00 1 2 3 4 5 6 0.5 1.0 1.5 15 10 5 29 de Ruiter Δ = R/31208 Δ = R/15604 Δ = R/8192 Δ = R/4096 Δ = R/1024 Δ = R/256 30 τ St2/3 31 2.0 10 20 30 40 00.5 1.00 1.5 2.52.0 3.53.0 (×104) (×10–3) (×10–4) (×10–3) (a) (b) (c) (d) (e) (f) Figure 11. Plots of (a)hmin/R(τ),(b)p/(ρU2)(τ),and(c) time evolution of (∂p/∂r)R/(ρU2)(τ ) at the spatio-temporal region where the gas pressure is maximum for different values of the discretization level Δ. The numerical results have been calculated for We =4andSt =3×104, which correspond to the values of the dimensionless parameters characterizing a water drop of radius R=1.05 mm impacting a wall with velocity U=0.52 m s−1reported by de Ruiter et al. (2012). The numerical results are compared with the experimental measurements in de Ruiter et al. (2012)intheinsetof(a). The values of Δin the legend indicate the minimum grid size. Numerical values at τ∗≈12 are given for (d)hm/R,(e)p/(ρU2)and ( f)(∂p/∂r)R/(ρU2)as functions of R/Δ,withΔindicating the minimum grid size. pressure gradient and the minimum gas film thickness become independent of the grid size and converge towards well-defined values. Figure 11 also shows that our numerical results reproduce, with small relative errors, the experimental value of the minimum gas film thickness reported by de Ruiter et al. (2012),andtable 1 reports the minimum number of cells in the zdirection used to compute the gas velocity field within the lubrication layer; see also figures 16 and 17. 980 A35-23 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press P. García-Geijo, G. Riboux and J.M. Gordillo We St Size 43.03 ×10415 12 1.20 ×10423 12 2.60 ×10411 22 2.12 ×10410 36 2.60 ×1047 48 1.84 ×1049 48 3.60 ×1045 60 1.80 ×1049 Table 1. Minimum number of numerical cells used to describe the flow in the lubrication layer at the radial position where the thickness of the gas film is minimum. In consequence, the number of cells in the zdirection for the rest of the radial positions and instants of times is larger than the values provided in the table. 0.010 0.015 0.020 0.025 0.030 36 We = 4, St = 3 × 104 We = 12, St = 1.2 × 104 We = 12, St = 2.6 × 104 We = 22, St = 2.1 × 104 We = 36, St = 2.6 × 104 We = 48, St = 1.2 × 104 We = 48, St = 1.8 × 104 We = 48, St = 3.6 × 104 We = 48, St = 4.5 × 104 We = 60, St = 1.8 × 104 10 12 –1 20 30 40 τ pm St–2/3 Figure 12. Comparison between the maximum liquid pressure predicted in (3.5) and the maximum gas pressure at the wall for different instants of time and different values of the Weber and Stokes numbers. The dashed line indicates the transition at τ∗=12 between the pre-impact and post-impact stages. Appendix B. Comparison between the predicted and calculated values of the gas pressure and the gas pressure gradient In this contribution, we do not restrict ourselves to validate our physical model with the results shown in figures 6 and 7; indeed, notice here that the equations for the minimum gas film thickness hmhave been deduced through (3.2), which depends on the value of the local pressure gradient at the spatio-temporal region where the pressure is maximum. Hence our predictions for hmare linked to the correctness of the approximations to the time-dependent values of both the pressure and the pressure gradient given, respectively, in (3.5)and(3.17). This is the reason why figure 12 compares the result in (3.5)with the numerical values of the maximum gas pressure at the wall. Notice that the agreement between predictions and the numerical results improves for increasing values of We due to the fact that the numerical values for the gas pressure depicted in figure 12 are the result of subtracting the capillary pressure from the value of the maximum liquid pressure predicted by (3.5). Moreover, figure 13 shows the time evolution of the pressure gradient at 980 A35-24 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press The skating of drops impacting over gas or vapour layers 0 3 6 10 12 20 30 40 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.5 1.0 1.5 2.0 ξ τ 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.2 0.4 0.8 0.6 1.0 ξ Πc ∗Πi ∗ (×10–3) 102 103 104 105 (×10–4) –∂p/∂r We = 12, St = 1.2 × 104 We = 12, St = 2.6 × 104 We = 22, St = 2.1 × 104 We = 36, St = 2.6 × 104 We = 48, St = 1.2 × 104 We = 48, St = 1.8 × 104 We = 48, St = 3.6 × 104 We = 48, St = 4.5 × 104 We = 60, St = 1.8 × 104 (a) (b)(c) Figure 13. (a) Values of the pressure gradient at the spatio-temporal region where the pressure is maximum for the range of We and St considered in this study. The pressure gradient has been calculated as the slope of the lines tangent to the gas pressure distribution at the wall; see figure 2.(b) The values of the maximum pressure gradients, attained at τ=τ∗≈12, can be predicted well for all values of ξ using the expression for −∂p/∂rin (3.17) corresponding to the capillary scaling. (c) The values of the pressure gradient at τ=τ∗≈12 can be predicted well for ξ4 using the expression for −∂p/∂rin (3.17) corresponding to the inertial scaling. Here, Π∗ i=Πi(τ∗)=(−∂p/∂r)(τ∗)St−5/3/(ρU2/R)and Π∗ c= Πc(τ∗)=(−∂p/∂r)(τ∗)We−2/3St−14/9/(ρU2/R). the region where the pressure is maximum. In both this figure and figure 14, the numerical values of the pressure gradient have been calculated as illustrated in figure 2, namely, as the slope of the lines tangent to the gas pressure distribution at the wall. The results in figures 13(b,c) show that the maximum values of the pressure gradients depicted in figure 13(a) can be predicted using the equations for −∂p/∂rgiven in (3.17) particularized at the instant τ=τ∗≈12. Moreover, figure 14 validates further the predictions given in (3.17) for the time-varying values of the pressure gradient at the spatio-temporal region where the pressure is maximum. Indeed, figure 14(a) shows that the numerical values corresponding to the smaller values of We follow the capillary scaling of −∂p/∂rprovided in (3.17), whereas figure 14(b) confirms that the values of the calculated pressure gradient for We ≥36 are well captured, specially for the larger values of τ, by the equation corresponding to the inertial limit given in (3.17), as expected from the fact that the transition from the capillary to the inertial regimes is controlled by the value of the parameter ¯ ξ=τξ;see(3.15)–(3.16). 980 A35-25 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press P. García-Geijo, G. Riboux and J.M. Gordillo ZHANG,B.,SANJAY,V.,SHI,S.,ZHAO,Y.,LV,C.,FENG, X.-Q. & LOHSE, D. 2022 Impact forces of water drops falling on superhydrophobic surfaces. Phys. Rev. Lett. 129, 104501. ZHANG,P.&LAW, C.K. 2011 An analysis of head-on droplet collision with large deformation in gaseous medium. Phys. Fluids 23 (4), 042102. 980 A35-32 https://doi.org/10.1017/jfm.2024.20 Published online by Cambridge University Press