The skating of drops impacting over gas or vapour layers

J. Fluid Mech. (2024), ol.980, A35, doi:10.1017/j m.2024.20 The ska ing o d ops impac ing o e gas o apou laye s P. Ga cía- Geijo1,G.Riboux 1and J.M. Go dillo1,† 1Á ea de Mecánica de Fluidos, Depa amen o de Ingenie ía Ae oespacial y Mecánica de Fluidos, Uni e sidad de Se illa, A enida de los Descub imien os s/n, 41092 Se illa, Spain (Recei ed 12 June 2023; e ised 22 Decembe 2023; accep ed 30 Decembe 2023) We epo nume ical simula ions con i ming he p edic ions in Go dillo & Riboux (J. Fluid Mech., ol. 941, 2022, A10), whe e we elucida ed he lub ica ion mechanism by which a d op o a low- iscosi y liquid impac ing o e a smoo h solid subs a e ska es o e a hin gas ilm ha p e en s con ac wi h he wall. Mo eo e , wi h he pu pose o explaining he so-called li -o mechanism epo ed in Kolinski e al. (Phys.Re .Le ., ol. 112, issue 13, 2014, 134501), we ex end ou p e ious indings and de i e exp essions o he ime- a ying hickness o he gas laye a he egion whe e he dis ance o he wall is minimum, inding good ag eemen wi h he nume ical esul s. In addi ion, we epo ha ou p edic ions o he minimum hickness o he gas ilm sepa a ing a alling d op om a wall a oom empe a u e ollow closely he expe imen al alues when gas kine ic e ec s a e e ained in he analysis, and also epo ha he analogous equa ion o he minimum hickness o he apou laye o med a e a d op impac s a supe hea ed wall p edic s well he expe imen al measu emen s. Key wo ds: boiling, d ops 1. In oduc ion The impac o a d op o e a subs a e has been he subjec o in ense esea ch e o s du ing he pas decades as a consequence o i s ele ance in a my iad o na u al and echnological p ocesses;see e.g. Josse and & Tho oddsen (2016). One o he many open ques ions ha emains o be sol ed and has ecei ed subs an ial a en ion in ecen yea s e e s o he desc ip ion and quan i ica ion o he condi ions unde which a d op impac ing a supe hea ed subs a e ska es o e a apou ilm, a phenomenon e e ed o as he dynamic †Email add ess o co espondence: [email p o ec ed] © The Au ho (s), 2024. Published by Camb idge Uni e si y P ess. This is an Open Access a icle, dis ibu ed unde he e ms o he C ea i e Commons A ibu ion licence (h p://c ea i ecommons.o g/ licenses/by/4.0), which pe mi s un es ic ed e-use, dis ibu ion and ep oduc ion, p o ided he o iginal a icle is p ope ly ci ed. 980 A35-1 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess P. Ga cía-Geijo, G. Riboux and J.M. Go dillo Leiden os e ec , which limi s he cooling a e o solids in hea ans e echnological applica ions (T an e al. 2012,2013;Shi o ae al. 2016). Indeed, as a consequence o he smallness o he he mal conduc i i y o gases, he exis ence o a s able apou ilm benea h he d op educes he hea lux and he e o e he cooling capaci y o he liquid ( an Limbeek e al. 2017). The heo e ical and nume ical s udies aimed a desc ibing he e ec o a gas o apou laye on he impac o a d op o e a solid subs a e we e pionee ed by Smi h, Li & Wu (2003)and Ko obkin, Ellis & Smi h (2008), who coupled he equa ions o he in iscid mo ion o a wo-dimensional d op impac ing ei he a wall o a hin liquid laye wi h he lub ica ion equa ions o he gas. La e , Mand e, Mani & B enne (2009), Mani, Mand e & B enne (2010)and Mand e & B enne (2012) made use o he heo e ical and nume ical amewo k al eady in oduced by Smi h e al. (2003)and Ko obkin e al. (2008)wi h he pu pose o de i ing scaling ela ionships o he dimple heigh – namely, he hickness o he pancake-shaped bubble en apped a he cen e o he impac ing d op – and also o he minimum dis ance be ween he d op and he wall, which is a ained no a he axis o symme y, bu a an o -cen e posi ion. The scaling o he dimple heigh deduced by Mand e e al. (2009) in he wo-dimensional case and by Hicks & Pu is (2010) o he mo e ealis ic case o sphe ical d ops was con i med ia expe imen s and also ia nume ical simula ions by Bouwhuis e al. (2012) who, in addi ion, disco e ed a new egime domina ed by capilla i y ha desc ibes he en apmen o bubbles a low impac eloci ies. Le us poin ou he e ha he analysis o he en apmen o a gas pocke benea h a alling liquid mass is analogous o he cushioning e ec o he ai en apped a e a solid impac s a ee su ace, a physical si ua ion ha has been analysed by e.g. Wilson (1989), Ross & Hicks (2019)and Moo e (2021), using Wagne ’s o iginal ideas (Wagne 1932). Again, back in he con ex o d ops alling o e a wall, Duchemin & Josse and (2011) de eloped a bounda y in eg al me hod coupled wi h simpli ied lub ica ion equa ions o he gas low, and epo ed nume ical esul s o he minimum ilm hickness ha did no ollow he p edic ions in Mand e e al. (2009), Mani e al. (2010)and Mand e & B enne (2012). F om he pu ely expe imen al poin o iew, and making use o high-speed imaging echniques, Chand a & A edisian (1991)and Tho oddsen e al. (2005)we e he i s o epo he shape and hickness o he bubble en apped a he cen e o he d op, whe eas Kolinski, Mahade an & Rubins ein (2014b) epo ed expe imen al da a on he ime- a ying minimum ilm hickness, and desc ibed wha hey called he li -o mechanism, which akes place when a d op ska es o e a nanome ic gas ilm, inding ha his e ec depends on he gas o liquid iscosi y a io. Ve y ecen ly, Chan elo & Lohse (2021,2023) ex ended he p e ious expe imen al s udies o he case o supe hea ed subs a es, epo ing measu emen s o he minimum ilm hickness as a unc ion o he impac eloci y and o he subs a e empe a u e, and hey scaled and in e p e ed hei own da a using he ideas in Mand e e al. (2009)and Mand e & B enne (2012). Recen ly, Go dillo & Riboux (2022) p esen ed a physical model based on he idea ha bo h he liquid p essu e g adien and he componen o he liquid eloci y along he angen di ec ion o he wall d i e he gas low wi hin he spa io- empo al egion whe e he dis ance be ween he liquid and he solid is minimum. One o he main esul s o Go dillo & Riboux (2022), who made ex ensi e use o Wagne ’s heo e ical amewo k (Wagne 1932), is ha he classical lub ica ion mechanism, by which he Coue e and Poiseuille low a es a e in balance in he sligh ly con e ging geome y o med be ween he d op and he wall, p e en s he liquid con ac ing he solid p o ided ha he minimum gas laye 980 A35-2 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess The ska ing o d ops impac ing o e gas o apou laye s hickness calcula ed in his way is la ge han he heigh o he subs a e aspe i ies o o he in e acial co uga ions (Kim e al. 2011; Kolinski, Mahade an & Rubins ein 2014a; Chan elo & Lohse 2021). The s udy by Go dillo & Riboux (2022) p esen esul s ha di e om hose in Mand e e al. (2009), Mand e & B enne (2012)and Chan elo & Lohse (2021,2023), hence one o he main pu poses in his con ibu ion will be o deciphe which o he wo p edic ions o he minimum ilm hickness co esponding o he case o non-hea ed subs a es (i.e. ei he he one gi en in Mand e e al. (2009), Mani e al. (2010)and Mand e & B enne (2012), o he one deduced in Go dillo & Riboux (2022)) is in be e ag eemen wi h nume ical simula ions ca ied ou using Basilisk (Popine 2015) in he limi in which he Knudsen numbe Kn de ined in e ms o he gas ilm hickness is ze o, namely, when gas kine ic e ec s a e absen (Sp i les 2024). I could appea ha he main con ibu ion he e is no hing bu a con i ma ion o al eady published esul s, bu his is no he case. Indeed, he e we will also p o ide a quan i a i e explana ion o he so-called li -o mechanism i s desc ibed by Kolinski e al. (2014b), by which he minimum hickness o he gas ilm inc eases in ime a e he dis ance o he wall has eached a minimum. In spi e o se e al con ibu ions in he li e a u e on he subjec , some o which a e e y ecen (see e.g. Mish a, Rubins ein & Ryc o 2022), we ha e no ound any physical desc ip ion o heo y aimed a explaining and quan i ying he o iginal obse a ions made by Kolinski e al. (2014b). Then he e we deduce equa ions o he ime-dependen wid h o he gas laye whe e he liquid p essu e is maximum, inding ha ou p edic ions ag ee wi h he nume ical esul s,which –as has been poin ed ou in he pa ag aph abo e – ha e been ob ained in he ideal limi in which he Knudsen numbe cha ac e izing he low in he hin gas ilm is ze o. This con ibu ion also con ains a compa ison be ween ou p edic ions and he expe imen s epo ed by de Rui e e al. (2012)and Chan elo & Lohse (2023) o he case o a non-hea ed subs a e, wi h such a compa ison con aining wo undamen al di e ences om he analogous analysis epo ed in Go dillo & Riboux (2022). Indeed, he minimum ilm hickness is calcula ed he e using an equa ion ha has been alida ed p e iously by means o nume ical simula ions ca ied ou in he ideal limi Kn =0. Bu , in addi ion, in o de o compa e wi h expe imen s, we e ain in he algeb aic exp ession o he minimum ilm hickness he e ec o Kn by eplacing he ac ual gas iscosi y wi h he exp ession o he e ec i e gas iscosi y deduced by Zhang & Law (2011). The good ag eemen be ween ou p edic ions –which do no include any kind o adjus ing pa ame e since he only ee cons an will be de e mined using idealized nume ical simula ions – and expe imen s indica e ha , as was poin ed ou al eady by Li (2016)and Chubynsky e al. (2020), gas kine ic e ec s a e essen ial o p edic he dynamics o impac ing d ops ha ska e o e a gas laye ; in addi ion, hese esul s p o ide u he suppo o ou physical desc ip ion which, as i was poin ed ou abo e, di e om he one gi en in Mand e e al. (2009), Mand e & B enne (2012)and Chan elo & Lohse (2021,2023). Mo eo e , he p edic ions in Go dillo & Riboux (2022) o he cases o d ops impac ing a supe hea ed subs a e – namely, a subs a e wi h a empe a u e la ge han he boiling empe a u e o he liquid – will be compa ed wi h he expe imen al da a epo ed by Chan elo & Lohse (2021,2023) once gas kine ic e ec s a e aken in o accoun h ough he exp essions o he e ec i e gas iscosi y and he e ec i e he mal conduc i i y epo ed by Zhang & Law (2011) and Sha ipo , Cumin & Kalempa (2007), espec i ely. Le us poin ou clea ly he e ha i is no he pu pose o his con ibu ion o desc ibe he con ac be ween he liquid and he solid, i.e. he so-called ouchdown p oblem, which 980 A35-3 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess P. Ga cía-Geijo, G. Riboux and J.M. Go dillo Domain 2.1R Pa, Ta Ts 2.1R z0 ρ, μ R U z Figu e 1. Ske ch showing he axisymme ic domain whe e he nume ical simula ions ha e been ca ied ou using Basilisk (Popine 2015); see Appendix A o de ails. He e, and z, espec i ely, indica e he dis ance o he axis o symme y and he dis ance o he wall in a cylind ical coo dina e sys em. The nume ical box is a squa e o leng h 2.1R. emains an open ques ion due o he ac ha ei he he con ac o he ebound o a d op impac ing a solid depends on a numbe o ac o s, such as he subs a e oughness, he p esence o con aminan s a he ee in e ace, asymme ies, an de Waals o ces,o e en elec os a ic e ec s (Kim e al. 2011; Kolinski e al. 2014a; Sp i les 2024), which lead o di e ences on he ins an a which he gas ilm des abilizes unde e y simila expe imen al condi ions;see e.g. Kolinski e al. (2014a)and de Goede e al. (2019). The explana ion o hese di e ences is ou side he scope o his con ibu ion, which is hen ocused in he desc ip ion o hose egimes in which a d op impac ing a solid subs a e, which migh be hea ed abo e he boiling poin o he liquid o no , ska es o e a hin gas ilm, his being a subjec o ecen in e es in he li e a u e (Sp i les 2024). The manusc ip is s uc u ed as ollows.Sec ion 2is de o ed o p esen ing he esul s o nume ical simula ions ca ied ou using Basilisk (Popine 2015). In § 3,we e iew he physical model p esen ed in Go dillo & Riboux (2022), and compa e ou own p edic ions and hose in Mand e & B enne (2012)and Chan elo & Lohse (2023) wi h he nume ical esul s. Taking in o accoun gas kine ic e ec s, in § 4we compa e ou p edic ions o he minimum gas ilm hicknesses wi h he expe imen al alues gi en in de Rui e e al. (2012)and Chan elo & Lohse (2023). Finally, §5summa izes he main esul s in his con ibu ion. 2. Nume ical esul s co esponding o he case o iso he mal subs a es This sec ion is de o ed o p esen ing he esul s o simula ions ca ied ou using Basilisk (Popine 2015) in he nume ical domain depic ed in igu e 1, which shows a d op o adius Ro a liquid wi h densi y ρ, iscosi y μ, and in e acial ension coe icien σ, alling wi h uni o m eloci y Uagains a wall whose empe a u e Tsis equal o ha o he gas, Ts=Ta (iso he mal subs a e);see Appendix A o de ails on he nume ical implemen a ion. Using R,R/Uand ρU2as he cha ac e is ic alues o leng h, ime and p essu e, he nume ical esul s in his sec ion will be exp essed in e ms o he S okes and Webe numbe s de ined as S =ρUR μa ,We =ρU2R σ,(2.1a,b) 980 A35-4 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess The ska ing o d ops impac ing o e gas o apou laye s (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 = aˉ 0.2 0.3 0.4 0 0.2 0.4 /R 0.6 0.8 1.0 0 0.05 0.10 0.15 /R 0.20 0.25 0.30 0.35 0.40 15 10 5 0.8 0.6 0.4 0.2 = 3s R τ = 5 τ = 8 τ = 12 τ = 20 τ = 40 = aˉ Figu e 2. (a) Time-e ol ing shapes o a d op impac ing a wall wi h We =12 and S =2.6×104.Theinse shows he geome y o bo h he gas pocke and he egion, loca ed a /R=√3s(solid line), whe e he d op ska es o e a hin gas laye . No ice ha he adial posi ion whe e he maximum gas p essu e is a ained, =¯a (dashed line), e i ies ¯a≈R√3s.(b) Spa io- empo al e olu ion o he gas p essu e a he wall co esponding o he alues o We and S o (a). The alues o ∂p/∂ a he spa io- empo al egion whe e he gas p essu e is maximum a e calcula ed as he slopes o he solid lines in he igu e. and a ied wi hin he anges 9 ×103≤S ≤4.5×104,4≤We ≤60, while keeping ixed he alues o he a ios ρa/ρ =10−3and μa/μ =1.8×10−2,wi hρaand μaindica ing he gas densi y and iscosi y, espec i ely. Fo simplici y, nei he comp essibili y no gas kine ic e ec s ha e been e ained in he nume ical simula ions,which, as has been poin ed ou abo e, ha e been ca ied ou o he case o iso he mal subs a es. The nume ical esul s depic ed in igu e 2(a) e eal ha as he d op app oaches he wall, a dimple is o med a he axis o symme y, en apping a nea ly cylind ical gas pocke wi h adius ∝√Rhdand hickness hd.Figu e 2 also shows ha he liquid does no ouch he solid bu , ins ead, ska es o e a hin gas ilm whose minimum hickness hmin hdis a ained a ∝√Rhd. The hickness o he en apped bubble, hd, is deduced om he mass balance (Mand e e al. 2009; Bouwhuis e al. 2012) πU(Rhd)∼2πRhd h3 d 12μa pd √Rhd ,(2.2) whe e i has been aken in o accoun ha he gas low a e pe uni leng h induced by he p essu e jump pd=p( =z=0, )−Pais pd∝(h3 d/μa)(pd/√Rhd).Since in a i s app oxima ion he liquid eloci y ield wi hin he d op is i o a ional, he Eule –Be noulli equa ion pa icula ized a =0, z=hdyields (Bouwhuis e al. 2012) pd∼ρ∂φ ∂ ,(2.3) whe e φ∝U√Rhdis he alue o he eloci y po en ial a =0 c ea ed by a disk o adius √Rhdmo ing in o he liquid wi h a eloci y U;see e.g. Pe e s, an de Mee & Go dillo (2013). Then since he dimple is o med in a cha ac e is ic ime hd/U, he p essu e jump deduced om (2.3) eads pd∝ρU2√R/hd, om which, using (2.2), we ob ain hd∝RS −2/3;(2.4) 980 A35-5 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess P. Ga cía-Geijo, G. Riboux and J.M. Go dillo (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 0.15 0.20 0.25 00.05 0.10 /R 0.15 0.20 0.300.25 τ = 20 = aˉ (hmin)/R = 3s R Figu e 3. (a) Gas laye hickness h( )/Ra τ=20 o We =12 and S =2.6×104. The adial posi ion whe e he minimum ai ilm hickness is a ained, (hmin)/R(do ed e ical line), he adial posi ion whe e he maximum p essu e is eached, /R=¯a/R(dashed e ical line), and he we ing adius, /R=√3s(solid e ical line), a e indica ed. Values o he gas p essu e jump a he wall o he ollowing ins an s o ime: (b) τ=12, (c)τ=15, and (d)τ=20. hen he dimple is o med in a cha ac e is ic ime gi en by  d∝hd U→ dU R∝hd R∝S −2/3.(2.5) Hence, aking he o igin o imes a he ins an when he d op would con ac he subs a e i he gas we e no p esen , and de ining he dimensionless imes sand τas s= U R,τ=S 2/3s=S 2/3 U R,(2.6a,b) he esul in (2.5) indica es ha he dimple is o med a he ins an o ime τ=τ∗,wi h τ∗≈12 (Go dillo & Riboux 2022). Figu e 2(b) shows he spa io- empo al e olu ion o he gas p essu e a he wall i.e. a z=0. Fo a ixed alue o τ, he esul s in igu e 2(b) e eal ha he gas p essu e inc eases adially, eaching a maximum a an o -cen e posi ion ha mo es owa ds la ge alues o as ime p og esses. In addi ion, igu e 2 shows ha he maximum gas p essu e a he wall, and also he alues o he local p essu e g adien a he adial posi ion whe e he maximum p essu e is loca ed, inc ease wi h τ, eaching a maximum a τ=τ∗≈12. Mo eo e , he esul s in igu e 3 e eal ha he maximum gas p essu e and he maximum p essu e g adien , calcula ed as he slope o he lines depic ed in igu e 2(b), a e eached a he adial posi ion =a(s)=R√3s,(2.7) namely, a he adius o he ci cula egion ha is we ed by a d op impac ing wi h eloci y Uo e a wall, a esul ha was checked ca e ully agains expe imen s and was deduced in Riboux & Go dillo (2014) in he con ex o d op impac using Wagne ’s 980 A35-6 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess The ska ing o d ops impac ing o e gas o apou laye s 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 P e-impac We = 12, S = 1.2 × 104 We = 4, S = 3 × 104 We = 12, S = 2.6 × 104 We = 22, S = 2.1 × 104 We = 36, S = 2.6 × 104 We = 48, S = 1.8 × 104 We = 48, S = 3.6 × 104 We = 60, S = 1.8 × 104 τ (×10–3) (×10–3) (a)(b) Figu e 4. Time e olu ion o (a) he minimum gas laye hickness hmin,and(b) he hickness o he gas laye measu ed a /R=√3s,hm=h( =R√3s), co esponding o he alues o We and S indica ed in he legend. The p e-impac s age τ<12 is highligh ed in blue. heo e ical amewo k (Wagne 1932). Indeed, Wagne ’s heo y, which has been known o nea ly a cen u y in he con ex o wa e en y p oblems, pe mi s us o deduce he esul in(2.7) in a qui e s aigh o wa d manne ; see also Wilson (1989), whe e (2.7) was deduced in he con ex o ship slamming. Reade s in e es ed in he igo ous applica ion o Wagne ’s heo y (Wagne 1932) o di e en physical phenomena using ma ched asymp o ic echniques a e di ec ed o Ko obkin & Pukhnacho (1988), Wilson (1989), Howison, Ockendon & Wilson (1991), Scolan & Ko obkin (2001), Oli e (2002), Ko obkin & Scolan (2006)and Moo e (2014). The esul s in igu e 3 also sugges ha in spi e o he minimum gas ilm hickness hmin(τ) is no a ained a he adial posi ion whe e he maximum gas p essu e is eached, namely, hm(τ) /=hmin(τ ),wi h hm(τ) =h( =a(s)), (2.8) hm(τ) ≈hmin(τ);see igu e 2(a). This esul is app ecia ed mo e clea ly in igu e 4, which p o ides he alues o hm(τ) and hmin(τ) o di e en alues o he Webe and S okes numbe s explo ed in his s udy. Indeed, igu e 4 e eals ha hmin(τ) =hm(τ ) o he ins an s o ime (highligh ed in blue in igu e 4)τ≤τ∗, and also ha hm(τ) > hmin(τ ) o τ>τ ∗. Due o he ac ha he alues o he maximum gas p essu e, he maximum p essu e g adien and also he minimum o hm(τ) a e a ained a τ=τ∗, he ins an s o ime τ<τ ∗will be e med, in wha ollows, as p e-impac s age, whe eas hose co esponding o τ>τ ∗will be e e ed he e as pos -impac s age; see also he ime-e ol ing dimple shapes included as an inse o igu e 2(a). Hence he e we will conside ha he ‘impac ’ akes place when he maximum p essu e is a ained a τ=τ∗,so, using he esul in (2.6a,b)–(2.7), his e en is localized a he adial posi ion =R√3τ∗S −2/3≈6RS −1/3; he e o e he minimum hickness o he gas ilm is a ained a he dimensionless ins an τ=τ∗≈12 and a he dimensionless adial posi ion /R≈6S −1/3. The di ision o he impac p ocess in o wo well-de ined s ages is c ucial o unde s anding he di e ences be ween he p esen analysis and he p edic ions in Mand e & B enne (2012)and Chan elo & Lohse (2021,2023), who desc ibe he ins an s p e ious o he ‘impac ’, namely, he p e-impac s age aking place o τ<τ ∗. In con as , he p edic ions in Go dillo & Riboux (2022) we e deduced wi h he pu pose o desc ibing bo h he impac and he pos -impac s ages, and he e o e should be applicable o quan i y he di e en e en s aking place o τ≥τ∗. 980 A35-7 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess P. Ga cía-Geijo, G. Riboux and J.M. Go dillo 3.0 0 1 2 ||u /U|| = 5 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 ( – 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) Figu e 5. Radial componen o he dimensionless gas eloci y ield, u /U, ep esen ed in he ame o e e ence mo ing wi h he we ing eloci y Vmgi enin(3.3) o h ee di e en alues o ( −¯a)/¯ haand wo di e en ins an s o ime: (a)τ=20 and (b)τ=40. He e, ¯a≈R√3sindica es he adial posi ion whe e he maximum gas p essu e is a ained (see igu e 3)and¯ ha=2/(9π)¯a3≈ha,m,wi hha,mgi en in (3.4). He e, We =12, S =2.60 ×104, whe eas z(1)indica es he e ical coo dina e o he in e ace a he minimum alue o ( −¯a)/¯ ha ep esen ed in he igu e. The nex sec ion is de o ed o checking which o he p edic ions o he minimum ilm hickness – ei he he ones deduced in Mand e & B enne (2012) o hose in Go dillo & Riboux (2022)– a e in be e ag eemen wi h he nume ical esul s. 3. Modelling he impac o d ops o e iso he mal subs a es 3.1. Re iew o p e ious esul s The p edic ions in Mand e & B enne (2012), la e ex ended by Chan elo & Lohse (2021,2023) o he case o supe hea ed subs a es, a e based on he ollowing idea: he minimum gas ilm hickness is a ained when a sel -simila solu ion desc ibing he p e-impac s age a he egion whe e he dis ance o he wall is minimum, ails o p edic he low o τ≥τ∗because he capilla y and con ec i e e ms in he momen um equa ion, ini ially neglec ed, become o he o de o he dominan e ms in he app oxima e solu ion. Hence, as a as we unde s and, he esul s in Mand e e al. (2009), Mand e & B enne (2012)and Chan elo & Lohse (2021,2023) ha e been deduced using an a gumen ha nei he desc ibes no iden i ies he physical mechanism ha p e en s he con ac be ween he liquid and he wall. In con as , he physical model in Go dillo & Riboux (2022) desc ibes he lub ica ed impac o a d op o e a wall o ins an s o ime τ≥τ∗. The physical idea behind he p edic ions in Go dillo & Riboux (2022) elies on he well-known lub ica ion mechanism depic ed in igu e 5, whe e he gas eloci y ield calcula ed nume ically along he egion whe e he gas p essu e is maximum, is ep esen ed in a ame o e e ence mo ing wi h he we ing eloci y a which he local maximum p essu e p opaga es adially ou wa ds, 980 A35-8 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess The ska ing o d ops impac ing o e gas o apou laye s namely (see igu es 2 and 3) Vm(s)=da d =U 23 s,(3.1) whe e he subsc ip mis used o deno e he alues o quan i ies pa icula ized a =a(s); see (2.7). Figu e 5 shows ha in he mo ing ame o e e ence, he gas eloci y ield can be exp essed as he supe posi ion o he Poiseuille (pa abolic) eloci y p o ile induced by he a ou able p essu e g adien poin ing adially ou wa ds, owa ds he a mosphe e (see igu e 2b), plus he Coue e (linea ) eloci y p o ile caused by he ela i e mo ion be ween he poin o maximum p essu e and he wall, which is di ec ed owa ds he axis o symme y. No ice ha he poin o maximum p essu e is a ained a he adial posi ion whe e he liquid in e acial eloci y in he labo a o y ame o e e ence equals he eloci y o he mo ing ame o e e ence; see e.g. Wagne (1932)and Go dillo & Riboux (2022). Hence, o he case o iso he mal subs a es, he dis ance be ween he d op and he wall a =a(s)du ing he ins an s close o he one o which he minimum ilm hickness is a ained, can be quan i ied h ough he equa ion (Go dillo & Riboux 2022) −h3 m 12μa ∂p ∂ −Vmhm 2≈0.(3.2) Non-con inuum e ec s, which could ha e been e ained in (3.2) by conside ing alues o he slip leng hs a he in e ace and a he wall di e en om ze o (see e.g. Duchemin & Josse and (2012)and Riboux & Go dillo (2014)), will be conside ed in § 4using he app oach de ailed in Li (2016) in his nume ical s udy o he head-on collision o d ops. This app oxima ion consis s in modi ying he alue o he ac ual iscosi y by using he equa ion o he e ec i e iscosi y deduced in Zhang & Law (2011), which depends explici ly on he Knudsen numbe de ined in e ms o he gas ilm hickness. While gas kine ic e ec s need o be e ained in o de o compa e ou p edic ions wi h expe imen s, an de Waals e ec s can be neglec ed sa ely in he modelling because hese o ces become ele an only when hm20 nm (Sp i les 2024), namely, o alues o he gas ilm hicknesses ha a e well below hose measu ed expe imen ally by de Rui e e al. (2012) o he case o iso he mal impac s, and by Chan elo & Lohse (2021,2023) o he case o d ops impac ing he wall in he dynamic Leiden os egime. No ice ha (3.2) exp esses ha he alue o he Poiseuille low a e pe uni leng h induced by he la ge p essu e g adien gene a ed a ound =a(s)(see igu e 3) needs o be balanced by he Coue e low because, o he wise, he gas benea h he egion whe e he p essu e is maximum would low adially ou wa ds, emp ying his olume, and consequen ly he liquid would make con ac wi h he wall. In o de o deduce an equa ion o hm, he nex s ep ha we ollowed in Go dillo & Riboux (2022) was o make use o he ac ha he e ical in e acial eloci ies du ing τ≥τ∗a e much smalle han he impac eloci y,i.e.(1/U)∂h/∂ 1, and also ha he gas ilm is slende , ∂h/∂ 1. In his way, since he Reynolds numbe e i ies Re =S μa/μ 1, and hence he p oduc ion o o ici y a he gas–liquid in e ace is con ined wi hin small bounda y laye s, he liquid eloci y and p essu e ields can be app oxima ed by he i o a ional alues calcula ed using Wagne ’s heo e ical amewo k Wagne (1932). A his poin , no ice ha he condi ion (1/U)∂h/∂ 1 also implies ha dhm/d Vmhm–wi h indica ing he cha ac e is ic leng h along which hm a ies,o he o de o ∼hm– due o he ac ha >hmand also because Vm∝US 1/3U, his being he eason why he e m dhm/d has been neglec ed in he mass balance (3.2). 980 A35-9 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess P. Ga cía-Geijo, G. Riboux and J.M. Go dillo in Go dillo & Riboux (2022) ha he hea lux ac oss he liquid he mal bounda y laye can be neglec ed o he pa icula case o he expe imen s epo ed in Chan elo & Lohse (2021,2023). The e o e, he equa ion analogous o (3.2) o he case o d ops impac ing o e supe hea ed subs a es eads −h3 m 12μ ∂p ∂ −Vmhm 2≈k T ρ L  hm ,(4.2) whe e we ha e made use o (4.1). The subs i u ion o (3.3)–(3.6)in o(4.2) yields he ollowing equa ion o hm/R: y−1+1 6 μa μ y=β∗,(4.3) wi h y=S 2(/R) Vm Uhm R2 and β∗=βρ ρ μ μa,whe e β=k T μ L.(4.4) Consequen ly, hm R=2yS −1Vm U−1 R1/2 ,(4.5) wi h y=3μ μa1+1+2β∗ 3μa μ .(4.6) The subs i u ion o (4.6) and o ei he (3.4)o (3.8)in o(4.5)p o ides he same exp essions as hose gi en in (3.17), which a e hen alid o desc ibe he ska ing o d ops o e ei he a gas o a apou laye , he only di e ence being ha y=6 o he case o iso he mal impac s, whe eas yis gi en by (4.4)and(4.6) o he cases o d ops impac ing a subs a e in he dynamic Leiden os egime. Hence ou physical desc ip ion di e s subs an ially om ha o Chan elo & Lohse (2021,2023), who deduce di e en equa ions depending on whe he he subs a e is supe hea ed o no , he e o e he iso he mal case canno be eco e ed using hei esul s co esponding o supe hea ed subs a es in he limi in which he p oduc ion o apou ends o ze o. The ecen e iew on he subjec by Sp i les (2024)s a es clea ly ha gas kine ic e ec s canno be neglec ed in he desc ip ion o he head-on collision o d ops (Li 2016)o in he impac o d ops on a subs a e (Riboux & Go dillo 2014; Chubynsky e al. 2020)i he alue o he Knudsen numbe , de ined as Kn =λ hm ,(4.7) wi h λdeno ing he mean ee pa h o he gas, becomes Kn 0.1, which is he case o in e es he e. Indeed, he ypical alues o he gas ilm hickness in he expe imen s epo ed by de Rui e e al. (2012) and Chan elo & Lohse (2021,2023) a y be ween 103 980 A35-16 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess The ska ing o d ops impac ing o e gas o apou laye s and 102nanome es, whe eas he alues o he mean ee pa h o ai and e hanol a no mal condi ions a e, espec i ely, λa≈69 nm and λe h ≈50 nm.(4.8a,b) Then, in o de o accoun o gas kine ic e ec s, we de ine an e ec i e gas iscosi y μ∗(Kn)(Sp i les 2024) and make use o he exp ession deduced by Zhang & Law (2011), employed success ully by Li (2016) in his s udy o he head-on collision o d ops: μ∗=μ 1+6.0966 Kn +0.9650 Kn2+0.6967 Kn3,(4.9) whe e μ e e s o he alue o he ac ual gas iscosi y. No ice ha o hecaseo d ops impac ing a non-hea ed subs a e, μ in (4.6) e e s o he e ec i e ai iscosi y, namely, μ =μ∗ a(see (4.9)), whe eas he Knudsen numbe de ined in (4.7) is calcula ed using he alue o he mean ee pa h gi en by λ=λa Pa Pa+pm ,(4.10) wi h λaand pmgi en, espec i ely, in (4.8a,b)and(3.5). Fo he case o d ops impac ing a supe hea ed subs a e, μ =μ∗ e h and λ=λe h Pa Pa+pm 273 +0.5(Tb+Ts) Ta ,(4.11) wi h Ta=298 K, and Tb=78 and Ts e e o he alues in Celsius o he boiling empe a u e o e hanol and o he subs a e empe a u e, espec i ely. Fo he case o d ops impac ing a supe hea ed subs a e, he e we also include gas kine ic e ec s in he hea ans e om he wall in o he liquid, making use o he e ec i e hea conduc i i y o he apou gi en by (Sha ipo e al. 2007) k∗ =k 1+3.91 Kn.(4.12) As poin ed ou in he In oduc ion, an de Waals e ec s a e no e ained in he analysis because in all he expe imen al esul s epo ed by de Rui e e al. (2012)and Chan elo & Lohse (2021,2023), he gas ilm hickness is well abo e 20 nm, which is he leng h scale below which hese o ces become ele an (Sp i les 2024). 4.2. Compa ison wi h expe imen s The compa ison wi h he nume ical esul s in §3con i ms ou physical desc ip ion,andi is now ou pu pose in his subsec ion o check whe he ou esul s, once gas kine ic e ec s a e aken in o conside a ion, can also be used o p edic he expe imen al da a epo ed by de Rui e e al. (2012) and Chan elo & Lohse (2021,2023) o d ops impac ing o e ei he iso he mal o supe hea ed subs a es. The equa ions gi en in (4.6)and(3.17) can be used o p edic he minimum gas ilm hickness o a bi a y alues o Ts. Howe e , he uni ied desc ip ion o he ska ing o a d op o e a gas o apou ilm p o ided by (4.6)and(3.17) also equi es us o in oduce he e ec o he apou p oduced a he dimple on he ins an o ime τ∗a which he cen al bubble is o med, i.e. a he ins an when he minimum ilm hickness is a ained a ≈RS −1/3√3τ∗. 980 A35-17 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess P. Ga cía-Geijo, G. Riboux and J.M. Go dillo Then, o he case o supe hea ed subs a es, he e m on he le -hand side o he mass conse a ion equa ion (2.2) needs o be modi ied in o de o ake in o accoun he e apo a ion o he liquid. Consequen ly, in his case, he low a e ha needs o be e acua ed adially ou wa ds om he axis o symme y is π(Rhd)(U+k T/(ρ hdL)). The e o e, exp essing hd/R=τ∗S −2/3and ollowing he same s eps as hose de ailed in §3, he mass balance (2.2)p o ides he ollowing equa ion o τ∗(Go dillo & Riboux 2022): τ∗5/2=12.43/2τ∗+β∗S −1/3,(4.13) wi h β∗de ined in (4.4). No ice ha (4.13) eco e s he alue o τ∗≈12 co esponding o he case o iso he mal subs a es, o which β∗=0. Be o e compa ing ou p edic ions wi h he expe imen al alues p o ided in de Rui e e al. (2012) and Chan elo & Lohse (2021,2023), i is i s necessa y o exp ess he eal empe a u e-dependen ma e ial p ope ies o he gas and o he apou as unc ions o he subs a e empe a u e. This is done he e using he Py hon ou ines p o ided as supplemen a y ma e ial a ailable a h ps://doi.o g/10.1017/j m.2024.20, which implemen (3.17), (4.6)and (4.7)–(4.12) and make use o he alues σ=22 ×10−3Nm −1o σ=17 ×10−3Nm −1 o he in e acial ension coe icien o e hanol a ei he oom o boiling empe a u e, wi h hese alues aken om www.ddbs .com. No ice ha all he ma e ial p ope ies a e quan i ied a he mean empe a u e (Tb+Ts)/2as de ailed in Go dillo & Riboux (2022),and his is e lec ed in he Py hon ou ines p o ided as supplemen a y ma e ial. Le us poin ou he e ha we ha e no conside ed he e ec o Ma angoni s esses in ou physical model because he liquid loca ed a a dis ance hm om he wall is e apo a ing along a egion o leng h , he e o e he in e acial empe a u e emains cons an and equal o he boiling empe a u e o he liquid a he egion o in e es he e, namely, whe e he minimum ilm hickness is a ained. Figu e 8 e eals ha he minimum gas ilm hickness can be p edic ed using he equa ions co esponding o he capilla y o ine ial limi s in (3.17)and (4.4)–(4.12)wi h ela i e e o s ∼30%. Fo he case o iso he mal subs a es, he p edic ed alue o he minimum ilm heigh has been calcula ed in igu e 8 e aining gas kine ic e ec s and using he capilla y limi in (3.17)wi hA=3.5, namely, he alue deduced om igu e 6 o he case o d ops impac ing a wall a oom empe a u e in he ideal case Kn =0. Hence o he case o subs a es a oom empe a u e, we conclude ha he expe imen al da a can be app oxima ed using he equa ion (see (3.17)) hm R≈3.5×6μ∗ a μa2/3 122/3We−1/3S −10/9,(4.14) wi h μ∗ a=μ∗de ined in (4.9). Figu e 8 also compa es he expe imen al measu emen s in Chan elo & Lohse (2021,2023) wi h ou p edic ions. In his case, since he apou laye p e en s con ac be ween he liquid and he wall, a ansi ion be ween he capilla y and ine ial egimes in (3.17) is obse ed when he impac eloci y inc eases. In his case, he equa ions o he minimum ilm hickness used in he compa isons o igu e 8 a e (see (3.17)) hm R≈2.3y2/3τ∗2/3We−1/3S −10/9(4.15) 980 A35-18 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess The ska ing o d ops impac ing o e gas o apou laye s 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/h h hm/h h 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) Figu e 8. (a) Compa ison be ween he p edic ions gi en in (4.14)–(4.16) and he alues o he minimum hickness o he ai o apou laye s measu ed by de Rui e e al. (2012) (DeR) and Chan elo & Lohse (2023) (C&L) o di e en alues o he impac eloci y Uand o di e en alues o he subs a e empe a u e Ts. He e, he alues o he e ec i e gas iscosi y and he e ec i e gas conduc i i y ha e been modi ied aking in o accoun kine ic e ec s h ough (4.7)–(4.12). Solid lines ep esen he p edic ions o hmco esponding o he capilla y egime, whe eas he p edic ions o hmin he ine ial egime a e ep esen ed using dashed lines. (b) Compa ison be ween he p edic ed and measu ed alues o hmin (a) as a unc ion o ξ.(c) The expe imen al da a a e compa ed he e only wi h he p edic ions gi en by (4.16). In his case, he alue o he p e ac o is 1.15 ins ead o 1.25. (d) Compa ison be ween he p edic ed and measu ed alues o hmin (c) as a unc ion o ξ.The dashed ho izon al lines in (b,d) a e placed a 1 ±0.3. and hm R≈1.25τ∗S −7/6y1/2(4.16) o he capilla y and ine ial egimes, espec i ely. In (4.15)–(4.16), yis gi en by (4.6), τ∗has been calcula ed using (4.13), and gas kine ic e ec s ha e been quan i ied h ough (4.7)–(4.12). The alue o he p e ac o in (4.15), co esponding o he capilla y limi in (3.17), di e s om ha in (4.14). This could be due o he di e ences in he local geome y o he in e ace o ≈a; indeed, he expe imen s in igu e 3 o Chan elo & Lohse (2021) show ha he cu a u e o he in e ace nea ≈a o he case o Leiden os d ops inc eases wi h Ts(see also Kolinski e al. 2014b), a ac implying ha he p e ac o a ec ing in (4.5) o he case o supe hea ed subs a es should be smalle han o he case o iso he mal impac s. No ice also ha since he local cu a u e is e y much dependen on whe he he subs a e is supe hea ed o no , i could also be he case ha deRui e e al. (2012)and Chan elo & Lohse (2023)p o ide he expe imen al alues o hmin(τ =12) o he case o iso he mal subs a es, whe eas Chan elo & Lohse (2021,2023)p o ide hm(τ ≈12) o he case o supe hea ed subs a es; in ha case, he alue o he cons an in (4.15) would be e y simila o he one in (3.19). Figu es 8(c,d) 980 A35-19 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess P. Ga cía-Geijo, G. Riboux and J.M. Go dillo also show a compa ison be ween he p edic ed and measu ed minimum ilm hicknesses unde he app oxima ion ollowed in Go dillo & Riboux (2022), whe e we conside ed ha he minimum ilm hickness could be p edic ed using he ine ial app oxima ion gi en in (4.16). In his case, igu es 8(c,d)show ha he expe imen al measu emen s by Chan elo & Lohse (2021,2023) can be p edic ed easonably well, wi h ela i e e o s ±30 %, using a alue1.15 o he p e ac o in (4.16), which is e y simila o ha deduced om igu e 6. No ice also ha igu e 8 also includes he p edic ed minimum ilm hickness co esponding o he la ges empe a u e, Ts=295 ◦C, when he alue o he p e ac o in (4.16)is a ied om 1.25 o 1.7. A possible eason why an inc ease in he alue o he p e ac o imp o es he compa ison wi h he expe imen al da a o he case o he highes subs a e empe a u e could be he ac ha he slende app oxima ion unde which (3.17)a ededuced b eaks o su icien ly la ge alues o Ts. Indeed, he pa ame e exp essing he a io be ween he minimum ilm hickness and he leng h along which he p essu e g adien s in he liquid ake place, namely, hm ha,m≈y1/2S −1/6,(4.17) whe e we ha e made use o (3.4)and(4.16), could become la ge han uni y o subs a e empe a u es Tsexceeding he h eshold alue gi en by he condi ion (see (4.17)) y2S 2/3⇒k (Ts−Tb) μ Lρ ρ S 2/3,(4.18) whe e use o (4.6) has been made. The esul s depic ed in igu e 9 e eal ha , indeed, he spa ial egion whe e he maximum p essu e g adien is a ained is clea ly no slende o he la ge alue o Tsbecause he ilm hickness is la ge han he leng h along which he liquid p essu e g adien akes place. In hese cases, since hm≈ha,m, he capilla y p essu e is ∼σhm/h2 a,m∼σ/ha,m, wi h his alue being simila o he liquid o e p essu e pmgi en in (3.5) because pm σ/ha,m∼We S −1/3,(4.19) which happens o be o o de uni y o he expe imen s co esponding o he la ges empe a u es epo ed by Chan elo & Lohse (2023). Then he loss o slende ness o he la ges alue o Tsimplies la ge capilla y p essu es because hmis no much smalle han ha,m, a ac also implying ha he alues o he gas p essu e g adien a e smalle han hose co esponding o he slende limi in which (3.17) ha e been deduced. A educ ion in he p essu e g adien implies la ge alues o hm(see (4.2)), and his ac could be behind he esul depic ed in igu e 8 o he case Ts=295 ◦C. Le us also poin ou ha he disc epancies be ween he p edic ions and expe imen al measu emen s o he case Ts=295 ◦C could also be o igina ed as a consequence o limi a ions in he empo al esolu ion o he expe imen s ca ied ou by Chan elo & Lohse (2023) o he highes impac eloci ies. 5. Conclusions In his con ibu ion, we ha e p esen ed nume ical simula ions alida ing he p edic ions in Go dillo & Riboux (2022), whe e we desc ibed he lub ica ion mechanism by which a d op alling o e a subs a e ska es o e a gas o apou laye . Wi h he pu pose o explaining and quan i ying he li -o mechanism epo ed empi ically by Kolinski e al. (2014b)and 980 A35-20 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess The ska ing o d ops impac ing o e gas o apou laye s 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 Figu e 9. Ra io be ween he measu ed minimum ilm hickness and he leng h ha,mgi enin(3.4) along which p essu e g adien s ake place. The meanings o he symbols a e he same as in igu e 8. in es iga ed nume ically by Mish a e al. (2022), we ha e ex ended ou p e ious esul s and ha e p o ided equa ions, including he alues o he p e ac o s, ha ep oduce closely he ime e olu ion o he minimum ilm hickness calcula ed nume ically. Mo eo e , wi h he pu pose o compa ing ou esul s wi h he measu ed alues o he gas o apou ilm hicknesses, we ha e included gas kine ic e ec s in he algeb aic equa ions o he minimum hickness o he gas laye , inding ha ou p edic ions, up o p e ac o s, a e in good ag eemen wi h he esul s epo ed by de Rui e e al. (2012)and Chan elo & Lohse (2021,2023). Supplemen a y ma e ial. Supplemen a y ma e ial is a ailable a h ps://doi.o g/10.1017/j m.2024.20. Acknowledgemen s. J.M.G. hanks P. Chan elo and D. Lohse o ui ul discussions. Funding. This wo k has been suppo ed by he g an PID2020-115655GB-C21, inanced by he Spanish MCIN/AEI/10.13039/501100011033, and all he simula ions we e un on he High Pe omance Compu ing clus e p o ided by he Cen o In o má ico Cien í ico de Andalucía (CICA). Decla a ion o in e es s. The au ho s epo no con lic o in e es . Au ho ORCIDs. P. Ga cía-Geijo h ps://o cid.o g/0000-0001-8608-4804; G. Riboux h ps://o cid.o g/0000-0003-2395-1653; J.M. Go dillo h ps://o cid.o g/0000-0003-1431-3780. Au ho con ibu ions. P.G.-G. ca ied ou he nume ical simula ions and analysed he da a, G.R. analysed he da a and pa ly w o e he pape , and J.M.G. designed he esea ch and he heo y, and w o e he pape . All au ho s e iewed he esul s and app o ed he inal e sion o he manusc ip . Appendix A. Nume ical simula ions The nume ical esul s in § 2, which ha e been calcula ed using he ee so wa e Basilisk (Popine 2015), ex ending he ideas in he codes by Sanjay (2022)and Zhang e al. (2022), simula e he axisymme ic impac o a d op o adius Rini ially cen ed a z=1.006R, alling wi h a uni o m eloci y Uo e an impe meable ho izon al subs a e. The nume ical esul s ha e been ob ained by imposing symme y condi ions a he axis, he impene abili y and no-slip bounda y condi ions a he wall limi ing he compu a ional domain depic ed in igu e 1,and ou low bounda y condi ions a he wo emaining 980 A35-21 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess P. Ga cía-Geijo, G. Riboux and J.M. Go dillo 0.360 0.25 0 0.05 0.10 0.15 0.20 0.30 0.35 0.40 0.361 /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) Figu e 10. Each black do ep esen s he e ex o a nume ical cell, whe eas he in e ace is plo ed using a blue solid line. (a) The nume ical domain is di ided in o egions wi h di e en disc e iza ion le els: ed box, h/R<0.1and /R<1.0, wi h cell sizes Δ≥R/1950; yellow box, h/R<0.01 and /R<√3s+0.15, wi h cell sizes Δ≥R/31208. Fo he es o he nume ical domain, Δ≥R/975. (b) A close iew o he egion a ound he ad ancing on . (c) De ailed iew o he spa ial egion a ound he we ing adius =√3s( ed dashed line). He e, We =12, S =2.60 ×104and τ=0.0435 ≈38 S −2/3. bounda ies, whe e bo h he p essu e and he g adien o no mal eloci ies a e se o ze o. We ha e checked ha he nume ical esul s a e una ec ed by he dimensions o he compu a ional domain. We ha e made use o an adap i e Ca esian mesh, which e ines he solu ion bo h a he in e ace and a he egions wi h he la ges eloci y g adien s. The ole ances o he olume ac ion ield, o he eloci y ield and o he cu a u e a e se o 10−3,10 −2and 10−6, espec i ely, and he maximum le el o g id e inemen is de ined in each o he di e en egions in o which he compu a ional domain has been di ided; indeed, he g id size is Δ≥R/975 o z>0.1, bu Δ≥R/31208 nea he wall, whe e he eloci y and p essu e g adien s each he la ges alues;see he igu e 10 cap ion o de ails. Mo eo e , we ha e ca ied ou a sensi i i y analysis in o de o e i y ha he esul s epo ed a e independen o he g id size; indeed, igu e 11 shows ha he alues o he p essu e, he 980 A35-22 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess The ska ing o d ops impac ing o e gas o apou laye s τ 0 1000 2000 (∂p/∂ )/(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 Rui e Δ = R/31208 Δ = R/15604 Δ = R/8192 Δ = R/4096 Δ = R/1024 Δ = R/256 30 τ S 2/3 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) ( ) Figu e 11. Plo s o (a)hmin/R(τ),(b)p/(ρU2)(τ),and(c) ime e olu ion o (∂p/∂ )R/(ρU2)(τ ) a he spa io- empo al egion whe e he gas p essu e is maximum o di e en alues o he disc e iza ion le el Δ. The nume ical esul s ha e been calcula ed o We =4andS =3×104, which co espond o he alues o he dimensionless pa ame e s cha ac e izing a wa e d op o adius R=1.05 mm impac ing a wall wi h eloci y U=0.52 m s−1 epo ed by de Rui e e al. (2012). The nume ical esul s a e compa ed wi h he expe imen al measu emen s in de Rui e e al. (2012)in heinse o (a). The alues o Δin he legend indica e he minimum g id size. Nume ical alues a τ∗≈12 a e gi en o (d)hm/R,(e)p/(ρU2)and ( )(∂p/∂ )R/(ρU2)as unc ions o R/Δ,wi hΔindica ing he minimum g id size. p essu e g adien and he minimum gas ilm hickness become independen o he g id size and con e ge owa ds well-de ined alues. Figu e 11 also shows ha ou nume ical esul s ep oduce, wi h small ela i e e o s, he expe imen al alue o he minimum gas ilm hickness epo ed by de Rui e e al. (2012),and able 1 epo s he minimum numbe o cells in he zdi ec ion used o compu e he gas eloci y ield wi hin he lub ica ion laye ; see also igu es 16 and 17. 980 A35-23 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess P. Ga cía-Geijo, G. Riboux and J.M. Go dillo We S 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 numbe o nume ical cells used o desc ibe he low in he lub ica ion laye a he adial posi ion whe e he hickness o he gas ilm is minimum. In consequence, he numbe o cells in he zdi ec ion o he es o he adial posi ions and ins an s o imes is la ge han he alues p o ided in he able. 0.010 0.015 0.020 0.025 0.030 36 We = 4, S = 3 × 104 We = 12, S = 1.2 × 104 We = 12, S = 2.6 × 104 We = 22, S = 2.1 × 104 We = 36, S = 2.6 × 104 We = 48, S = 1.2 × 104 We = 48, S = 1.8 × 104 We = 48, S = 3.6 × 104 We = 48, S = 4.5 × 104 We = 60, S = 1.8 × 104 10 12 –1 20 30 40 τ pm S –2/3 Figu e 12. Compa ison be ween he maximum liquid p essu e p edic ed in (3.5) and he maximum gas p essu e a he wall o di e en ins an s o ime and di e en alues o he Webe and S okes numbe s. The dashed line indica es he ansi ion a τ∗=12 be ween he p e-impac and pos -impac s ages. Appendix B. Compa ison be ween he p edic ed and calcula ed alues o he gas p essu e and he gas p essu e g adien In his con ibu ion, we do no es ic ou sel es o alida e ou physical model wi h he esul s shown in igu es 6 and 7; indeed, no ice he e ha he equa ions o he minimum gas ilm hickness hmha e been deduced h ough (3.2), which depends on he alue o he local p essu e g adien a he spa io- empo al egion whe e he p essu e is maximum. Hence ou p edic ions o hma e linked o he co ec ness o he app oxima ions o he ime-dependen alues o bo h he p essu e and he p essu e g adien gi en, espec i ely, in (3.5)and(3.17). This is he eason why igu e 12 compa es he esul in (3.5)wi h he nume ical alues o he maximum gas p essu e a he wall. No ice ha he ag eemen be ween p edic ions and he nume ical esul s imp o es o inc easing alues o We due o he ac ha he nume ical alues o he gas p essu e depic ed in igu e 12 a e he esul o sub ac ing he capilla y p essu e om he alue o he maximum liquid p essu e p edic ed by (3.5). Mo eo e , igu e 13 shows he ime e olu ion o he p essu e g adien a 980 A35-24 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess The ska ing o d ops impac ing o e gas o apou laye s 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/∂ We = 12, S = 1.2 × 104 We = 12, S = 2.6 × 104 We = 22, S = 2.1 × 104 We = 36, S = 2.6 × 104 We = 48, S = 1.2 × 104 We = 48, S = 1.8 × 104 We = 48, S = 3.6 × 104 We = 48, S = 4.5 × 104 We = 60, S = 1.8 × 104 (a) (b)(c) Figu e 13. (a) Values o he p essu e g adien a he spa io- empo al egion whe e he p essu e is maximum o he ange o We and S conside ed in his s udy. The p essu e g adien has been calcula ed as he slope o he lines angen o he gas p essu e dis ibu ion a he wall; see igu e 2.(b) The alues o he maximum p essu e g adien s, a ained a τ=τ∗≈12, can be p edic ed well o all alues o ξ using he exp ession o −∂p/∂ in (3.17) co esponding o he capilla y scaling. (c) The alues o he p essu e g adien a τ=τ∗≈12 can be p edic ed well o ξ4 using he exp ession o −∂p/∂ in (3.17) co esponding o he ine ial scaling. He e, Π∗ i=Πi(τ∗)=(−∂p/∂ )(τ∗)S −5/3/(ρU2/R)and Π∗ c= Πc(τ∗)=(−∂p/∂ )(τ∗)We−2/3S −14/9/(ρU2/R). he egion whe e he p essu e is maximum. In bo h his igu e and igu e 14, he nume ical alues o he p essu e g adien ha e been calcula ed as illus a ed in igu e 2, namely, as he slope o he lines angen o he gas p essu e dis ibu ion a he wall. The esul s in igu es 13(b,c) show ha he maximum alues o he p essu e g adien s depic ed in igu e 13(a) can be p edic ed using he equa ions o −∂p/∂ gi en in (3.17) pa icula ized a he ins an τ=τ∗≈12. Mo eo e , igu e 14 alida es u he he p edic ions gi en in (3.17) o he ime- a ying alues o he p essu e g adien a he spa io- empo al egion whe e he p essu e is maximum. Indeed, igu e 14(a) shows ha he nume ical alues co esponding o he smalle alues o We ollow he capilla y scaling o −∂p/∂ p o ided in (3.17), whe eas igu e 14(b) con i ms ha he alues o he calcula ed p essu e g adien o We ≥36 a e well cap u ed, specially o he la ge alues o τ, by he equa ion co esponding o he ine ial limi gi en in (3.17), as expec ed om he ac ha he ansi ion om he capilla y o he ine ial egimes is con olled by he alue o he pa ame e ¯ ξ=τξ;see(3.15)–(3.16). 980 A35-25 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess P. Ga cía-Geijo, G. Riboux and J.M. Go dillo ZHANG,B.,SANJAY,V.,SHI,S.,ZHAO,Y.,LV,C.,FENG, X.-Q. & LOHSE, D. 2022 Impac o ces o wa e d ops alling on supe hyd ophobic su aces. Phys. Re . Le . 129, 104501. ZHANG,P.&LAW, C.K. 2011 An analysis o head-on d ople collision wi h la ge de o ma ion in gaseous medium. Phys. Fluids 23 (4), 042102. 980 A35-32 h ps://doi.o g/10.1017/j m.2024.20 Published online by Camb idge Uni e si y P ess