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David by Edited and Netherlands; The Netherlands Enschede, The AE 7500 Twente, of University France; Paris, 75005 Diderot, a Clanet Christophe and www.pnas.org/cgi/doi/10.1073/pnas.1616904114 with solution soap a For a scale: millimetric the ρ H Milieux bubbles soap Cohen giant Caroline of shape the On aoaor ’yrdnmqed ’,UR74 CNRS, 7646 UMR l’X, de d’Hydrodynamique Laboratoire 0 ' stelqi est) yial,ti rniini bevdat observed is transition this Typically, density). liquid the is fw okfrtesm rniini opbbls eexpect we bubbles, soap in transition same the for look we If to drop cap spherical a from transition the drops, liquid For o ml peia rp,wietehih flrepuddles large of height the while drops, spherical small for stema hcns ftesa l and film soap the of thickness mean the is Nmand mN/m ic oetHoe()fis aldteatnino h Royal the of history attention scientific the called long first a (1) had Hooke have Robert since bubbles soap and films oap 1. 7 a γ m h w smttcrgmsmyb distinguished be may regimes asymptotic two The mm. et ´ b = R erog ´ p 2 | hti,frado ieo h re ftecapillary the of order the of size drop a for is, That . aagn stress Marangoni γ ns(MH,UR73 uCR,EPIPrsPrsSine tLtrs(S)Rsac nvriySron Universit University/Sorbonne Research (PSL) Lettres et Sciences Paris/Paris ESPCI CNRS, du 7636 UMR (PMMH), enes ` e b 0 γ /ρg a,1 1 = b b R atseDrosTexier Darbois Baptiste , tnsfrvprlqi ufc eso,and tension, surface vapor–liquid for stands 2 µm, (γ ttetpclsize typical the at , b stelqi–ao ufc eso,and tension, surface liquid–vapor the is a c ` ρgR hsc fFud ru,Uiest fTet,70 EEshd,TeNetherlands; The Enschede, AE 7500 Twente, of University Group, Fluids of ' | 3.1 h 3 self-similarity 0 e ≈ 0 m. obcm fteodro the of order the of become to , a . R ≈ h 0 with `, ihvolume: with a,1 a = ` ten Reyssat Etienne , cl oyehiu,918PlieuCdx France; Cedex, Palaiseau 91128 Polytechnique, Ecole ρgR γ ´ = p b ` a 30 = γ 2 = b 4 / e /ρg eateto ple hsc,EnhvnUiest fTcnlg,50 B Eindhoven, MB, 5600 Technology, of University Eindhoven Physics, Applied of Department e a (R 0 2 where , /e mN/m, sthe is sthe is 0 γ h 0 b (e = ≈ ρ 0 b,2 cne n yes(7,tetikeso h ubei mea- film soap is punctured a bubble in the hole Following A of technique: mass. thickness bursting the liquid the via the sured (17), determines Mysels it and minimize as McEntee to weight, bubble, bubble the the of center the as height errors. same parallax the at diam- is the measure we particular, placed In 1. Fig. eter, in shown as images form above to sticks water the the into pulled loop bubble. and the the the dipping air before of in surface, water loop sides soap the the the opposite opened in on slowly solu- immersed located sticks pool, soap were experimenters, wood of strings two Two cm The with 20 solution. strings. assembled to cotton was cm two wand 10 and bubble with large filled A 1), tion. (Fig. diameter m the 4 using be measured to found was was mixtures It sur- method. different of The drop experiments. the pendant volume before of h one 10 tension and for face aside water, left of was Dreft and of volumes glycerol volumes two two mixing liquid, by dishwashing prepared is solution soap The Setup Experimental presented is structures inflated with analogy in the and shape totic on in presented is bubbles 1073/pnas.1616904114/-/DCSupplemental at online information supporting contains article This 2 Submission. 1 Direct PNAS a is article paper.This interest. the of wrote C. conflict Clanet no C. and declare and D.Q., authors J.H.S., J.H.S., The E.R., B.D.T., E.R., B.D.T., Cohen, Cohen, C. and C. data; research; analyzed performed Clanet Clanet C. and J.H.S., E.R., uhrcnrbtos .Chn ... n .Cae eindrsac;C oe,B.D.T., Cohen, C. research; designed Clanet C. and B.D.T., Cohen, C. contributions: Author owo orsodnesol eadesd mi:[email protected]. Email: addressed. be should correspondence work. whom this To to equally contributed B.D.T. and Cohen C. ac .Snoeijer H. Jacco , xc nlg hw httesaeo in opbblsis An bubbles soap 2/3. structures. giant inflatable power large of the by shape realized to the nevertheless radius that the shows analogy as regimeexact grows self-similar height this to the access where the for limit prop- height physicochemical of the maximum erties practice, a in but, impose bubbles, not soap We large does shapes. their mechanics compute that demonstrate we find and we weight, their However, under flatten also sizes. bubbles meter-sized that larger theoretically and experimentally much remain at bubbles soap millimeter- drops, spherical with into contrast In drops puddles. larger flat thick flattens gravity sessile whereas small of drops, shape cap spherical the dictates tension Surface Significance h l thickness film The view side by analyzed is shape the rest, at is bubble the Once of pool swimming inflated round a in formed are bubbles The large such of study the to dedicated setup experimental The nlg ihIfltbeStructures. Inflatable with Analogy xeietlResults Experimental 2R 4 n h height, the and , rmtecne ftepo,adtecne ftelens the of center the and pool, the of center the from m PNAS | c,d,e e 0 ac ,2017 7, March and sipratwe tdigteefc of effect the studying when important is eoeinformation before Setup, Experimental ai Qu David , h b aoaor ePyiu tM et Physique de Laboratoire 0 d h icsino h asymp- the on discussion The Model. ftegatbbl.Tecmr is camera The bubble. giant the of , EA nttt o Nanotechnology, for Institute MESA+ . | er ´ o.114 vol. www.pnas.org/lookup/suppl/doi:10. e ´ b , γ b es/Universit ´ 26 = | o 10 no. ± cnqedes ecanique 1 ´ Paris e ´ | mN/m. 2515–2519 ©

PHYSICS bles’ shapes remain approximately spherical (h0/` = R/`, black dashed line) only up to R/` ≈ 0.3. For larger sizes, the bubble height is significantly lower than that of a . The largest value of h0/` reached experimentally is ∼1.2, with correspond- ing radius R/` = 1.6. Two points must be underlined at this stage: (i) The experimental data in Fig. 3B show no sign of a height satu- ration for increased bubble volumes, and (ii) despite our efforts, we never managed to make bubbles larger than R = 1 m. Both observations will be explained in Model.

Model We now consider the mechanical equilibrium of the soap film and predict how gravity affects the bubble shape. Bubbles are axisymmetric, and we assume a uniform film thickness e0. The Fig. 1. Presentation of a “giant” and definition of its radius, bubble shape is described using the parametrization shown in R, and height, h0. (Here, R = 1.09 m, and h0 = 0.97 m.) Fig. 4 A and B: The local height of the soap film is h(s), and the local angle of the membrane relative to the horizontal is θ(s) (defined as positive everywhere). The height and angle are func- grows because of unbalanced forces at the edge tions of the curvilinear coordinate s that measures the arclength of the hole. The opening velocity v is constant and is given by the starting from the top of the bubble; ϕ is the azimuthal angle Dupr´e–Taylor–Culick law (18–20), around the vertical axis. 2γ v 2 = b = 2g`, [1] The equilibrium of an infinitesimal part of the membrane of ρe0 surface rdsdϕ is first considered along the s direction. To −3 account for the experimentally observed slow draining and long where γb ' 26 mN/m, and ρ ' 1,000 kg·m . Examples of thick- bubble lifetimes, the air–liquid interface must strongly moderate ness measurements are presented in Fig. 2: In Fig. 2A and Fig. the flow and behave as partially rigid, in contrast with the no- 2B, we present two sequences of four pictures showing the open- stress behavior of surfactant-free interfaces. The main effect is ing of a hole in two soap bubbles of different sizes. We use such that gradients in surface tension γ(s), due to the presence of sur- sequences to extract the bursting velocity plotted as a function factants, have to balance viscous stresses applied by the flowing of time in Fig. 2C. We observe that v is almost constant and liquid along the interface. In reaction, viscous stresses balance takes the value of 8 m/s for sequence in Fig. 2A and 2.8 m/s the weight of the liquid in the film. Finally, the weight of liquid for sequence in Fig. 2B. From the value of v, we deduce is fully transmitted to the walls through viscous stresses and bal- e0 ' 0.81 µm and ` ≈ 3.2 m for Fig. 2A and e0 ' 6.6 µm and anced by surface tension gradients (15). The contribution due to ` ≈ 0.4 m for Fig. 2B, using Eq. 1. Although it may seem surpris- surface tension on each side of the infinitesimal element gives ing to find that the film thickness is homogeneous, earlier studies a force 2γrdϕ, where γ is a function of position s. The weight on the drainage of almost spherical liquid shells have shown that of the liquid inside the film is ρge0rdϕds sin θ, when projected the thickness approaches a profile with little spatial variations along the s direction. The balance of surface tension and weight (21, 22). thus gives

Experimental Results 2rdϕ [γ(s) − γ(s + ds)] = ρge0rdϕds sin θ, [2] Once ` is determined, we use it to rescale the experimental which, using dh/ds = − sin θ, yields shapes. An example of a nonspherical bubble is presented in 1 1 reduced scale in Fig. 3A. A systematic analysis of the effect of dγ = ρge0dh ⇒ γ(s) = γb + ρge0h(s). [3] gravity on the bubble shape is shown in Fig. 3B, where we plot 2 2 the reduced height h0/` as a function of the bubble reduced Here γb is the surface tension of the soap solution at the base radius R/` for all experiments. Fig. 3B reveals that the bub- of the bubble (h = 0), and the surface tension is found to increase

A C

B

Fig. 2. Bursting of bubbles used to determine the soap film thickness. (A) Image sequence of bursting bubbles, with a time step of 60 ms between images. Red arrows indicate the boundary of the opening hole on each image. (Scale bar, 50 cm.) (B) Bursting sequence with a time step of 50 ms between images. (Scale bar, 10 cm.) (C) The bursting velocity corresponding to A and B is plotted versus time.

2516 | www.pnas.org/cgi/doi/10.1073/pnas.1616904114 Cohen et al. Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 qainfrtesaeo h bubble, the of shape the for equation 1/2ρge id,w bantesaeeuto ndmninesform, dimensionless in equation shape the obtain we tilde, a where oe tal. et Cohen of height dimensionless the (s bubble the of top the at value its that by remind Eq. We by surface. given axisymmetric is an for pal where limit spherical the represents line black dashed the and prediction, model the for height scaled the Eq. by of quantified integration numerical the through obtained 3. Fig. n oue sepce,salbblsaedmntdb sur- by dominated are bubbles small expected, As volume. ing bath). the ( conditions reads weight interfaces. and liquid–air curved of two balance the The to due film pressure the to Laplace normal the projected weight the balance to pres- The membrane. the difference sure to normal equilibrium the considering the of top mN/m. and 5 base typically the is between contrast bubble tension thickness surface of the m, bubble 1 a For (23). height with AB cln l egh with lengths all Scaling nqebbl hp sfudnmrclyfrec au of value each for numerically found is shape bubble unique A i.4C Fig. next when obtained is shape bubble the for equation closed A d˜ dθ/ (dθ/ dθ/ (2γ 0 xml ffltee opbbl feutra imtr2 m h lc ahdln sacrl,adterdsldln stetertclshape theoretical the is line solid red the and circle, a is line dashed black The cm. 20 diameter equatorial of bubble soap flattened of Example (A) s h (˜  0 s b ds Combining . + 2 hw h orsodn ubesae o increas- for shapes bubble corresponding the shows ds ∆ 0) = + h − ˜ P stecrauealong curvature the is = ρ sin + ρge − safia tp ti ovnett eliminate to convenient is it step, final a As 3. h ˜ ∆ ge 2 =  yasotn loih omthteboundary the match to algorithm shooting a by h ˜  P 0 0 0  1 h  and ewe h nieadotieo h bubble, the of outside and inside the between γ θ/ ) dθ d˜ − (s 1  s r cos − ) + dθ ds )| θ ∆  cos s 0 = ` h ˜ P →0 sin θ dθ ds + = 0 h r ˜ hsi oeb dutn h value the adjusting by done is this ; 2( + (s 0 θ a /` ttetp with top, the at θ → sin + 2(h + 2 0) = r − /e safnto ftesae radius scaled the of function a as ds 2dθ/ h sin ˜ θ 0 0 r 2 − eoigsae aibe by variables scaled denoting , − s dθ d˜ ihEqs. with θ 0 s and  2 h − ˜

dθ ds ) ˜ s + | 0 = h =0 s dθ d˜ =0 sin )

s ρge s  e dθ ds 6 ,

=0 0 and ˜ s htgvstesm setratio aspect same the gives that θ/ θ θ =0 1 = 0

 3 0 = = s cos r =0  and γ 2 π/ sohrprinci- other is n height and µm ρge (s and , 0. =  θ, )= 0) = 0 = 4 0 at cos ie the gives h h ˜ . = 0 = θ γ γ ∆P h and where R/`, [4] [5] [6] b (s 0 at + ), ) laiylength illarity rdcinfrteheight the (Eq. for solution model prediction the The superim- with we distinguished. bubble where be diameter 3A, 20-cm Fig. Eq. a in of of presented picture is the bubble pose shape real theoretical a the of with comparison direct A shape. spherical the get bubbles as ( However, larger spherical. perfectly are and tension face yrsai rsueblne(3 n sdfeetfo Eq. classical from different the is from and found liq- (13) be for balance can result pressure droplets classical hydrostatic of the shape with The contrast drops. stark uid in is saturation, 4E of Fig. in scaling the to verti- leads and and Information horizontal bubble Supporting the the determines for edge scales the cal at scaling 2/3 The balance dominant the univer- from reads the edge the of near scaling shape The sal shape. universal single, a to collapsed 4C Fig. in shapes bubble in large shown is as Indeed, heights. bubble dimen- Eq. with simple scale to on gravity Interestingly, According derived height. grounds. be bubble cannot sional the features of asymptotic saturation these no is i.e., there ratio, time, aspect decaying log–log in that a a detail find implies on we more height volumes, in large bubble For highlighted dimensionless plot. the is showing feature 4E, This Fig. volume. large of height: bubble the of ration gravity. the to due parameter, flattening adjustable observed any experimentally without well, 3B very Fig. describes in result data experimental the with compared h ee h egh eemd iesols sn h cap- the using dimensionless made were lengths the Here, hssaiglwfrlrebbls n,i atclr h lack the particular, in and, bubbles, large for law scaling This satu- a predict not does solution numerical the Unexpectedly, 0 ` = = h 0 R. a /R 6 2 h ˜ /e 0 hthstesm ratio same the has that lteigo opbblsdet gravity, to due bubbles soap of Flattening (B) experiment. the as 2 > 0  ice ersn xeietldt,tesldrdln stands line red solid the data, experimental represent Circles . ,te hwatnec oflte ihrsetto respect with flatten to tendency a show they 1), dθ dˆ s a + = ρ ge p sin PNAS 0 r ˆ γ hc onst cl nainea large at invariance scale a to points which , θ b o ealdanalysis). detailed for /ρg − | h ˜ h 0 ˜ ac ,2017 7, March 2 n eoe yhte variables. hatted by denoted and , h ˜ dθ/ = dθ dˆ 0 h xii cl naineadcnbe can and invariance scale exhibit ˜ s h otne oices ntelimit the in increase to continues ≈

0 ds ˆ s /` =0 h ( 0 R (cos = ˜ ohsraetninand tension surface both 3, logvsaquantitative a gives also 6) /R  versus h h the Information, Supporting ˜ ˜ − 0 − / h w hpscannot shapes two The . r R ˜ ≈ | ˜ ( ) h ˆ 2/3  o.114 vol. θ R ˜ 0 R − ˜ 2/3 − u,a h same the at but, 1, scnb inferred be can as , 1) = hssaiglaw scaling This . h ˆ ' R sldln) The line). (solid 0. = ) /` | h 0 o 10 no. 2 htcnbe can that /2 nEq. in | 6, (see 2517 [7] 6.

PHYSICS A B

C E

D

Fig. 4. (A) Sketch of a giant bubble of radius R and height h0. The local height is h. The gray area shows an infinitesimal surface element of the bubble. (B) Zoom on the infinitesimal part of the bubble located at distance r from the axis of the bubble. The surface area of this surface element is rdsdϕ; tan θ is the local slope of the soap film with respect to the horizontal. (C) Shapes of soap bubbles with dimensionless radius R/` = 0.3, 1, 5, and 10. Although large bubbles tend to flatten (i.e., h0/R → 0), the height of giant bubbles shows no sign of saturation. (D)√ Shapes of liquid drops with θ = 90◦ and dimensionless radius R/a = 0.3, 1, 5, and 10. The dimensionless height of large drops saturate to 2. (E) Experimental dimensionless height of soap bubbles as a function of their dimensionless radius (circles). The theoretical dimensionless height h0/` of bubbles (full line) is plotted as a function of their dimensionless radius R/`. Small bubbles (R/` < 0.3) are insensitive to gravity and remain hemispherical, thus minimizing their surface area for the given volume. Giant bubbles flatten, but there is no saturation height as exists for drops larger than the : For large bubbles, one 2/3 finds h0 ≈ R . The physical chemistry of surfactants, however, limits the range of accessible surface tension, setting the actual upper limit for the size of giant bubbles (dashed line): hmax /` = 2∆γ/γb.

Fig. 4D shows the corresponding numerical solutions: Large maximal height of a bubble of thickness e0 = 5 µm is of order drops develop toward√ puddles, which, for θ = π/2, saturate to 2 m, close to the size of the biggest bubbles we experimentally the height hˆ0 = 2. produced (Fig. 1). The height of soap bubbles may, however, be limited by phys- ical chemistry of surfactants. The water that constitutes the bub- ble is prevented from draining quickly by gradients in the sur- face tension. The larger surface tension at the top of the bubble supports the weight of water in the liquid shell (15). In prac- tice, the surface tension of a soap solution cannot be higher ∗ than that of pure water, γ ; γ also has a minimum, γb , set by the surfactant concentration of the solution used in the experi- ments. Eq. 3 thus gives a criterion for the maximal height hmax of the bubble, 1 γ∗ = γ + ρge h [8] b 2 0 max

so that h 2∆γ max = , [9] ` γb ∗ where ∆γ = γ −γb is the highest achievable surface tension con- trast between the top and bottom of a bubble; γ∗ ' 70 mN/m, and γb may typically be as low as 20 mN/m, so that the expected Fig. 5. Festo’s Airquarium, 31 m in diameter.

2518 | www.pnas.org/cgi/doi/10.1073/pnas.1616904114 Cohen et al. Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 0 isnL,MlsA 15)Apiaino h ebaeaaoyt h ouinof solution the to analogy membrane the of Application (1950) AJ Miles LH, Wilson 10. 1 yesK,SioaK rne (1959) S Frankel K, Shinoda KJ, Mysels 11. nry h racntan,adtewr oeb h pres- the by done work functional (see the this gives for equation and Euler–Lagrange The constraint, difference. sure area the energy, as based inside shape imposed analogy ric condition develops multiplier this no-stretch Lagrange that confirm a the through to with tension minimization, interesting is the energy is on sheet by It thin replaced membrane. 4 the the is Fig. of in tension element that face infinitesimal to difference equivalent an pressure and strictly on stretched, a be analysis by cannot structures ical assume inflated These we is that 5. large sheet Fig. which thin of in a context shown of the as consist in to such relevant correspond structures, is discussed inflatable that just problem have minimization we a shapes the Interestingly, Structures Inflatable with Analogy with reads minimized oe tal. et Cohen .GifihA,Tyo I(97 h s fsa lsi ovn oso problems. torsion solving in films soap of use The (1917) GI st Taylor prismatischen AA, von Griffith torsion Zur 8. (1903) L Prandtl 7. .WnC,LiJ 20)Aaoybtensa l n a yais .Eutosand Equations I. dynamics. gas and film soap between Analogy (2003) JY Lai CY, Wen 9. films. soap of hydrodynamics the On (1989) M Rabaud JM, Chomaz Y, Couder electricity. 6. in researches Experimental (1941) (1851) M H Faraday 5. Robbins R, Courant 4. (1873) J Plateau 3. bubbles. soap (1730) in SI film) (black Newton holes On 2. (1672) R Hooke 1. L(h h he em epcieyrpeettegaiainlfree gravitational the represent respectively terms three The hc upconditions. jump shock Rep Tech Aeronaut Comm D 141:7–28. London). Mol Forces Light of Colours and etcnuto problems. conduction heat o,Oxford). mon, 37:384–405. , h 0 = ) naal Structures): Inflatable eculaires ´ h ρge (r ttqeExp Statique n thus and ), pik r raieo h eetos ercin,Inflections, , Reflections, the of Treatise A or, Opticks 0 Guhe-ilr,Paris). (Gauthier-Villars, RSc London). Soc, (R h F 1+ (1 x Fluids Exp [h 333(3):920–937. plPhys Appl J = ] rmnaeE Th Et erimentale ´ h 0 2 h Z 34:107–114. ) 0 1/2 htI ? Is What = dr hrceiigteaxisymmet- the Characterizing λ. 21:532–535. opFls tde fTerThinning Their of Studies Films: Soap dh + 2π /dr 1+ λ(1 r oiu e iudsSui u Seules Aux Soumis Liquides Des eorique ´ L(h A aben. ¨ h functional the , and , h h omnRSoc R Commun hsZ Phys 0 0 2 ), h oeo sur- of role The B: ) hlsTasRScLondon Soc R Trans Philos ∆ 1/2 P 4:758–770. Ofr nvPress, Univ (Oxford − h mechan- The . ∆ a 28. Mar h P F [ h . ] Physica (Perga- obe to [11] [10] Advis 2 e ,e l 21)Fbiaino lne lsi hlsb h otn fcre sur- curved films. of soap coating “Young” the (2001) by PG Gennes shells de elastic slender 23. of Fabrication Non- (2016) floating: al. to et bouncing A, From Lee (2005) A 22. Boudaoud CH, Gauthier film. E, soap Fort ruptured Y, a Couder on sheets. Comments 21. fluid (1960) of F Disintegration Culick III. fluid 20. of sheets thin of dynamics The (1959) GI Taylor study. experimental 19. An I. Dupr films. bubble soap air 18. of an bursting of The shape (1969) the KJ Mysels is WR, What McEntee (2015) PI 17. Teixeira SJ, Cox films. S, soap Arscott of rupture MA, Teixeira and generation 16. of study A (2014) al. et L, Saulnier 15. Qu F, Brochard-Wyart PG, Gennes de 13. (1992) C Isenberg 12. 4 ei ,SietJ attI ineA(05 ntegnrto fafa l during film a of generation the On (2015) A Biance I, Cantat J, Seiwert P, Petit 14. esuytesaeo ag opbblsadso htgravity that show scale and the bubbles soap at large important of becomes shape the study We the Conclusion increase that help seams should the structures. design such on of This and lifetime parts. sheets various reduc- the and the in wrinkles connect exerted avoiding stresses sheets, tensile the of ing parts various compres- and of stretching sion avoid naturally will shapes optimal these the dictates (Eq. that shapes equation the bubble to identical strictly indeed, is, which opbblsi o hrceie yastrto fteheight the of which in saturation giant behavior a self-similar of a by shape by characterized the but drops, not to is contrary bubbles that, bubbles, soap out soap of point size we the to Finally, sur- limit that of physicochemical is a consequence evolution is direct distinct there A the two stresses. and in Marangoni via matters term tension gravity face hydrostatic that expected show the and terms: shape the for tion lsadfrhrcntutv rtcs fteiiilvrino u work. our of version soap initial of the stability of the criticism on constructive input her her for for and Cantat films Isabelle and thank Paris between also collaboration We the Twente. initiated that School Summer Krogerup ACKNOWLEDGMENTS. faces. bath. fluid a on drops of coalescence A Soc R Proc Chem Phys surface? liquid a on 10:2899–2906. oooia rearrangement. topological a York). New (Springer, (1867) A e ´ a Commun Nat ∆ P 73(9):3018–3028. 253:313–321. ( = nae eCii td Physique de et Chimie de Annales h cec fSa im n opBubbles Soap and Films Soap of Science The λ 7:11155. Langmuir + PNAS .Dsgigteifltbesrcue along structures inflatable the Designing 4). etakTmsBh o raiigte2013 the organizing for Bohr Tomas thank We ρge 0 li Mech Fluid J 31:13708–13717. h | ) ac ,2017 7, March  er ´ hsRvLett Rev Phys dθ ds (2004) D e ´ ` Langmuir + = 763:286–301. sin a 2 r ailrt n etn Phenomena and Capillarity h /e 11(4):194. θ 0 94:177801. plPhys Appl J | 17:2416–2419. /`  0 edrv h equa- the derive We . o.114 vol. + ≈ (R ρge Dvr iel,NY). Mineola, (Dover, /`) 31:1128–1129. 0 | cos 2/3 o 10 no. θ, . otMatter Soft | h [12] max 2519 J .

PHYSICS