Floating with Surface Tension from Archimedes to Keller

FLOATING WITH SURFACE TENSION FROM ARCHIMEDES TO KELLER

Dominic Vella Mathematical Institute, University of Oxford 1

GFD Newsletter 2006 Faculty of Walsh College JBK @ WHOI

Joe was a regular ﬁxture of the Geophysical Fluid Dynamics programme (run at Woods Hole since 1959 – last attended in 2015) usual magic in the Lab. We continue to be endebted to W.H.O.I. Education, who once more provided a perfect atmosphere. Jeanne Fleming, Penny Foster and Janet Fields all contributed importantly to the smooth running of the program.

The 2006 GFD Photograph. GFD Photo – 2006

A Sketch of the Summer Ice was the topic under discussion at Walsh Cottage during the 2006 Geophysical Fluid Dynamics Summer Study Program. Professor Grae Worster (University of The dragon enters the ice field. Cambridge) was the principal lecturer, and navigated our path through the fluid dynamics of icy processes in GFD. Towards the end of Grae’s lectures, we also held the 2006 GFD Public Lecture. This was given by Schedule of Principal Lectures Greg Dash of the University of Washington, on matters Week 1: of ice physics and a well-known popularization: “Nine Monday, June 19: Introduction to Ice Ices, Cloud Seeding and a Brother’s Farewell; how Kurt Tuesday, June 20: Diffusion-controlled solidification Vonnegut learned the science for Cat’s Cradle (but con- Wednesday, June 21: Interfacial instability in super- veniently left some out).” We again held the talk at Red- cooled fluid field Auditorium, and relaxed in the evening sunshine Thursday, June 22: Interfacial instability in two- at the reception afterwards. As usual, the principal lec- component melts tures were followed by a variety of seminars on topics icy and otherwise. We had focussed sessions on sea ice, Friday, June 24: Formation of mushy layers the impact of ice on climate, and glaciology. Week 2: John Wettlaufer, nimbly assisted by Neil Balm- Monday, June 26: Idealized mushy layers forth, directed the summer program, and almost single- Tuesday, June 27: Convection in mushy layers handedly took on the supervision of the fellows. Some important acknowledgements: Young-Jin Kim (Univer- Thursday, June 29: Interfacial pre-melting sity of Chicago) helped out with the computers dur- Friday, June 30: Thermomolecular flow, thermal ing the first few weeks, and Keith Bradley worked his regelation and frost heave 1

GFD Newsletter 2006 Faculty of Walsh College JBK @ WHOI

The 2006 GFD Photograph. GFD Photo – 2006

Chats on the porch of Walsh Cottage are a unique feature of GFD A Sketch of the Summer Ice was the topic under discussion at Walsh Cottage during the 2006 Geophysical Fluid Dynamics Summer Study Program. Professor Grae Worster (University of The dragon enters the ice field. Cambridge) was the principal lecturer, and navigated our path through the fluid dynamics of icy processes in GFD. Towards the end of Grae’s lectures, we also held the 2006 GFD Public Lecture. This was given by Schedule of Principal Lectures Greg Dash of the University of Washington, on matters Week 1: of ice physics and a well-known popularization: “Nine Monday, June 19: Introduction to Ice Ices, Cloud Seeding and a Brother’s Farewell; how Kurt Tuesday, June 20: Diffusion-controlled solidification Vonnegut learned the science for Cat’s Cradle (but con- Wednesday, June 21: Interfacial instability in super- veniently left some out).” We again held the talk at Red- cooled fluid field Auditorium, and relaxed in the evening sunshine Thursday, June 22: Interfacial instability in two- at the reception afterwards. As usual, the principal lec- component melts tures were followed by a variety of seminars on topics icy and otherwise. We had focussed sessions on sea ice, Friday, June 24: Formation of mushy layers the impact of ice on climate, and glaciology. Week 2: John Wettlaufer, nimbly assisted by Neil Balm- Monday, June 26: Idealized mushy layers forth, directed the summer program, and almost single- Tuesday, June 27: Convection in mushy layers handedly took on the supervision of the fellows. Some important acknowledgements: Young-Jin Kim (Univer- Thursday, June 29: Interfacial pre-melting sity of Chicago) helped out with the computers dur- Friday, June 30: Thermomolecular flow, thermal ing the first few weeks, and Keith Bradley worked his regelation and frost heave JBK @ WHOI

PHYSICS FLUIDS VOLUME 10, NUMBER 11 NOVEMBER 1998

BRIEF COMMUNICATIONS The purpose of this Brief Communications section is to present important research results of more limited scope than regular articles appearing in Physics of Fluids. Submission of material of a peripheral or cursory nature is strongly discouraged. Brief Communications cannot exceed three printed pages in length, including space allowed for title, ﬁgures, tables, references, and an abstract limited to about 100 words.

Surface tension force on a partly submerged body Joseph B. Keller Departments of Mathematics and Mechanical Engineering, Stanford University, Stanford, California 94305-2125 ~Received 26 January 1998; accepted 29 July 1998! The vertical component of the surface tension force on a body partly submerged in a liquid is shown to equal the weight of liquid displaced by the meniscus. It is upward if the meniscus is depressed and downward if the meniscus is elevated. Previously this was known for vertical axially symmetric bodies and for two-dimensional vertical plates. The vertical component of the pressure force on the body is shown to equal the weight of liquid which would ﬁll the volume bounded by the wetted surface of the body, a vertical cylinder through the waterline, and the original horizontal free surface. © 1998 American Institute of Physics. @S1070-6631~98!02111-4#

A surface tension force FT and a pressure force FP act on a body partly submerged in a liquid at rest. The vertical T n•ndt5WM . ~2! EG component of FT will be shown to equal the weight of liquid displaced by the meniscus, directed upward if the meniscus W is rg times the volume between the interface z is depressed and downward if it is elevated. This extends to M 5h(x,y) and the surface z50, counted negative if h.0. bodies of any shape the two-dimensional result for the force Thus W is the weight of the liquid displaced by the menis- on a vertical flat plate ~Rusanov and Prokhorov1! and the M cus. axially symmetric results of Laplace2 for the volume of fluid To interpret the left side of ~2! we note that raised in a vertical capillary tube and of Schulze3 for the force on any vertical axially symmetric body. The vertical component of F will be shown to equal the weight of liquid P FT5T r˙~s!3n@x~s!,y~s!#ds. ~3! which would fill a volume bounded by the wetted surface of EC the body, a vertical cylinder through the waterline, and the original horizontal free surface ~see Fig. 1!. The horizontal components of FT and FP will be considered also. The interface z5h(x,y) between a liquid and gas at rest satisfies the Young–Laplace equation

2T~]xn11]yn2!5rgh~x,y!. ~1!

Here T is the coefficient of surface tension, n(x,y) 2 2 21/2 [(n1 ,n2 ,n3)5@2hx ,2hy,1#(11hx 1hy) is the unit normal to the interface pointing out of the liquid, r is the density of the liquid, and g is the acceleration of gravity which acts in the 2z direction. We assume that the interface extends from a curve C on the surface S of a partly submerged body to infinity, where h50. C is given by r(s) FIG. 1. The vertical component of the force FT due to surface tension is positive in ~a! and negative in ~b!. Its magnitude is the weight of liquid where s is arclength on C. which would fill the cross-hatched regions. The vertical component of the The projection of C on the (x,y) plane is a curve G with pressure force FP is positive in both cases. In ~a! it is equal to the weight of arclength t and inward normal n(t). We integrate ~1! over the liquid which would fill the unhatched region between the body surface the (x,y) plane outside G, apply the divergence theorem to and the x axis. In ~b! the area of the two small triangular regions above the axis must be subtracted from the area below the axis. Thus in both cases, the the left side, and assume that n1 and n2 vanish sufficiently force is less than the weight of the liquid displaced by the body. In ~c! both rapidly at infinity. Then we get vertical forces are differences between volumes below and above the axis.

Downloaded 04 Dec 2004 to 131.111.18.82. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp JBK @ WHOI 3010 Phys. Fluids, Vol. 10, No. 11, November 1998 Brief Communications

We multiply ~3! by zˆ• and observe that zˆ–r˙3n5n–zˆ3r˙ face of the body, and above by the undisturbed interface z PHYSICS FLUIDS5n–n sin w where w is the angle betweenVOLUMEzˆ and r˙ 10,. Since NUMBER5 110, when h(x,y) is single valued. If this regionNOVEMBER were ﬁlled 1998 sin w ds5dt, we get from ~3! with liquid, it would obviously be held at rest by the liquid below it. This provides a direct proof of ~9!. Subtracting ~8! ~ ! ~ ! BRIEF COMMUNICATIONSzˆ–FT5T n–n sin w ds5T n–ndt. ~4! from 9 gives a direct proof of 5 . Weights or volumes EC EG The purpose of this Brief Communications section is to present important researchabove resultsz50 of must more be limited counted scope as thannegative. regular Similarly, the total articles appearingNow ~2! inand Physics~4! yield of Fluids. the result Submission of material of a peripheral orhorizontal cursory nature force is on strongly the body discouraged. is equal to Brief the horizontal force Communications cannot exceed three printed pages in length, including space allowedon a for vertical title, ﬁgures, cylinder tables, of large references, radius and surrounding an the body. abstract limitedˆ to about 100 words. z–FT5WM . ~5! That force is zero, so we have

The force FP due to hydrostatic pressure is given by xˆ•~FT1FP!50, yˆ•~FT1FP!50. ~10! h F , the horizontal component of FP , is the same as the FP5 rgzN~x!dA. ~6! P E horizontal component of the force exerted by the external Surface tensionHere dA is the force area element on of the a wetted partly surface submergedS and N~x! fluid body on the vertical cylinder which extends downward from is the unit normal to S directed out of the body. To compute C. We assume that the wetted surface of the body lies inside this cylinder. The forces are equal because there is no net Josephthe B. vertical Keller component of FP we note that zˆ–N(x)dA Departments56dx ofdy Mathematics. This is the signed and Mechanical area element Engineering, on the x,y plane Stanfordhorizontal University, force Stanford, on the fluid bounded by S and this cylinder. California 94305-2125 Thus into which dA projects, where the sign is that of zˆ–N. When ~Receivedthe sign 26 is January negative 1998; at all accepted points of 29S, S Julyprojects 1998 onto! the z~t! rg interior A of G and we obtain Fh 5rg n~t!zdzdt5 z2~t!n~t!dt. ~11! G P E E 2 E The vertical component of the surface tension force on a body partly submergedG z0 in a liquid is shownG to equal thezˆ–FP weight52rg ofz liquid~x,y!dx displaced dy. by the meniscus.~7 It! isHere upwardz(t) if is the the meniscusz coordinate is of depressedC at arclength position t on EA and downward if the meniscusG is elevated. Previously this was knownG. The lower for vertical limit z0 , axially which issymmetric chosen below all points of S, Was this paperThe integral a inresult~7! is just of minus Walsh the volume CottageV of the domain chat? drops out of the calculation because the integral of n(t) bodies and for two-dimensional vertical plates. The vertical component of the pressure force on the h 5 around the closed curve G vanishes. Thus F is determined bodybounded is shown below to equal by S, abovethe weight by the of plane liquidz 0, which and laterally would fill the volume bounded by the wetted P by the vertical cylinder through C, which intersects the hori- by the depression of the waterline due to surface tension. surfacezontal of plane the body, in the curvea verticalG. Thus cylinder the right through side of ~7 the! is waterline,just and the original horizontal free surface.the weight © 1998WV Americanof the fluid Institute contained of in Physics. this domain,@S1070-6631 so we ACKNOWLEDGMENTS~98!02111-4# have I thank George Veronis for asking the question that led zˆ–FP5WV . ~8! to this work. This work supported in part by the AFOSR and NSF. A surfaceThis tension result force remainsFT trueand when a pressurezˆ–N changes force sign,FP act as one can on a body partlyshow submerged by considering in a separately liquid at the rest. regions The where vertical it is posi- T n•ndt5WM . ~2! E1 G tive and where it is negative. A. I. Rusanov and V. A. Prokhorov, ‘‘Interfacial tensiometry,’’ Studies in component of FT will be shown to equal the weight of liquid Interface Science; 3 ~Elsevier, Amsterdam, The Netherlands, 1996!. displaced by the meniscus,To obtain the directed total vertical upward force if on the the meniscus body, we add ~8! 2P. S. Laplace, ‘‘Traite´ de mechanique celeste; Supplements au livre X, W is1805rg andtimes 1806,’’ thein Oeuvres volume Complete between~Gauthier-Villars, the interface Paris, 1839!,z is depressed andand downward~5!: if it is elevated. This extends to M 5h(x,yVol.) and 4. English the translation surface byz5 N.0, Bowditch, counted reprinted negative by Chelsea, if h New.0. bodies of any shapezˆ–~F theT1 two-dimensionalFP!5WM1WV . result for the force ~9! York, 1966. 1 Thus W3H.M J.is Schulze, the weight ‘‘Physico-chemical of the liquid elementary displaced processes inby flotation,’’ the menis-De- on a verticalThe flat right plate side~Rusanov of ~9! is the and weight Prokhorov of liquid! thatand would the fill the velopments in Mineral Processing; 4 ~Elsevier, Amsterdam, The Nether- 2 cus. axially symmetricregion results bounded of Laplace below byfor the interfacethe volume and of the fluid wetted sur- lands, 1984!. To interpret the left side of ~2! we note that raised in a vertical capillary tube and of Schulze3 for the force on any vertical axially symmetric body. The vertical component of F will be shown to equal the weight of liquid P FT5T r˙~s!3n@x~s!,y~s!#ds. ~3! which would fill a volume bounded by the wetted surface of EC the body, a vertical cylinder through the waterline, and the original horizontal free surface ~see Fig. 1!. The horizontal components of FT and FP will be considered also. The interface z5h(x,y) between a liquid and gas at rest satisfies the Young–Laplace equation

2T~]xn11]yn2!5rgh~x,y!. ~1!

Here T is the coefficient of surface tension, n(x,y) 2 2 21/2 [(n1 ,n2 ,n3)5@2hx ,2hy,1#(11hx 1hy) is the unit normal to the interface pointing out of the liquid, r is the density of the liquid, and g is the acceleration of gravity which acts in the 2z direction. We assume that the interface extends from aDownloaded curve C 04on Dec the 2004 surface to 131.111.18.82.S of a partly Redistribution sub- subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp merged body to infinity, where h50. C is given by r(s) FIG. 1. The vertical component of the force FT due to surface tension is positive in ~a! and negative in ~b!. Its magnitude is the weight of liquid where s is arclength on C. which would fill the cross-hatched regions. The vertical component of the The projection of C on the (x,y) plane is a curve G with pressure force FP is positive in both cases. In ~a! it is equal to the weight of arclength t and inward normal n(t). We integrate ~1! over the liquid which would fill the unhatched region between the body surface the (x,y) plane outside G, apply the divergence theorem to and the x axis. In ~b! the area of the two small triangular regions above the axis must be subtracted from the area below the axis. Thus in both cases, the the left side, and assume that n1 and n2 vanish sufficiently force is less than the weight of the liquid displaced by the body. In ~c! both rapidly at infinity. Then we get vertical forces are differences between volumes below and above the axis.

Downloaded 04 Dec 2004 to 131.111.18.82. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp A BRIEF HISTORY OF FLOATING

“Any object, wholy or partaly immersed in a ﬂuid, is buoyed up by a force equal t te weight of te ﬂuid displaced by te object.” Archimedes, On Floating Bodies (c. 250 BC) A BRIEF HISTORY OF FLOATING

“Any object, wholy or partaly immersed in a ﬂuid, is buoyed up by a force equal t te weight of te ﬂuid displaced by te object.” Archimedes, On Floating Bodies (c. 250 BC)

Archimedes’ principle seems to rule out dense objects (e.g. drawing pins) ﬂoating on water A BRIEF HISTORY OF FLOATING

“Any object, wholy or partaly immersed in a ﬂuid, is buoyed up by a force equal t te weight of te ﬂuid displaced by te object.” Archimedes, On Floating Bodies (c. 250 BC)

FL47CH06-Vella ARI 14 November 2014 9:50

abc GAS

γγ

mg mg

LIQUID

Figure 2 (a) Illustration showing Galileo’s understanding that a dense object (CIOS) may float because of the volume of liquid displaced by the meniscus (AICB). Panel a reproduced from Galilei (1663). (b,c) The equivalence demonstrated by Keller’s (1998) generalization of Archimedes’ principle to incorporate surface tension. (b)Therestoringforcesthatcounteractthefloatingbody’sweight,mg,comprise a surface tension force per unit length γ,whichactsparalleltothesurfaceandnormaltothecontactline,andthebuoyancyforce generated by hydrostatic pressure p acting normal to the wetted surface of the body. It is natural that the resultant buoyancy force from hydrostatic pressure is equal in magnitude to the weight of liquid displaced by the wetted body ( gray shaded area). (c)Keller(1998) showed that the total restoring force on a floating body (including that due to surface tension) is the weight of the entire volume of liquid displaced by that body ( gray shaded area).

Whether to think of the additional force as coming from the weight of liquid displaced by the meniscus or from the surface tension force acting on the body is therefore a matter of preference. However, it is important to bear both points of view in mind. For example, if one thinks only in terms of the force per unit length provided by surface tension, it is tempting to believe that having a long, convoluted contact line will allow surface tension to provide arbitrarily large supporting forces. Thinking instead in terms of the weight of liquid displaced makes it clear that this cannot be the case, for reasons discussed below. Although impressive for its time, Galileo’s intuitive understanding that the weight of liquid displaced by the meniscus must balance the excess weight of a dense ﬂoating object misses an important part of the story: Objects cannot lower themselves arbitrarily deep into the liquid and hence displace an arbitrary amount of liquid in their meniscus. Instead, the shape of a meniscus, z h(x, y), for example, of a liquid with interfacial tension γ , density ρ , and subject to a gravi- = ℓ tational acceleration g, must be given by a solution of the Laplace-Young equation:

n h/ℓ2, (1) −∇· = c where ℓ (γ /ρ g)1/2 is the capillary length, and n ( h, 1)/(1 h 2)1/2 is the unit (outward c = ℓ = −∇ +|∇ |

Annu. Rev. Fluid Mech. 2015.47:115-135. Downloaded from www.annualreviews.org pointing) normal to the liquid surface (Finn 1986). There are no general results that limit the maximum depth of a meniscus; indeed, for a meniscus Access provided by University of Oxford - Bodleian Library on 01/06/15. For personal use only. contained within a sufﬁciently tight corner, the meniscus depth may be unbounded (e.g., see Finn 1986, p. 116). Nevertheless, in cases of practical interest, the maximum depth of a static

meniscus is a multiple of either the capillary length, ℓc , or the typical radius of the object, R, whichever is smaller (Finn 1986, Lo 1983). Because the horizontal length scale over which a

meniscus decays is the capillary length ℓc , the typical weight of liquid displaced in the meniscus scales as ρ g min R, ℓ ℓ2. Hence, we expect that a solid object of density ρ and typical radius R ℓ { c } c s can float only if ρ R3 αρ min R, ℓ ℓ2 for some dimensionless constant α. s ≤ ℓ { c } c From this scaling, it is clear that objects with solid densities significantly larger than that of the liquid can only float if R ℓ . In this case, flotation is possible provided that ≪ c 2 1 ρ /ρ α(R/ℓ )− αBo− , (2) s ℓ ≤ c =

118 Vella A BRIEF HISTORY OF FLOATING

“Any object, wholy or partaly immersed in a ﬂuid, is buoyed up by a force equal t te weight of te ﬂuid displaced by te object.” Archimedes, On Floating Bodies (c. 250 BC)

“When ever te excess of te Gravit of te Solid above te Gravit of FL47CH06-Vella ARI 14 November 2014 9:50 te Watr shal have te same proporton t te Gravit of te Watr tat te Alttude of te Rampart, hat t te tickness of te Solid, tat Solid shal not sink, but being never so litle ticker it shal.” abc Galileo Galilei, Discourse on Floating Bodies (1663) GAS γγ

mg mg

LIQUID

n h/ℓ2, (1) −∇· = c where ℓ (γ /ρ g)1/2 is the capillary length, and n ( h, 1)/(1 h 2)1/2 is the unit (outward c = ℓ = −∇ +|∇ |

meniscus is a multiple of either the capillary length, ℓc , or the typical radius of the object, R, whichever is smaller (Finn 1986, Lo 1983). Because the horizontal length scale over which a

118 Vella A BRIEF HISTORY OF FLOATING

“Any object, wholy or partaly immersed in a ﬂuid, is buoyed up by a force equal t te weight of te ﬂuid displaced by te object.” Archimedes, On Floating Bodies (c. 250 BC)

“When ever te excess of te Gravit of te Solid above te Gravit of te Watr shal have te same proporton t te Gravit of te Watr tat te Alttude of tFL47CH06-Vellae Rampart, ARI 14ha Novembert 2014t t 9:50e tickness of te Solid, tat Solid shal not sink, but being never so litle ticker it shal.”

Galileo Galilei,abc Discourse on Floating Bodies (1663) GAS Galileo’s picture γγ p

Extra force arises from weight of liquid displaced by meniscusmg mg

LIQUID

n h/ℓ2, (1) −∇· = c where ℓ (γ /ρ g)1/2 is the capillary length, and n ( h, 1)/(1 h 2)1/2 is the unit (outward c = ℓ = −∇ +|∇ |

meniscus is a multiple of either the capillary length, ℓc , or the typical radius of the object, R, whichever is smaller (Finn 1986, Lo 1983). Because the horizontal length scale over which a

118 Vella A BRIEF HISTORY OF FLOATING

“Any object, wholy or partaly immersed in a ﬂuid, is buoyed up by a force equal t te weight of te ﬂuid displaced by te object.” Archimedes, On Floating Bodies (c. 250 BC)

Galileo Galilei,abc Discourse on Floating Bodies (1663) FL47CH06-Vella ARI 14 November 2014 9:50 GAS Galileo’s picture γγ p

Extra force arises from weight of liquid displaced by meniscusmg mg

abcLIQUID GAS

Figure 2 γγ (a) Illustration showing Galileo’s understanding that a dense object (CIOS) may float because of the volume of liquid displaced by the meniscus (AICB). Panel a reproduced from Galilei (1663). (b,c) The equivalence demonstrated by Keller’s (1998) generalizationp of What about the force fromArchimedes’ surface principle to incorporatetension? surface tension. (b)Therestoringforcesthatcounteractthefloatingbody’sweight,mg,comprise a surface tension force per unit length γ,whichactsparalleltothesurfaceandnormaltothecontactline,andthebuoyancyforcemg mg generated by hydrostatic pressure p acting normal to the wetted surface of the body. It is natural that the resultant buoyancy force from hydrostatic pressure is equal in magnitude to the weight of liquid displaced by the wetted body ( gray shaded area). (c)Keller(1998) LIQUID showed that the total restoring force on a floating body (including that due to surface tension) is the weight of the entire volume of liquid displaced by that body ( gray shaded area). Figure 2 (a) Illustration showing Galileo’s understanding that a dense object (CIOS) may float because of the volume of liquid displaced by the Whether to thinkmeniscus of the additional (AICB). Panel forcea reproduced as coming from from Galilei the weight (1663). (ofb,c) liquid The equivalence displaced demonstratedby the by Keller’s (1998) generalization of meniscus or from theArchimedes’ surface tension principle force to incorporate acting on thesurface body tension. is therefore (b)Therestoringforcesthatcounteractthefloatingbody’sweight, a matter of preference. mg,comprise However, it is importanta surface to tensionbear both force points per unit of length view inγ,whichactsparalleltothesurfaceandnormaltothecontactline,andthebuoyancyforce mind. For example, if one thinks only in generated by hydrostatic pressure p acting normal to the wetted surface of the body. It is natural that the resultant buoyancy force from terms of the force perhydrostatic unit length pressure provided is equal by in surface magnitude tension, to the it weight is tempting of liquid to displaced believe that by the having wetted body ( gray shaded area). (c)Keller(1998) a long, convoluted contactshowed that line the will total allow restoring surface force tension on a floating to provide body (including arbitrarily that large due supporting to surface tension) is the weight of the entire volume of forces. Thinking insteadliquid in displaced terms of by the that weight body ( gray of liquid shadedarea displaced). makes it clear that this cannot be the case, for reasons discussed below. Although impressive for its time, Galileo’sWhether intuitive to understanding think of the additional that the weight force as of coming liquid from the weight of liquid displaced by the displaced by the meniscus must balance themeniscus excess weight or from of the a surface dense floating tension forceobject acting misses on an the body is therefore a matter of preference. important part of the story: Objects cannotHowever, lower themselves it is important arbitrarily to bear deep both into points the liquid of view and in mind. For example, if one thinks only in hence displace an arbitrary amount of liquidterms in their of the meniscus. force per Instead, unit length the providedshape of aby meniscus, surface tension, it is tempting to believe that having z h(x, y), for example, of a liquid with interfaciala long, convoluted tension γ , contact density lineρ , and will subject allow surface to a gravi- tension to provide arbitrarily large supporting = ℓ tational acceleration g, must be given by a solutionforces. Thinking of the Laplace-Young instead in terms equation: of the weight of liquid displaced makes it clear that this cannot be the case, for reasons discussed below. Althoughn h/ℓ2, impressive for its time, Galileo’s intuitive(1) understanding that the weight of liquid −∇· = c displaced by the meniscus must balance the excess weight of a dense floating object misses an where ℓ (γ /ρ g)1/2 is the capillary length, and n ( h, 1)/(1 h 2)1/2 is the unit (outward c = ℓ important= part−∇ of the story:+|∇ Objects| cannot lower themselves arbitrarily deep into the liquid and Annu. Rev. Fluid Mech. 2015.47:115-135. Downloaded from www.annualreviews.org pointing) normal to the liquid surface (Finnhence 1986). displace an arbitrary amount of liquid in their meniscus. Instead, the shape of a meniscus, There are no general results that limit the maximum depth of a meniscus; indeed, for a meniscus

Access provided by University of Oxford - Bodleian Library on 01/06/15. For personal use only. z h(x, y), for example, of a liquid with interfacial tension γ , density ρ , and subject to a gravi- = ℓ contained within a sufﬁciently tight corner,tational the meniscus acceleration depthg, must may be be given unbounded by a solution (e.g., seeof the Laplace-Young equation: Finn 1986, p. 116). Nevertheless, in cases of practical interest, the maximum depth of a static meniscus is a multiple of either the capillary length, ℓ , or the typical radius of the object,n hR/ℓ,2, (1) c −∇· = c whichever is smaller (Finn 1986, Lo 1983). Because the horizontal length scale over which a 1/2 2 1/2 where ℓc (γ /ρℓg) is the capillary length, and n ( h, 1)/(1 h ) is the unit (outward meniscus decays is the capillary length ℓc , the typical= weight of liquid displaced in the meniscus= −∇ +|∇ |

Annu. Rev. Fluid Mech. 2015.47:115-135. Downloaded from www.annualreviews.org 2 pointing) normal to the liquid surface (Finn 1986). scales as ρℓg min R, ℓc ℓc . Hence, we expect that a solid object of density ρs and typical radius R { 3 } 2 There are no general results that limit the maximum depth of a meniscus; indeed, for a meniscus can float only if ρ Access provided by University of Oxford - Bodleian Library on 01/06/15. For personal use only. s R αρℓ min R, ℓc ℓ for some dimensionless constant α. ≤ { } c contained within a sufficiently tight corner, the meniscus depth may be unbounded (e.g., see From this scaling, it is clear that objects with solid densities significantly larger than that of the Finn 1986, p. 116). Nevertheless, in cases of practical interest, the maximum depth of a static liquid can only float if R ℓc . In this case, flotation is possible provided that ≪ meniscus is a multiple of either the capillary length, ℓc , or the typical radius of the object, R, 2 1 ρ /ρ αwhichever(R/ℓ )− isα smallerBo− , (Finn 1986, Lo 1983). Because(2) the horizontal length scale over which a s ℓ ≤ c = meniscus decays is the capillary length ℓc , the typical weight of liquid displaced in the meniscus scales as ρ g min R, ℓ ℓ2. Hence, we expect that a solid object of density ρ and typical radius R 118 Vella ℓ { c } c s can float only if ρ R3 αρ min R, ℓ ℓ2 for some dimensionless constant α. s ≤ ℓ { c } c From this scaling, it is clear that objects with solid densities significantly larger than that of the liquid can only float if R ℓ . In this case, flotation is possible provided that ≪ c 2 1 ρ /ρ α(R/ℓ )− αBo− , (2) s ℓ ≤ c =

118 Vella KELLER’S THEOREM

“Te vertcal component of te surface tnsion force on a body partly submerged in a liquid is shown t equal te weight of liquid displaced by te meniscus.” Keller, Phys. Fluids (1998) 2.2 The contribution of surface tension

2.2 The contribution of surface tension

The quotation at the beginning of this chapter clearly shows that Galileo believed that 2.2 The contribution of surface tension small, dense objects could ﬂoat because of the substantial volume of liquid displaced in 2.2 The contribution of surface tension what we now call the menisci.Thishypothesisiseasilyrationalizedbecausetheliquidhas

F st = γAB r˙(s)×n ds. (2.1)

γAB !C Since we shall be interested primarily in the vertical component of this force, we take the r˙ C (s) scalar product of (2.1) with the vertical unit vector, k.Now,theprojectionof onto the C x y plane (denoted by )hasinwardpointingnormalν and a diﬀerent arc length, s . − Cπ ′

Figure 2.3: Schematic diagram of an irregularly shaped object ﬂoating ataliquid–ﬂuidinterface. The notation here is that used in the proof of the generalized Archimedes’ principle. The contact line, ,isgivenintermsofarclengths by x = r(s). C

γAB 9

r˙ C (s)

9 2.2 The contribution of surface tension

2.2 The contribution of surface tension

To progress further, wethe must balance relate betweenγAB∇ · then,thecurvaturepressureduetosurfaceten- hydrostatic and curvature pressuresmaybeexpressedbythe r˙(s) sion, to the deﬂection ofLaplace–Young the contact line. equation In equilibrium,C (Finn, 1986) the curvature pressure is precisely

balanced by the hydrostatic pressure in the liquid. If fluids of density ρA and ρB (ρA < ρB) meet at the interface, then the pressure just beneath( theρB intρerfaceA)gh = is (γρAB∇ρ· n)gh, ,where (2.4) − − A − B h is the displacement of the interface from horizontal (see figure 2.2). Mathematically, n the balance between thewhere hydrostaticis the unit and vector curvature normal pressure to thesmaybeexpressedbythe interface. Substituting thisresultinto(2.3),we Figure 2.3: Schematicobtain diagram of an irregularly shaped object floating ataliquid–fluidinterface. Laplace–YoungThe notation equation here (Finn,is that used1986) in the proof of the generalized Archimedes’ principle. The contact k · F st = (ρB ρA)g h dA. (2.5) line, ,isgivenintermsofarclengths by x = r(s). 2 − − R Wπ C ! \ (ρB ρA)gh = γAB∇ · n, (2.4) Physically, (2.5)− shows that− the vertical force contributedbysurfacetensionissimplythe where n is the unit vector(modified) normal weight to the of interface. liquid displaced Substituting in the thi meniscisresultinto(2.3),we exteriortotheobject. obtain For small objects, such as the spider’s legs shown in figure 2.1b, the vertical contribution k · F st = (ρB ρA)g h dA. (2.5) 9 of surface tension dominates the2 contribution from the more conventional Archimedean − − R Wπ ! \ upthrust caused by the excluded volume of the body itself. However, for large, thin objects Physically, (2.5) shows that the vertical force contributedbysurfacetensionissimplythe (such as the drawing pin of figure 2.1a) the situation is more subtle. In this case, there (modified) weight of liquid displaced in the menisci exteriortotheobject. is also a substantial volume of fluid displaced directly abovetheobjectitself.Itisa For small objects, suchrelatively as the spider’s simple matterlegs shown to show in figure that 2. the1b, the vertical vertical force contribution contribution due to hydrostatic

of surface tension dominatespressure, thek · contributionF hp,isgivenby from the more conventional Archimedean upthrust caused by the excluded volume of the body itself. However, for large, thin objects (such as the drawing pin of figure 2.1a) the situationk · F is more= (ρ subtle.ρ In)g thish case,dA, there (2.6) hp − B − A is also a substantial volume of fluid displaced directly abovetheobjectitself.Itisa!Wπ relatively simple matterwhich to show is simply that the the (modified) vertical force weight con oftribution water that due would to hydrostatic fill the hatched region in figure

pressure, k · F hp,isgivenby2.4 for a circular cylinder lying horizontally at the interface. Combining this with the physical interpretation of (2.5), we see that the total vertical force acting on an interfacial objectk is· F indeedhp = the(ρB modifiedρA)g weighth dA, of liquid that its presence(2.6) at the interface displaces − − !Wπ which is simply the (modified) weight of water that would fill the hatched region in figure 2.4 for a circular cylinder lying horizontally at the interface. Combining this with the 10 physical interpretation of (2.5), we see that the total vertical force acting on an interfacial object is indeed the modified weight of liquid that its presence at the interface displaces

10 2.2 The contribution of surface tension

2.2 The contribution of surface tension

10 2.2 The contribution of surface tension

2.2 The contribution of surface tension

10 2.2 The contribution of surface tension

2.2 The contribution of surface tension

The quotation at the beginning of this chapterno clearly knowledge shows that Galileo of thebelieved precise that shape of the object away from the wetted surface. The liquid small, dense objects could float because of the substantial volume of liquid displaced in what we now call the menisci.Thishypothesisiseasilyrationalizedbecausetheliquidmust therefore supplyhas a vertical force equal to the total weight of displaced liquid, by Geometry then gives k × r˙ ds = ν ds′ and so we have that no knowledge of the precise shape of the object awaythe from familiar the wetted Archimedes’ surface. The liquid principle. At the same time, the notion of an interfacial tension must therefore supply a vertical force equal to the totalKELLER’S weight of displaced liquid, byTHEOREM the familiar Archimedes’ principle. At the samesuggests time, the notion that of an theinterfacial vertical tension force supplied by the external liquid must be the result of this suggests that the vertical force supplied by the external liquid must be the result of this k · F st = γAB n · ν ds′. (2.2) tension acting around the contact line. Keller (1998)tension was thefirsttoshowthatthesetwo acting around the contact line. Keller (1998) was thefirsttoshowthatthesetwo π physical pictures are, in fact, equivalent. Since ourphysical consideration pictures of when objects are, can in float fact, equivalent. Since our consideration!C of when objects can float will rely heavily on this equivalence, we repeat the proof“T givene byver Kellert incal this section. component of te surface tnsion force willDenoting rely heavily the on this projection equivalence, of the we repeat wetted the surface proof given onto by the Kellerx iny thisplane section. by Wπ,thedivergence Consider an irregularly shaped object floating at an interface, as in figure 2.3. The − contact line, ,isgivenintermsofarclengths by xon= r(s )andthenormaltotheliquida body partly submerged2 in a liquid is shown t C theorem applied on R Wπ gives interface (pointing from liquid to fluid) is n.Atthecontactline,thesurfacetensionConsider an irregularlyγAB shaped\ object floating at an interface, as in figure 2.3. The acts perpendicular to both the normal to the interface and thetangenttothecontactline. Summing the contributions of this force aroundcontact theequal closed loop line, of t the contacte,isgivenintermsofarclength weight line, ,we of liquid displaceds by x =byr( st)andthenormaltotheliquide C C 2.2 The contribution of surface tension find that the total force on the object due to surface tension, F ,is interface (pointingst from liquid to fluid)k is· Fn.Atthecontactline,thesurfacetensionst = γAB ∇ · n dA. γAB (2.3) R2 × meniscus.” Wπ F st = γAB r˙(sacts) n ds. perpendicular to both(2.1) the normal× to the interface and! th\etangenttothecontactline. !C Geometry then gives k r˙ ds = ν ds′ and so we have that Summing the contributions of this force aroundKeller, the closed Phys. loFluidsop of (1998) the contact line, ,we Since we shall be interested primarily in the vertical component of this force, we take the ∇ · n C scalar product of (2.1) with the vertical unit vector, kTo.Now,theprojectionof progress further,onto the we must relate γAB ,thecurvaturepressureduetosurfaceten- find that the totalC force on the object due to surface tension, F st,is x y plane (denoted by )hasinwardpointingnormalν and a different arc length,Surfaces . tension force:k · F st = γAB n · ν ds′. (2.2) − Cπ ′ sion, to the deflection of the contact line. In equilibrium,π the curvature pressure is precisely 2.2 The contribution!C of surface tension F r˙ ×n balancedDenoting by the hydrostatic the projectionst = pressure ofγAB the wetted( ins) the surfaced liquid.s. onto theIf fluidsx y planeof density by W(2.1),thedivergenceρA and ρB (ρA < ρB) − π γAB R2 !C Geometry thenmeet gives at thektheorem× r interface,˙ ds = appliedν ds′ then onand so the weWπ have pressuregives that just beneath the interface is (ρA ρB)gh,where Since we shall be interestedLooking primarily at the\ in vertical the vertical component: component of this force, we take the − r˙ C (s) h is the displacement of the interface from horizontal (see figure 2.2). Mathematically, scalar product of (2.1) with the vertical unit vector, k.Now,theprojectionof onto the k · F st = γAB n k· ν· Fdsst′.= γAB ∇ · n dA. (2.2) (2.3) 2 C the balance between the hydrostaticπ and curvatureR Wπ pressuresmaybeexpressedbythe x y plane (denoted by π)hasinwardpointingnormal!C normal to ! ν \and a different arc length, s′. − C Laplace–Youngprojection equation of contact (Finn, 1986)contact line Denoting the projectionTo of progress the wetted further, surface we onto must the relatex γy plane∇ · n by,thecurvaturepressureduetosurfaceten-W ,thedivergence Figure 2.3: Schematic diagram of an irregularly shaped object floating ataliquid–fluidinterface.line to 2-d plane − AB π The notation here is that used in the proof of the generalized Archimedes’R principle.2 The contact line, ,isgivenintermsofarclengthstheoremby x = r(s). applied on sion,Wπ togives the deflection of the contact line. In equilibrium, the curvature pressure is precisely C \ ∇ · n To eliminate curvature, use Laplace-Youngbalanced by theequation: hydrostatic( pressureρB ρ inA) thegh liquid.= γ IfAB fluids of density, ρA and ρB (ρA < ρB) (2.4) · ∇ · − − meet atk theF st interface,= γAB then the pressuren dA.γAB just beneath the interface(2.3) is (ρ ρ )gh,where 2 A B R Wπ − where n ish is the the unit displacement9 vector! normal of\ the interface to the from interface. horizontal Substituting (see figure 2.2). thi Mathematically,sresultinto(2.3),we To progress further, wethe must balance relate betweenγAB∇ · then,thecurvaturepressureduetosurfaceten- hydrostatic and curvature pressuresmaybeexpressedbythe obtain r˙(s) sion, to the deflection ofLaplace–Young the contact line. equation In equilibrium,C (Finn, 1986) the curvature pressure is precisely k · F st = (ρB ρA)g h dA. (2.5) balanced by the hydrostatic pressure in the liquid. If fluids of density ρA and ρ2B (ρA < ρB) − − R Wπ ! \ meet at the interface, then the pressure just beneath( theρB intρerfaceA)gh = is (γρAB∇ρ· n)gh, ,where (2.4) − − A − B h is the displacementPhysically, of (2.5) the interface shows from that horizontal the vertical (see fig forceure 2.2). contribute Mathematically,dbysurfacetensionissimplythe n the balance(modified) between thewhere weight hydrostaticis the of liquid unit and vector curvature displaced normal pressure to in the thesmaybeexpressedbythe interface. menisci Substituting exteriortotheobject. thisresultinto(2.3),we Figure 2.3: Schematicobtain diagram of an irregularly shaped object floating ataliquid–fluidinterface. Laplace–YoungThe notation equation here (Finn,is that used1986) in the proof of the generalized Archimedes’ principle. The contact k · F st = (ρB ρA)g h dA. (2.5) line, For,isgivenintermsofarclength small objects, such ass theby x spider’s= r(s). legs shown2 in figure 2.1b, the vertical contribution − − R Wπ C ! \ (ρB ρA)gh = γAB∇ · n, (2.4) of surfacePhysically, tension (2.5) dominates− shows that− the the contribution vertical force contribute from thedbysurfacetensionissimplythe more conventional Archimedean where n isupthrust the unit vector(modified) caused normal by weight to the the excluded of interface. liquid displaced volume Substituting in of the the thi meniscisresultinto(2.3),we body exterio itself.rtotheobject. However, for large, thin objects obtain (such as the drawing pin of figure 2.1a) the situation is more subtle. In this case, there For small objects, such as the spider’s legs shown in figure 2.1b, the vertical contribution k · F st = (ρB ρA)g h dA. (2.5) 9 is also aof substantial surface tension volume dominates of fluid the2 contribution displaced from directly the more abovconventionaletheobjectitself.Itisa Archimedean − − R Wπ ! \ upthrust caused by the excluded volume of the body itself. However, for large, thin objects Physically,relatively (2.5) shows that simple the vertical matter force to contribute show thatdbysurfacetensionissimplythe the vertical force contribution due to hydrostatic (such as the drawing pin of figure 2.1a) the situation is more subtle. In this case, there (modified)pressure, weight of liquidk · F displacedhp,isgivenby in the menisci exteriortotheobject. is also a substantial volume of fluid displaced directly abovetheobjectitself.Itisa For small objects, suchrelatively as the spider’s simple matterlegs shown to show in figure that 2. the1b, the vertical vertical force contribution contribution due to hydrostatic k · F = (ρB ρA)g h dA, (2.6) of surface tension dominatespressure, thek · contributionF hp,isgivenby fromhp the more conventional Archimedean − − Wπ upthrust caused by the excluded volume of the body itself. However, for large,! thin objects (such as the drawing pin of figure 2.1a) the situationk · F is more= (ρ subtle.ρ In)g thish case,dA, there (2.6) which is simply the (modified) weighthp of− waterB − thatA would fill the hatched region in figure is also a substantial volume of fluid displaced directly abovetheobjectitself.Itisa!Wπ 2.4 for a circular cylinder lying horizontally at the interface. Combining this with the relatively simple matterwhich to show is simply that the the (modified) vertical force weight con oftribution water that due would to hydrostatic fill the hatched region in figure physical interpretation of (2.5), we see that the total vertical force acting on an interfacial pressure, k · F hp,isgivenby2.4 for a circular cylinder lying horizontally at the interface. Combining this with the object isphysical indeed interpretation the modified of (2.5), weight we see of thatliquid the that total vert itsical presen forcece acting at the on an interface interfacial displaces objectk is· F indeedhp = the(ρB modifiedρA)g weighth dA, of liquid that its presence(2.6) at the interface displaces − − !Wπ which is simply the (modified) weight of water that would fill the hatched region in figure 10 2.4 for a circular cylinder lying horizontally at the interface. Combining this with the 10 physical interpretation of (2.5), we see that the total vertical force acting on an interfacial object is indeed the modified weight of liquid that its presence at the interface displaces

10 4.11 brief comms NR 28/10/04 4:22 pm Page 36

ROUGHbrief communicationsSURFACES dissolving lead shavings in vinegar8. Stannic This finding suggests that the force exerted a oxide would have been an acceptable substi- Vessel wall by the strider’s legs could be due to a ‘super- Often think of surface tension as providingtute,with supplies abeing force available per through the hydrophobic’ effect (that is, the contact angle Cornish tin industry. The Romano–British Cantilever with water would be greater than 150°). We unit length – does this explain whych emistswater probably walking believed that they were verified that this was indeed the case by sessile using a new source of cerussa, so confused Leg water-drop measurements, which showed insects have hairy legs? were their encyclopaedias in distinguishing that that the contact angle of the insect’s legs lead from tin. with water was 167.6Ǆ4.4°(Fig.1a,inset). None ofthe surviving Greco–Roman med- The contact angle of water made with the ical corpora suggest that tin had pharmaceu- cuticle wax secreted on a strider’s leg is about Keller’s Theorem shows that this cannottical value9 .Abelthough the organic case tin compounds 105° (ref. 5), which is not enough to account are now well known for their toxicological for its marked water repellence.Knowing that – a hairy surface does not signiﬁcantlyproperties,inorganic alter the forms seem to be largely microstructures on an object with low surface 10 b 6 inert .Hence, unless SnO2 was added owing energy can enhance its hydrophobicity ,we volume of liquid displaced to some hitherto unrecognized medicinal investigated the physical features of the legs. attribute, we must conclude that its function Scanning electronic micrographs revealed was solely as a pigment.The non-toxic prop- numerous oriented setae on the legs.These are Gao & Jiang (2004) &Jiang Gao Key point: contact line length is increasederties of SnO but2 would also have been desir- needle-shaped, with diameters ranging from able, because by the second century AD the 3 micrometres down to several hundred presence of nearby hairs causes mini-meniscidangers of lead were becomingto recognized9. nanometres (Fig. 1b). Most setae are roughly R. P.Evershed*, R. Berstan*, F. Grew†, 50 Ȗm in length and arranged at an inclined ﬂatten (and hence provide less force)M. S. Copley*, A. J. H. Charmant*, angle of about 20° from the surface of leg. E. Barham†, H. R. Mottram*, G. Brown‡ Many elaborate, nanoscale grooves are evi- *School of Chemistry, University of Bristol, dent on each microseta, and these form a Cantock’s Close, Bristol BS8 1TS, UK c unique hierarchical structure (Fig.1c). e-mail:[email protected] According to Cassie’s law for surface wet- †Museum of London, London Wall, tability, such microstructures can be regard- London EC2Y 5HN, UK ed as heterogeneous surfaces composed of 7 ‡Pre-Construct Archaeology, Unit 54, Brockley solid and air .The apparent contact angle Ȓl Cross Business Centre, Endwell Road, of the legs is described by cosȒlǃf1cosȒwǁf2, London SE4 2PD, UK where f1 is the area fraction of microsetae 1. Durrani, N. Curr.Archaeol. 192, 540–547 (2004). with nanogrooves,f2 is the area fraction ofair 2. Beagrie, N. Britannia 20, 169–191 (1989). on the leg surface and Ȓw is the contact angle 3. Copley, M. S. et al. Proc. Natl Acad. Sci. USA 100, 1524–1529 (2003). of the secreted wax. Using measured values 4. Ralph, J. & Hatfield, R. D. J. Agric. Food Chem. 39, of Ȓl and Ȓw,we deduce from the equation 1426–1437 (1991). Figure 1 The non-wetting leg of a water strider. a, Typical side that the air fraction between the leg and the 5. Blakeney, A. B., Harris, P. J.,Henry,R.J.& Stone,B.A. view of a maximal-depth dimple (4.38Ǆ0.02 mm) just before the water surface corresponds to f ǃ96.86%. Carbohyd. Res. 113, 291–299 (1983). 2 6. PDF-2 Powder Diffraction Database (International Centre for leg pierces the water surface. Inset, water droplet on a leg; this Available air is trapped in spaces in the Diffraction Data, Pennsylvania, 1999). makes a contact angle of 167.6Ǆ4.4°. b, c, Scanning electron microsetae and nanogrooves to form a cush- 7. Cert, A., Lanzón, A., Carelli, A. A., Albi, T. & Amelitti, G. microscope images of a leg showing numerous oriented spindly ion at the leg–water interface that prevents Food Chem. 49, 287–293 (1994). microsetae (b) and the fine nanoscale grooved structures on a 8. Pliny Natural History (ed. Rackham, H.) 9 (Loeb Classical the legs from being wetted. Library, Cambridge, Massachusetts, 1944). seta (c). Scale bars: b, 20 Ȗm; c, 200 nm. This unique hierarchical micro- and 9. Riddle, J. M. in Dioscorides on Pharmacy and Medicine 153–154 nanostructuring on the leg’s surface there- (Univ. of Texas Press, 1985). water surface (for methods, see supplemen- fore seems to be responsible for its water 10. Toxicological Profile for Tin and Tin Compounds (Agency for Toxic Substances and Disease Registry, US Public Health Service, 2003). tary information). Surprisingly, the leg does resistance and the strong supporting force. Competing financial interests: declared none. not pierce the water surface until a dimple of This clever arrangement allows water strid- 4.38Ǆ0.02 mm depth is formed (Fig. 1a). ers to survive on water even if they are being The maximal supporting force of a single leg bombarded by raindrops,when they bounce Biophysics is 152 dynes (see supplementary informa- to avoid being drowned. Our discovery may tion),or about 15 times the total body weight be helpful in the design of miniature aquatic Water-repellent legs of the insect. The corresponding volume of devices and non-wetting materials. of water striders water ejected is roughly 300 times that of Xuefeng Gao*†, Lei Jiang*‡ the leg itself, indicating that its surface is *Key Laboratory of Organic Solids, Institute of ater striders (Gerris remigis) have strikingly water repellent. Chemistry, and the †Graduate School, Chinese remarkable non-wetting legs that For comparison, we made a hydrophobic Academy of Sciences, Beijing 100080, China Wenable them to stand effortlessly and ‘leg’ from a smooth quartz fibre that was ‡National Center for Nanoscience and move quickly on water, a feature believed to similar in shape and size to a strider’s leg. Its Nanotechnology, Beijing 100080, China be due to a surface-tension effect caused by surface was modified by a self-assembling e-mail: [email protected] 1–3 secreted wax .We show here, however, that monolayer of low-surface-energy hepta- 1. Caponigro, M. A. & Erilsen, C. H. Am.Midland Nat. 95, it is the special hierarchical structure of the decafluorodecyltrimethoxysilane (FAS-17), 268–278 (1976). 2. Dickinson, M. Nature 424, 621–622 (2003). legs, which are covered by large numbers of which makes a contact angle of 109° with a 3. Hu, D. L., Chan, B. & Bush, J. W. M. Nature 424, 663–666 (2003). oriented tiny hairs (microsetae) with fine water droplet on a flat surface4.Water sup- 4. Tadanaga, K., Katata, N. & Minami, T. J. Am. Ceram. Soc. 80, nanogrooves, that is more important in ports the artificial leg with a maximal force of 1040–1042 (1997). 5. Holdgate, M. W. J. Exp. Biol. 32, 591–617 (1955). inducing this water resistance. only 19.05 dynes (see supplementary infor- 6. Feng, L. et al. Adv. Mater. 14, 1857–1860 (2002). We used a high-sensitivity balance system mation), which is enough to support the 7. Cassie, A. B. D. & Baxter, S. Trans. Farad. Soc. 40, 546–551 (1944). Supplementary information accompanies this communication on to construct force–displacement curves for strider at rest but not to enable it to glide or Nature’s website. a water strider’s legs when pressing on the dart around rapidly on the surface. Competing financial interests: declared none.

ROUGHbrief communicationsSURFACES dissolving lead shavings in vinegar8. Stannic This finding suggests that the force exerted a oxide would have been an acceptable substi- Vessel wall by the strider’s legs could be due to a ‘super- Often think of surface tension as providingtute,with supplies abeing force available per through the hydrophobic’ effect (that is, the contact angle Cornish tin industry. The Romano–British Cantilever with water would be greater than 150°). We unit length – does this explain whych emistswater probably walking believed that they were verified that this was indeed the case by sessile using a new source of cerussa, so confused Leg water-drop measurements, which showed insects have hairy legs? were their encyclopaedias in distinguishing that that the contact angle of the insect’s legs lead from tin. with water was 167.6Ǆ4.4°(Fig.1a,inset). None ofthe surviving Greco–Roman med- The contact angle of water made with the ical corpora suggest that tin had pharmaceu- cuticle wax secreted on a strider’s leg is about Keller’s Theorem shows that this cannottical value9 .Abelthough the organic case tin compounds 105° (ref. 5), which is not enough to account are now well known for their toxicological for its marked water repellence.Knowing that – a hairy surface does not significantlyproperties,inorganic alter the forms seem to be largely microstructures on an object with low surface 10 b 6 inert .Hence, unless SnO2 was added owing energy can enhance its hydrophobicity ,we volume of liquid displaced to some hitherto unrecognized medicinal investigated the physical features of the legs. attribute, we must conclude that its function Scanning electronic micrographs revealed was solely as a pigment.The non-toxic prop- numerous oriented setae on the legs.These are Gao & Jiang (2004) &Jiang Gao Key point: contact line length is increasederties of SnO but2 would also have been desir- needle-shaped, with diameters ranging from able, because by the second century AD the 3 micrometres down to several hundred presence of nearby hairs causes mini-meniscidangers of lead were becomingto recognized9. nanometres (Fig. 1b). Most setae are roughly R. P.Evershed*, R. Berstan*, F. Grew†, 50 Ȗm in length and arranged at an inclined flatten (and hence provide less force)M. S. Copley*, A. J. H. Charmant*, angle of about 20° from the surface of leg. E. Barham†, H. R. Mottram*, G. Brown‡ Many elaborate, nanoscale grooves are evi- *School of Chemistry, University of Bristol, dent on each microseta, and these form a Cantock’s Close, Bristol BS8 1TS, UK c unique hierarchical structure (Fig.1c). e-mail:[email protected] According to Cassie’s law for surface wet- †Museum of London, LondonAlso Wall, allows us to understand floating oftability, such microstructures can be regard- London EC2Y 5HN, UK ed as heterogeneous surfaces composed of ‘inverse Leidenfrost drops’ (water on 7 ‡Pre-Construct Archaeology, Unit 54, Brockley solid and air .The apparent contact angle Ȓl Cross Business Centre, Endwell Road, of the legs is described by cos f cos f , liquid Nitrogen): Ȓlǃ 1 Ȓwǁ 2 London SE4 2PD, UK where f1 is the area fraction of microsetae 1. Durrani, N. Curr.Archaeol. 192, 540–547 (2004). with nanogrooves,f2 is the area fraction ofair 2. Beagrie, N. Britannia 20, 169–191 (1989). on the leg surface and Ȓw is the contact angle 3. Copley, M. S. et al. Proc. Natl Acad. Sci. USA 100, 1524–1529 (2003). of the secreted wax. Using measured values 4. Ralph, J. & Hatfield, R. D. J. Agric. Food Chem. 39, of Ȓl and Ȓw,we deduce from the equation 1426–1437 (1991). Figure 1 The non-wetting leg of a water strider. a, Typical side that the air fraction between the leg and the 5. Blakeney, A. B., Harris, P. J.,Henry,R.J.& Stone,B.A. view of a maximal-depth dimple (4.38Ǆ0.02 mm) just before the water surface corresponds to f ǃ96.86%. Carbohyd. Res. 113, 291–299 (1983). 2 6. PDF-2 Powder Diffraction Database (International Centre for leg pierces the water surface. Inset, water droplet on a leg; this Available air is trapped in spaces in the Diffraction Data, Pennsylvania, 1999). makes a contact angle of 167.6Ǆ4.4°. b, c, Scanning electron microsetae and nanogrooves to form a cush- 7. Cert, A., Lanzón, A., Carelli, A. A., Albi, T. & Amelitti, G. microscope images of a leg showing numerous oriented spindly ion at the leg–water interface that prevents Food Chem. 49, 287–293 (1994). microsetae (b) and the fine nanoscale grooved structures on a 8. Pliny Natural History (ed. Rackham, H.) 9 (Loeb Classical the legs from being wetted. Library, Cambridge, Massachusetts, 1944). seta (c). Scale bars: b, 20 Ȗm; c, 200 nm. This unique hierarchical micro- and 9. Riddle, J. M. in Dioscorides on Pharmacy and Medicine 153–154 nanostructuring on the leg’s surface there- (Univ. of Texas Press, 1985). water surface (for methods, see supplemen- fore seems to be responsible for its water 10. Toxicological Profile for Tin and Tin Compounds (Agency for Toxic Substances and Disease Registry, US Public Health Service, 2003). tary information). Surprisingly, the leg does resistance and the strong supporting force. Competing financial interests: declared none. not pierce the water surface until a dimple of This clever arrangement allows water strid- 4.38Ǆ0.02 mm depth is formed (Fig. 1a). ers to survive on water even if they are being The maximal supporting force of a single leg bombarded by raindrops,when they bounce Biophysics is 152 dynes (see supplementary informa- to avoid being drowned. Our discovery may tion),or about 15 times the total body weight be helpful in the design of miniature aquatic Water-repellent legs of the insect. The corresponding volume of devices and non-wetting materials. of water striders water ejected is roughly 300 times that of Xuefeng Gao*†, Lei Jiang*‡ the leg itself, indicating that its surface is *Key Laboratory of Organic Solids, Institute of ater striders (Gerris remigis) have strikingly water repellent. Chemistry, and the †Graduate School, Chinese remarkable non-wetting legs that For comparison, we made a hydrophobic Academy of Sciences, Beijing 100080, China Wenable them to stand effortlessly and ‘leg’ from a smooth quartz fibre that was ‡National Center for Nanoscience and move quickly on water, a feature believed to similar in shape and size to a strider’s leg. Its Nanotechnology, Beijing 100080, China be due to a surface-tension effect caused by surface was modified by a self-assembling e-mail: [email protected] 1–3 secreted wax .We show here, however, that monolayer of low-surface-energy hepta- 1. Caponigro, M. A. & Erilsen, C. H. Am.Midland Nat. 95, it is the special hierarchical structure of the decafluorodecyltrimethoxysilane (FAS-17), 268–278 (1976). 2. Dickinson, M. Nature 424, 621–622 (2003). legs, which are covered by large numbers of which makes a contact angle of 109° with a 3. Hu, D. L., Chan, B. & Bush, J. W. M. Nature 424, 663–666 (2003). oriented tiny hairs (microsetae) with fine water droplet on a flat surface4.Water sup- 4. Tadanaga, K., Katata, N. & Minami, T. J. Am. Ceram. Soc. 80, nanogrooves, that is more important in ports the artificial leg with a maximal force of 1040–1042 (1997). 5. Holdgate, M. W. J. Exp. Biol. 32, 591–617 (1955). inducing this water resistance. only 19.05 dynes (see supplementary infor- 6. Feng, L. et al. Adv. Mater. 14, 1857–1860 (2002). We used a high-sensitivity balance system mation), which is enough to support the 7. Cassie, A. B. D. & Baxter, S. Trans. Farad. Soc. 40, 546–551 (1944). Supplementary information accompanies this communication on to construct force–displacement curves for strider at rest but not to enable it to glide or Nature’s website. a water strider’s legs when pressing on the dart around rapidly on the surface. Competing financial interests: declared none.

ROUGHbrief communicationsSURFACES dissolving lead shavings in vinegar8. Stannic This finding suggests that the force exerted a oxide would have been an acceptable substi- Vessel wall by the strider’s legs could be due to a ‘super- Often think of surface tension as providingtute,with supplies abeing force available per through the hydrophobic’ effect (that is, the contact angle Cornish tin industry. The Romano–British Cantilever with water would be greater than 150°). We unit length – does this explain whych emistswater probably walking believed that they were verified that this was indeed the case by sessile using a new source of cerussa, so confused Leg water-drop measurements, which showed insects have hairy legs? were their encyclopaedias in distinguishing that that the contact angle of the insect’s legs lead from tin. with water was 167.6Ǆ4.4°(Fig.1a,inset). None ofthe surviving Greco–Roman med- The contact angle of water made with the ical corpora suggest that tin had pharmaceu- cuticle wax secreted on a strider’s leg is about Keller’s Theorem shows that this cannottical value9 .Abelthough the organic case tin compounds 105° (ref. 5), which is not enough to account are now well known for their toxicological for its marked water repellence.Knowing that – a hairy surface does not significantlyproperties,inorganic alter the forms seem to be largely microstructures on an object with low surface 10 b 6 inert .Hence, unless SnO2 was added owing energy can enhance its hydrophobicity ,we volume of liquid displaced to some hitherto unrecognized medicinal investigated the physical features of the legs. attribute, we must conclude that its function Scanning electronic micrographs revealed was solely as a pigment.The non-toxic prop- numerous oriented setae on the legs.These are Gao & Jiang (2004) &Jiang Gao Key point: contact line length is increasederties of SnO but2 would also have been desir- needle-shaped, with diameters ranging from able, because by the second century AD the 3 micrometres down to several hundred presence of nearby hairs causes mini-meniscidangers of lead were becomingto recognized9. nanometres (Fig. 1b). Most setae are roughly R. P.Evershed*, R. Berstan*, F. Grew†, 50 Ȗm in length and arranged at an inclined flatten (and hence provide less force)M. S. Copley*, A. J. H. Charmant*, angle of about 20° from the surface of leg. E. Barham†, H. R. Mottram*, G. Brown‡ Many elaborate, nanoscale grooves are evi- *School of Chemistry, University of Bristol, dent on each microseta, and these form a Cantock’s Close, Bristol BS8 1TS, UK c unique hierarchical structure (Fig.1c). e-mail:[email protected] According to Cassie’s law for surface wet- †Museum of London, LondonAlso Wall, allows us to understand floating oftability, such microstructures can be regard- London EC2Y 5HN, UK ed as heterogeneous surfaces composed of ‘inverse Leidenfrost drops’ (water on 7 ‡Pre-Construct Archaeology, Unit 54, Brockley solid and air .The apparent contact angle Ȓl Cross Business Centre, Endwell Road, of the legs is described by cos f cos f , liquid Nitrogen): Ȓlǃ 1 Ȓwǁ 2 London SE4 2PD, UK where f1 is the area fraction of microsetae 1. Durrani, N. Curr.Archaeol. 192, 540–547 (2004). with nanogrooves,f2 is the area fraction ofair 2. Beagrie, N. Britannia 20, 169–191 (1989). on the leg surface and Ȓw is the contact angle 3. Copley, M. S. et al. Proc. Natl Acad.• Sci.While USA 100, levitating, drop is 1524–1529 (2003). of the secreted wax. Using measured values 4. Ralph, J. & Hatfield, R. D. J. Agric. Foodeffectively Chem. 39, non-wetting sphere of Ȓl and Ȓw,we deduce from the equation 1426–1437 (1991). Figure 1 The non-wetting leg of a water strider. a, Typical side that the air fraction between the leg and the 5. Blakeney, A. B., Harris, P. J.,Henry,R.J.& Stone,B.A. view of a maximal-depth dimple (4.38Ǆ0.02 mm) just before the water surface corresponds to f ǃ96.86%. Carbohyd. Res. 113, 291–299 (1983). 2 6. PDF-2 Powder Diffraction Database (International Centre for leg pierces the water surface. Inset, water droplet on a leg; this Available air is trapped in spaces in the Diffraction Data, Pennsylvania, 1999). makes a contact angle of 167.6Ǆ4.4°. b, c, Scanning electron microsetae and nanogrooves to form a cush- 7. Cert, A., Lanzón, A., Carelli, A. A., Albi, T. & Amelitti, G. microscope images of a leg showing numerous oriented spindly ion at the leg–water interface that prevents Food Chem. 49, 287–293 (1994). microsetae (b) and the fine nanoscale grooved structures on a 8. Pliny Natural History (ed. Rackham, H.) 9 (Loeb Classical the legs from being wetted. Library, Cambridge, Massachusetts, 1944). seta (c). Scale bars: b, 20 Ȗm; c, 200 nm. This unique hierarchical micro- and 9. Riddle, J. M. in Dioscorides on Pharmacy and Medicine 153–154 nanostructuring on the leg’s surface there- (Univ. of Texas Press, 1985). water surface (for methods, see supplemen- fore seems to be responsible for its water 10. Toxicological Profile for Tin and Tin Compounds (Agency for Toxic Substances and Disease Registry, US Public Health Service, 2003). tary information). Surprisingly, the leg does resistance and the strong supporting force. Competing financial interests: declared none. not pierce the water surface until a dimple of This clever arrangement allows water strid- 4.38Ǆ0.02 mm depth is formed (Fig. 1a). ers to survive on water even if they are being The maximal supporting force of a single leg bombarded by raindrops,when they bounce Biophysics is 152 dynes (see supplementary informa- to avoid being drowned. Our discovery may tion),or about 15 times the total body weight be helpful in the design of miniature aquatic Water-repellent legs of the insect. The corresponding volume of devices and non-wetting materials. of water striders water ejected is roughly 300 times that of Xuefeng Gao*†, Lei Jiang*‡ the leg itself, indicating that its surface is *Key Laboratory of Organic Solids, Institute of ater striders (Gerris remigis) have strikingly water repellent. Chemistry, and the †Graduate School, Chinese remarkable non-wetting legs that For comparison, we made a hydrophobic Academy of Sciences, Beijing 100080, China Wenable them to stand effortlessly and ‘leg’ from a smooth quartz fibre that was ‡National Center for Nanoscience and move quickly on water, a feature believed to similar in shape and size to a strider’s leg. Its Nanotechnology, Beijing 100080, China be due to a surface-tension effect caused by surface was modified by a self-assembling e-mail: [email protected] 1–3 secreted wax .We show here, however, that monolayer of low-surface-energy hepta- 1. Caponigro, M. A. & Erilsen, C. H. Am.Midland Nat. 95, it is the special hierarchical structure of the decafluorodecyltrimethoxysilane (FAS-17), 268–278 (1976). 2. Dickinson, M. Nature 424, 621–622 (2003). legs, which are covered by large numbers of which makes a contact angle of 109° with a 3. Hu, D. L., Chan, B. & Bush, J. W. M. Nature 424, 663–666 (2003). oriented tiny hairs (microsetae) with fine water droplet on a flat surface4.Water sup- 4. Tadanaga, K., Katata, N. & Minami, T. J. Am. Ceram. Soc. 80, nanogrooves, that is more important in ports the artificial leg with a maximal force of 1040–1042 (1997). 5. Holdgate, M. W. J. Exp. Biol. 32, 591–617 (1955). inducing this water resistance. only 19.05 dynes (see supplementary infor- 6. Feng, L. et al. Adv. Mater. 14, 1857–1860 (2002). We used a high-sensitivity balance system mation), which is enough to support the 7. Cassie, A. B. D. & Baxter, S. Trans. Farad. Soc. 40, 546–551 (1944). Supplementary information accompanies this communication on to construct force–displacement curves for strider at rest but not to enable it to glide or Nature’s website. a water strider’s legs when pressing on the dart around rapidly on the surface. Competing financial interests: declared none.

ROUGHbrief communicationsSURFACES dissolving lead shavings in vinegar8. Stannic This finding suggests that the force exerted a oxide would have been an acceptable substi- Vessel wall by the strider’s legs could be due to a ‘super- Often think of surface tension as providingtute,with supplies abeing force available per through the hydrophobic’ effect (that is, the contact angle Cornish tin industry. The Romano–British Cantilever with water would be greater than 150°). We unit length – does this explain whych emistswater probably walking believed that they were verified that this was indeed the case by sessile using a new source of cerussa, so confused Leg water-drop measurements, which showed insects have hairy legs? were their encyclopaedias in distinguishing that that the contact angle of the insect’s legs lead from tin. with water was 167.6Ǆ4.4°(Fig.1a,inset). None ofthe surviving Greco–Roman med- The contact angle of water made with the ical corpora suggest that tin had pharmaceu- cuticle wax secreted on a strider’s leg is about Keller’s Theorem shows that this cannottical value9 .Abelthough the organic case tin compounds 105° (ref. 5), which is not enough to account are now well known for their toxicological for its marked water repellence.Knowing that – a hairy surface does not significantlyproperties,inorganic alter the forms seem to be largely microstructures on an object with low surface 10 b 6 inert .Hence, unless SnO2 was added owing energy can enhance its hydrophobicity ,we volume of liquid displaced to some hitherto unrecognized medicinal investigated the physical features of the legs. attribute, we must conclude that its function Scanning electronic micrographs revealed was solely as a pigment.The non-toxic prop- numerous oriented setae on the legs.These are Gao & Jiang (2004) &Jiang Gao Key point: contact line length is increasederties of SnO but2 would also have been desir- needle-shaped, with diameters ranging from able, because by the second century AD the 3 micrometres down to several hundred presence of nearby hairs causes mini-meniscidangers of lead were becomingto recognized9. nanometres (Fig. 1b). Most setae are roughly R. P.Evershed*, R. Berstan*, F. Grew†, 50 Ȗm in length and arranged at an inclined flatten (and hence provide less force)M. S. Copley*, A. J. H. Charmant*, angle of about 20° from the surface of leg. E. Barham†, H. R. Mottram*, G. Brown‡ Many elaborate, nanoscale grooves are evi- *School of Chemistry, University of Bristol, dent on each microseta, and these form a Cantock’s Close, Bristol BS8 1TS, UK c unique hierarchical structure (Fig.1c). e-mail:[email protected] According to Cassie’s law for surface wet- †Museum of London, LondonAlso Wall, allows us to understand floating oftability, such microstructures can be regard- London EC2Y 5HN, UK ed as heterogeneous surfaces composed of ‘inverse Leidenfrost drops’ (water on 7 ‡Pre-Construct Archaeology, Unit 54, Brockley solid and air .The apparent contact angle Ȓl Cross Business Centre, Endwell Road, of the legs is described by cos f cos f , liquid Nitrogen): Ȓlǃ 1 Ȓwǁ 2 London SE4 2PD, UK where f1 is the area fraction of microsetae 1. Durrani, N. Curr.Archaeol. 192, 540–547 (2004). with nanogrooves,f2 is the area fraction ofair 2. Beagrie, N. Britannia 20, 169–191 (1989). on the leg surface and Ȓw is the contact angle 3. Copley, M. S. et al. Proc. Natl Acad.• Sci.While USA 100, levitating, drop is 1524–1529 (2003). of the secreted wax. Using measured values 4. Ralph, J. & Hatfield, R. D. J. Agric. Foodeffectively Chem. 39, non-wetting sphere of Ȓl and Ȓw,we deduce from the equation 1426–1437 (1991). Figure 1 The non-wetting leg of a water strider. a, Typical side that the air fraction between the leg and the 5. Blakeney, A. B., Harris, P. J.,Henry,R.J.& Stone,B.A. view of a maximal-depth dimple (4.38Ǆ0.02 mm) just before the water surface corresponds to f ǃ96.86%. Carbohyd. Res. 113, 291–299 (1983). 2 leg pierces the water surface. Inset, water droplet on a leg; this Available air is trapped in spaces in the 6. PDF-2 Powder Diffraction Database•(InternationalOnce Centre forfrozen, stops levitating and Diffraction Data, Pennsylvania, 1999). makes a contact angle of 167.6Ǆ4.4°. b, c, Scanning electron microsetae and nanogrooves to form a cush- 7. Cert, A., Lanzón, A., Carelli, A. A., Albi, T. & Amelitti, G. microscope images of a leg showing numerous oriented spindly ion at the leg–water interface that prevents Food Chem. 49, 287–293 (1994). roughness causes sinking microsetae (b) and the fine nanoscale grooved structures on a 8. Pliny Natural History (ed. Rackham, H.) 9 (Loeb Classical the legs from being wetted. Library, Cambridge, Massachusetts, 1944). seta (c). Scale bars: b, 20 Ȗm; c, 200 nm. This unique hierarchical micro- and 9. Riddle, J. M. in Dioscorides on Pharmacy and Medicine 153–154 nanostructuring on the leg’s surface there- (Univ. of Texas Press, 1985). water surface (for methods, see supplemen- fore seems to be responsible for its water 10. Toxicological Profile for Tin and Tin Compounds (Agency for Toxic Substances and Disease Registry, US Public Health Service, 2003). tary information). Surprisingly, the leg does resistance and the strong supporting force. Competing financial interests: declared none. not pierce the water surface until a dimple of This clever arrangement allows water strid- 4.38Ǆ0.02 mm depth is formed (Fig. 1a). ers to survive on water even if they are being The maximal supporting force of a single leg bombarded by raindrops,when they bounce Biophysics is 152 dynes (see supplementary informa- to avoid being drowned. Our discovery may tion),or about 15 times the total body weight be helpful in the design of miniature aquatic Water-repellent legs of the insect. The corresponding volume of devices and non-wetting materials. of water striders water ejected is roughly 300 times that of Xuefeng Gao*†, Lei Jiang*‡ the leg itself, indicating that its surface is *Key Laboratory of Organic Solids, Institute of ater striders (Gerris remigis) have strikingly water repellent. Chemistry, and the †Graduate School, Chinese remarkable non-wetting legs that For comparison, we made a hydrophobic Academy of Sciences, Beijing 100080, China Wenable them to stand effortlessly and ‘leg’ from a smooth quartz fibre that was ‡National Center for Nanoscience and move quickly on water, a feature believed to similar in shape and size to a strider’s leg. Its Nanotechnology, Beijing 100080, China be due to a surface-tension effect caused by surface was modified by a self-assembling e-mail: [email protected] 1–3 secreted wax .We show here, however, that monolayer of low-surface-energy hepta- 1. Caponigro, M. A. & Erilsen, C. H. Am.Midland Nat. 95, it is the special hierarchical structure of the decafluorodecyltrimethoxysilane (FAS-17), 268–278 (1976). 2. Dickinson, M. Nature 424, 621–622 (2003). legs, which are covered by large numbers of which makes a contact angle of 109° with a 3. Hu, D. L., Chan, B. & Bush, J. W. M. Nature 424, 663–666 (2003). oriented tiny hairs (microsetae) with fine water droplet on a flat surface4.Water sup- 4. Tadanaga, K., Katata, N. & Minami, T. J. Am. Ceram. Soc. 80, nanogrooves, that is more important in ports the artificial leg with a maximal force of 1040–1042 (1997). 5. Holdgate, M. W. J. Exp. Biol. 32, 591–617 (1955). inducing this water resistance. only 19.05 dynes (see supplementary infor- 6. Feng, L. et al. Adv. Mater. 14, 1857–1860 (2002). We used a high-sensitivity balance system mation), which is enough to support the 7. Cassie, A. B. D. & Baxter, S. Trans. Farad. Soc. 40, 546–551 (1944). Supplementary information accompanies this communication on to construct force–displacement curves for strider at rest but not to enable it to glide or Nature’s website. a water strider’s legs when pressing on the dart around rapidly on the surface. Competing financial interests: declared none.

The static meniscus around a small vertical cylinder can be understood using Keller’s Theorem too

For small cylinders, ✏ = R/ ` c 1 , can perform matched asymptotic analysis of ⌧the Laplace–Young equation to ﬁnd meniscus height ⇣ 0 (see e.g. James 1974, Lo 1983). 1/2 ` = c ⇢g ✓ ◆ A SIMPLE APPLICATION

The static meniscus around a small vertical cylinder can be understood using Keller’s Theorem too

For small cylinders, ✏ = R/ ` c 1 , can perform matched asymptotic analysis of ⌧the Laplace–Young equation to ﬁnd meniscus height ⇣ 0 (see e.g. James 1974, Lo 1983). 1/2 Far away from the cylinder, ﬁnd the linearized ` = c ⇢g Laplace-Young equation ✓ ◆

`2 d d⇣ ⇣ = c r r dr dr ✓ ◆ A SIMPLE APPLICATION

The static meniscus around a small vertical cylinder can be understood using Keller’s Theorem too

`2 d d⇣ ⇣ = c r r dr dr = ⇣ = AK0(r/`c) ✓ ◆ ) A SIMPLE APPLICATION

The static meniscus around a small vertical cylinder can be understood using Keller’s Theorem too

`2 d d⇣ ⇣ = c r r dr dr = ⇣ = AK0(r/`c) ✓ ◆ ) Balancing weight of liquid displaced in outer meniscus with vertical force from surface tension ﬁnd that ⇣ 0 ✏ log(1/✏) `c ⇠ A SECOND PROBLEM

Keller & Miksis (KM83) consider inviscid retraction of ﬂuid wedge due to surface tension A SECOND PROBLEM

Keller & Miksis (KM83) consider inviscid retraction of ﬂuid wedge due to surface tension

KM83 has been very inﬂuential in drop breakup (though only studied 2D) ANOTHER RELATED PROBLEM

Analysis performed by KM83 also describes the early stages of meniscus onset

Keller & Miksis calculated similarity solutions for the interface shape with different contact angles

Clanet & Quéré (2002) studied ‘onset’ of menisci on cylinders of various radii

Key question is: How does meniscus height evolve with time? Christophe(Paris) Clanet

R = 0.225 mm duration ≃ 1 s ANOTHER RELATED PROBLEM

Analysis performed by KM83 also describes the early stages of meniscus onset air Keller & Miksis calculated similarity solutions for the interface shape with different contact angles water

Clanet & Quéré (2002) studied ‘onset’ of menisci on cylinders of various radii

Key question is: How does meniscus height evolve with time? Christophe(Paris) Clanet

R = 0.225 mm duration ≃ 1 s ANOTHER RELATED PROBLEM

Analysis performed by KM83 also describes the early stages of meniscus onset air Keller & Miksis calculated similarity solutions for the interface shape with different contact angles water

Clanet & Quéré (2002) studied ‘onset’ of menisci on cylinders of various radii

Key question is: How does meniscus height evolve with time? Christophe(Paris) Clanet

R = 0.225 mm duration ≃ 1 s ANOTHER RELATED PROBLEM

Analysis performed by KM83 also describes the early stages of meniscus onset air Keller & Miksis calculated similarity solutions for the interface shape with different contact angles water

Clanet & Quéré (2002) studied ‘onset’ of menisci on cylinders of various radii

Key question is: How does meniscus height evolve with time? Christophe(Paris) Clanet KM83 analysis is relevant to high Reynolds R = 0.225 mm (or Kapitsa) number and large cylinders duration ≃ 1 s Inertial meniscus rise

(a) (b) PLANES/LARGE(c) CYLINDERS ‘Large’ cylinders are effectively planar –R consider how meniscus rises up a wall d Newton II: (Mh˙ ) dt m ⇠ Geometry: M ⇢ h2 ⇠ l m t2 1/3 = h ) m ⇠ ⇢ ✓ l ◆

FIGURE 1. (a) Images from experiments reported by Clanet & Quéré (2002). Here, a ﬁbre of radius R 0.225 mm touches the interface between air and hexane at t 0 and a meniscus rises around= it. (Images courtesy of Christophe Clanet.) Sketches showing= the essential ingredients for determining the scaling of meniscus height hm with time t are shown for the cases of (b) rise next to a vertical wall and (c) rise outside a circular cylinder. The scaling laws for hm(t) are found by balancing the force due to the interfacial tension, lv, with the rate of change of momentum of the meniscus. For details, see text.

h ( R/⇢ )1/4t1/2. (2.2) m ⇠ lv ` While the form of the t1/2 scaling in (2.2) may be familiar from the problem of the surface-tension-driven coalescence of two liquid drops of radius R (see Duchemin, Eggers & Josserand 2003, for example), there is no clear analogy between the two problems. Indeed, in wedge retraction problems, the t2/3 scaling of (2.1) is almost generic and so the scaling (2.2) represents a novel twist. However, it is vital to note that a key step in the derivation of (2.2) is the simplifying assumption (made only 768 R2-3 Inertial meniscus rise

(a) (b) PLANES/LARGE(c) CYLINDERS ‘Large’ cylinders are effectively planar –R consider how meniscus rises up a wall d Newton II: (Mh˙ ) dt m ⇠ Geometry: M ⇢ h2 ⇠ l m t2 1/3 = h ) m ⇠ ⇢ ✓ l ◆ This is the classic “Keller & Miksis scaling” FIGURE 1. (a) Images from experiments reported by(though Clanet their & Quéré calculation (2002 also). Here, provides a the pre-factor) ﬁbre of radius R 0.225 mm touches the interface between air and hexane at t 0 and a meniscus rises around= it. (Images courtesy of Christophe Clanet.) Sketches showing= the essential ingredients for determining the scaling of meniscus height hm with time t are shown for the cases of (b) rise next to a vertical wall and (c) rise outside a circular cylinder. The scaling laws for hm(t) are found by balancing the force due to the interfacial tension, lv, with the rate of change of momentum of the meniscus. For details, see text.

(a) (b) PLANES/LARGE(c) CYLINDERS ‘Large’ cylinders are effectively planar –R consider how meniscus rises up a wall d Newton II: (Mh˙ ) dt m ⇠ Geometry: M ⇢ h2 ⇠ l m t2 1/3 = h ) m ⇠ ⇢ ✓ l ◆ This is the classic “Keller & Miksis scaling” FIGURE 1. (a) Images from experiments reported by(though Clanet their & Quéré calculation (2002 also). Here, provides a the pre-factor) ﬁbre of radius R 0.225 mm touches the interface between air and hexane at t 0 and = = a meniscus rises around it. (Images courtesy of Christophe Clanet.) Sketches1 showing00 the 3 essential ingredients for determining the scaling of meniscus height hm with time t are shown for the cases of (b) rise next to a vertical wall and (c) rise outside a2 circular cylinder. The scaling laws for hm(t) are found by balancing the force due to the10-1 interfacial R = 8mm tension, lv, with the rate of change of momentum of the meniscus. For details, see text. R = 4mm

10-2 &Quéré (2002) Clanet -2 -1 0 1 2 2.1. Scalings for the dynamics of inertial rise 10 10 10 10 10 Motivated by their experimental observations, CQ assumed that the contact angle of the liquid on the wetted body, ✓ in figure 1(b,c), is approximately constant during the motion. Using this assumption, they presented scaling analyses for the inertia- dominated dynamics in two particular geometries: (i) a meniscus next to a planar wall (illustrated schematically in figure 1b) and (ii) a meniscus outside a vertical cylinder (illustrated schematically in figure 1c). It is the latter case that is of primary interest here, but we consider the former case first for pedagogical reasons. Following CQ, we consider the force balance on a two-dimensional meniscus of unit width (into the page): the vertical force driving the capillary rise is that from surface tension, lv cos ✓, while the inertia of the meniscus resists rise. Since the mass 2 of the meniscus is not constant (m ⇢`hm by the assumption of a constant contact ⇠ ✓ ( )/ angle), Newton’s second law implies that lv cos d mh˙m dt. In scaling terms this immediately yields ⇠ h ( /⇢ )1/3t2/3. (2.1) m ⇠ lv ` This is the well known capillary–inertia scaling first found by Keller & Miksis (1983). In the case of an axisymmetric meniscus rising outside a circular cylinder, CQ follow an argument along the same lines. Because of the finite radius of the cylinder, however, the force from surface tension is now lvR cos ✓, while the mass of the meniscus is m ⇢`hm hm(R hm). Assuming that hm R and making use of Newton’s second⇠ law again,⇥ we+ find

(a) (b) PLANES/LARGE(c) CYLINDERS ‘Large’ cylinders are effectively planar –R consider how meniscus rises up a wall d Newton II: (Mh˙ ) dt m ⇠ Geometry: M ⇢ h2 ⇠ l m t2 1/3 = h ) m ⇠ ⇢ ✓ l ◆ This is the classic “Keller & Miksis scaling” FIGURE 1. (a) Images from experiments reported by(though Clanet their & Quéré calculation (2002 also). Here, provides a the pre-factor) ﬁbre of radius R 0.225 mm touches the interface between air and hexane at t 0 and = = a meniscus rises around it. (ImagesExperimentally, courtesy of observe Christophe KM scaling Clanet.) early Sketches 1 showing00 the 3 essential ingredients for determiningon (before the scaling meniscus of height meniscus saturates) height hm with time t are shown for the cases of (b) rise next to a vertical wall and (c) rise outside a2 circular cylinder. The scaling laws for h (t) are found by balancing the force due to the10-1 interfacial mNo dependence on cylinder size for R = 8mm tension, , with the rate of change of momentum of the meniscus. For details, see text. lv large enough cylinders R = 4mm

2.1. Scalings for the dynamics of inertial rise Motivated by their experimental observations, CQ assumed that the contact angle of the liquid on the wetted body, ✓ in figure 1(b,c), is approximately constant during the motion. Using this assumption, they presented scaling analyses for the inertia- dominated dynamics in two particular geometries: (i) a meniscus next to a planar wall (illustrated schematically in figure 1b) and (ii) a meniscus outside a vertical cylinder (illustrated schematically in figure 1c). It is the latter case that is of primary interest here, but we consider the former case first for pedagogical reasons. Following CQ, we consider the force balance on a two-dimensional meniscus of unit width (into the page): the vertical force driving the capillary rise is that from surface tension, lv cos ✓, while the inertia of the meniscus resists rise. Since the mass 2 of the meniscus is not constant (m ⇢`hm by the assumption of a constant contact ⇠ ✓ ( )/ angle), Newton’s second law implies that lv cos d mh˙m dt. In scaling terms this immediately yields ⇠ h ( /⇢ )1/3t2/3. (2.1) m ⇠ lv ` This is the well known capillary–inertia scaling first found by Keller & Miksis (1983). In the case of an axisymmetric meniscus rising outside a circular cylinder, CQ follow an argument along the same lines. Because of the finite radius of the cylinder, however, the force from surface tension is now lvR cos ✓, while the mass of the meniscus is m ⇢`hm hm(R hm). Assuming that hm R and making use of Newton’s second⇠ law again,⇥ we+ find h ( R/⇢ )1/4t1/2. (2.2) m ⇠ lv ` While the form of the t1/2 scaling in (2.2) may be familiar from the problem of the surface-tension-driven coalescence of two liquid drops of radius R (see Duchemin, Eggers & Josserand 2003, for example), there is no clear analogy between the two problems. Indeed, in wedge retraction problems, the t2/3 scaling of (2.1) is almost generic and so the scaling (2.2) represents a novel twist. However, it is vital to note that a key step in the derivation of (2.2) is the simplifying assumption (made only 768 R2-3 D. O’Kiely, J. P. Whiteley, J. M. Oliver and D. Vella The equilibrium shape of a meniscus is governed by the Laplace–Young equation, which arises from a balance between hydrostatic pressure and the normal pressure jump caused by the curvature of the interface (Finn 1986; de Gennes, Brochard-Wyart & Quéré 2003). In two dimensions, this equation can be solved analytically to describe the meniscus formed when a vertical plane pierces the interface (Landau & Lifshitz 1987). In other geometries, however, the problem becomes significantly more involved. For example, even the meniscus around a vertical circular cylinder is not known analytically, although asymptotic results in the limit of small cylinder radius are known (James 1974; Lo 1983). Inertial meniscus rise While the question of what shape a static meniscus takes has been well studied, the questionSMALL of how an CYLINDERS initially flat interface deforms to form such a meniscus has been neglected in comparison. An exception to this is the case of capillary rise (a) (b) Clanet(c) & Quéré repeat the same scaling argument, but with a finite radius cylinder: within a cylindrical tube, which was studied first by Washburn (1921). In the simplest cases, theR meniscus height grows diffusively,d h t1/2, at early times, but variants of Newton II: (Mh˙ m)m R Washburn’s scaling have been obtaineddt in a range⇠⇠ of geometries (see Reyssat et al. 2008; Ponomarenko, Quéré & Clanet 2011, for example). However, these studies have Geometry: M ⇢ h3 largely been confined to the case of slow⇠ viscousl m flows and so are not applicable to very early times (where they predict that the2 1 speed/4 becomes arbitrarily large) or to RtD. O’Kiely, J. P. Whiteley, J. M. Oliver and D. Vella inviscid liquids; in both of= theseh casesm inertial effects become important (Quéré 1997). ) ⇠ ⇢ Clanet & Quéré (2002) (CQ) reported✓ a comprehensivel ◆ study of the dynamic growth (a)(b) 100 of the meniscus outside a vertical cylinder.0 They performed a series of experiments 11000 in which a vertical cylinder of radius Rt increasingwas gently touched onto the interface4 4 between air and a liquid with surface tension lv, density ⇢` and kinematic1 viscosity ⌫ 1 (see figure 1a). The motion by which the initially flat interface transformed–1 to an FIGURE 1. (a) Images from experiments reported by Clanet & Quéré (2002). Here, a -1 10 equilibrium meniscus was characterized1010 –1 using two dimensionless parameters: the size . Clanet & Quéré’s experiments seem to fibre of radius R 0 225 mm touches the interfaceconfirm betweenof this the for air different cylinder and hexane radius cylinders at t 0 and a meniscus rises around= it. (Images courtesy of Christophe Clanet.) Sketches showing= the 1 ✏ R/`c, (1.1) essential ingredients for determining the scaling of meniscus height hm with time t are = &Quéré (2002) Clanet 3 1/2 10-2 –2 -4 -2 0 2 –2 where `c (lv/⇢`g) is the capillary10 10 length,10 and the10 Kapitsa10 number 10 shown for the cases of (b) rise next to a vertical wall and (c)= rise outside a circular 10–2 10–1 100 10–4 10–2 100 102 cylinder. The scaling laws for h (t) are found by balancing the force due to the interfacial 3 4 1/4 m Ka lv/(⇢` ⌫ g) , (1.2) tension, lv, with the rate of change of momentum of the meniscus. For details, see text. =[ ] (with g the acceleration dueFIGURE to gravity)4. The which utility is of the the characteristicfamily of similarity Reynolds solutions number, for understanding experimental Re, of capillary–gravity waves.results CQ (captured performed from figure careful 16 of experiments CQ). (a) Experimental in each of results the (points) and the results 5 four asymptotic regimes thatof result the full from numerical this two-dimensionalsimulations with ✏ parameter10 , ✓ space13 (solid and curve) may be collapsed 2.1. Scalings for the dynamics of inertial rise onto a master curve by using the similarity= rescaling= (5.1). (b) A plot of the same data determined scaling laws for the growth of the meniscus height, hm, with time t after Motivated by their experimental observations, CQ assumedthe cylinder that the touches contact the angle interface.points of as in (a) but in the rescaling suggested by (2.2) shows good initial collapse that breaks down later on. the liquid on the wetted body, ✓ in figure 1(b,c), is approximatelyIn the limit Ka constant1 (equivalent during to high Re) and large cylinders, ✏ 1, the problem the motion. Using this assumption, they presented scalingof the growthanalyses of for a meniscus the inertia- is mathematically equivalent to the retraction of a two- dominated dynamics in two particular geometries: (i) adimensional meniscus next wedge to a of planar fluidto with wall enable a fixed us to contact start with angle.R This problemwe use was the substitution first studiedS 1/R starting with by Keller & Miksis (1983)S but is0 important and solving in the the coalescence resulting= 1 problem of drops using and, first-order as such, finite= differences on a (illustrated schematically in figure 1b) and (ii) a meniscus outside a vertical cylinder= (illustrated schematically in figure 1c). It is the latter casehas continued that is of to primary generate interest interestrectangular (see Thompson grid in the & spatial Billingham variables2012⇢ and, for⇣. a We review then of step forward to consider > 0, using second-order differences in .) The solution of this linearized problem here, but we consider the former case first for pedagogicalthe literature). reasons. Previous studiesS of axisymmetric geometries have focusedS on a fixed wedge angle in the far fieldcan (Leppinen be rescaled & Lister to give2003 the results; Sierou for & any Lister contact2004 angle). In close this to ✓ p/2. The results = Following CQ, we consider the force balance on apaper, two-dimensional we focus on meniscus the caseof of thisof inviscid calculation motion, agreeKa well1, with and the small results cylinders, of our✏ full numerical1, simulations for unit width (into the page): the vertical force driving the capillary rise is that✓ from85 (see figure 3c). Again, we do not see any indication⌧ that a scaling of the with a fixed contact angle ✓ at= the cylinder wall. To the best of our knowledge, this surface tension, lv cos ✓, while the inertia of the meniscusdoes resists not have rise. a direct Since analogue the massform in (5.8 the) is literature present. on coalescence and breakup of drops. 2 of the meniscus is not constant (m ⇢`hm by the assumption of a constant contact angle), Newton’s second law implies⇠ that cos ✓ d(2.mhPrevious)/dt. In results scaling terms6. thisConclusions lv ⇠ ˙m immediately yields We have presented a fluid-mechanical model for the inertial growth of a meniscus 1/3 2/3 Before developing our own model of the inertial capillary rise around a small hm (lv/⇢`) t . external(2.1) to a circular cylinder. Our potential flow model with a constant contact ⇠ cylinder, we summarize existingangle results is able to for quantitatively this problem. reproduce We begin previously with the published scaling experimental data (CQ) This is the well known capillary–inertia scaling first foundlaws by given Keller by CQ & Miksis for the (1983 evolutionfor a). range of of the cylinder meniscus radii. height We havehm(t) assumed. that inertia dominates viscosity and, In the case of an axisymmetric meniscus rising outside768 R2-2 a circular cylinder,further, CQ that the dynamic contact angle is equal to the static contact angle throughout follow an argument along the same lines. Because of the finite radius of the cylinder,the motion. In reality these assumptions must break down at the very earliest times: 5/3 according to the similarity solution the viscous dissipation uxx/Re t /Re will, in however, the force from surface tension is now lvR cos ✓, while the mass of the 4/3 3 8⇠ fact, dominate the inertial terms ut, uux t for t . Re 10 (corresponding to meniscus is m ⇢`hm hm(R hm). Assuming that hm R and making use of 10 ⇠ ⇠ t . 10 s in dimensional terms). Similarly, the undeformed interface must first react ⇠ ⇥ + / Newton’s second law again, we find to the touch down of the cylinder. After these earliest transients, but with t . ✏3 2, the meniscus height grows like t2/3 – the radius of the cylinder does not play a role 1/4 1/2 hm (lvR/⇢`) t . and(2.2) the problem reduces to the two-dimensional one studied by Keller & Miksis ⇠ (1983). At later times the evolution of the meniscus height with time is governed While the form of the t1/2 scaling in (2.2) may be familiar from the problemby of a family of similarity solutions parametrized by the rescaled cylinder radius 2/3 2 1/3 the surface-tension-driven coalescence of two liquid drops of radius R (see Duchemin,R ✏/t R ⇢`/(lvt ) , provided that t 1 (so that gravity may be neglected). The= introduction= of the parameter R allows us⌧ to collapse previous experimental data Eggers & Josserand 2003, for example), there is no clear analogy between the two ⇥ ⇤ problems. Indeed, in wedge retraction problems, the t2/3 scaling of (2.1) is almostfor the meniscus height with different cylinder radii onto a master curve. Our analysis suggests that the scaling law (2.2) proposed by CQ rests on an implicit generic and so the scaling (2.2) represents a novel twist. However, it is vital to note assumption (that hm R) that is not consistent with the behaviour of menisci growing that a key step in the derivation of (2.2) is the simplifying assumption (madein only the presence of gravity. (We emphasize that this inconsistency occurs only in the 768 768R2-3R2-10 D. O’Kiely, J. P. Whiteley, J. M. Oliver and D. Vella The equilibrium shape of a meniscus is governed by the Laplace–Young equation, which arises from a balance between hydrostatic pressure and the normal pressure jump caused by the curvature of the interface (Finn 1986; de Gennes, Brochard-Wyart & Quéré 2003). In two dimensions, this equation can be solved analytically to describe the meniscus formed when a vertical plane pierces the interface (Landau & Lifshitz 1987). In other geometries, however, the problem becomes significantly more involved. For example, even the meniscus around a vertical circular cylinder is not known analytically, although asymptotic results in the limit of small cylinder radius are known (James 1974; Lo 1983). Inertial meniscus rise While the question of what shape a static meniscus takes has been well studied, the questionSMALL of how an CYLINDERS initially flat interface deforms to form such a meniscus has been neglected in comparison. An exception to this is the case of capillary rise (a) (b) Clanet(c) & Quéré repeat the same scaling argument, but with a finite radius cylinder: within a cylindrical tube, which was studied first by Washburn (1921). In the simplest cases, theR meniscus height grows diffusively,d h t1/2, at early times, but variants of Newton II: (Mh˙ m)m R Washburn’s scaling have been obtaineddt in a range⇠⇠ of geometries (see Reyssat et al. 2008; Ponomarenko, Quéré & Clanet 2011, for example). However, these studies have Geometry: M ⇢ h3 largely been confined to the case of slow⇠ viscousl m flows and so are not applicable to very early times (where they predict that the2 1 speed/4 becomes arbitrarily large) or to RtD. O’Kiely, J. P. Whiteley, J. M. Oliver and D. Vella inviscid liquids; in both of= theseh casesm inertial effects become important (Quéré 1997). ) ⇠ ⇢ Clanet & Quéré (2002) (CQ) reported✓ a comprehensivel ◆ study of the dynamic growth (a)(b) 100 of the meniscus outside a vertical cylinder.0 They performed a series of experiments 11000 in which a vertical cylinder of radius Rt increasingwas gently touched onto the interface4 4 between air and a liquid with surface tension lv, density ⇢` and kinematic1 viscosity ⌫ 1 (see figure 1a). The motion by which the initially flat interface transformed–1 to an FIGURE 1. (a) Images from experiments reported by Clanet & Quéré (2002). Here, a -1 10 equilibrium meniscus was characterized1010 –1 using two dimensionless parameters: the size . Clanet & Quéré’s experiments seem to fibre of radius R 0 225 mm touches the interfaceconfirm betweenof this the for air different cylinder and hexane radius cylinders at t 0 and a meniscus rises around= it. (Images courtesy of Christophe Clanet.) Sketches showing= the 1 ✏ R/`c, (1.1) essential ingredients for determining the scaling of meniscus height hm with time t are = &Quéré (2002) Clanet 3 1/2 10-2 –2 -4 -2 0 2 –2 See diffusivewhere growth,`c rather(lv/ ⇢than`g) t2/3 ofis KM83 the capillary10 10 length,10 and the10 Kapitsa10 number 10 shown for the cases of (b) rise next to a vertical wall and (c)= rise outside a circular 10–2 10–1 100 10–4 10–2 100 102 cylinder. The scaling laws for h (t) are found by balancing the force due to the interfacial 3 4 1/4 m Ka lv/(⇢` ⌫ g) , (1.2) tension, lv, with the rate of change of momentum of the meniscus. For details, see text. =[ ] (with g the acceleration dueFIGURE to gravity)4. The which utility is of the the characteristicfamily of similarity Reynolds solutions number, for understanding experimental Re, of capillary–gravity waves.results CQ (captured performed from figure careful 16 of experiments CQ). (a) Experimental in each of results the (points) and the results 5 four asymptotic regimes thatof result the full from numerical this two-dimensionalsimulations with ✏ parameter10 , ✓ space13 (solid and curve) may be collapsed 2.1. Scalings for the dynamics of inertial rise onto a master curve by using the similarity= rescaling= (5.1). (b) A plot of the same data determined scaling laws for the growth of the meniscus height, hm, with time t after Motivated by their experimental observations, CQ assumedthe cylinder that the touches contact the angle interface.points of as in (a) but in the rescaling suggested by (2.2) shows good initial collapse that breaks down later on. the liquid on the wetted body, ✓ in figure 1(b,c), is approximatelyIn the limit Ka constant1 (equivalent during to high Re) and large cylinders, ✏ 1, the problem the motion. Using this assumption, they presented scalingof the growthanalyses of for a meniscus the inertia- is mathematically equivalent to the retraction of a two- dominated dynamics in two particular geometries: (i) adimensional meniscus next wedge to a of planar fluidto with wall enable a fixed us to contact start with angle.R This problemwe use was the substitution first studiedS 1/R starting with by Keller & Miksis (1983)S but is0 important and solving in the the coalescence resulting= 1 problem of drops using and, first-order as such, finite= differences on a (illustrated schematically in figure 1b) and (ii) a meniscus outside a vertical cylinder= (illustrated schematically in figure 1c). It is the latter casehas continued that is of to primary generate interest interestrectangular (see Thompson grid in the & spatial Billingham variables2012⇢ and, for⇣. a We review then of step forward to consider > 0, using second-order differences in .) The solution of this linearized problem here, but we consider the former case first for pedagogicalthe literature). reasons. Previous studiesS of axisymmetric geometries have focusedS on a fixed wedge angle in the far fieldcan (Leppinen be rescaled & Lister to give2003 the results; Sierou for & any Lister contact2004 angle). In close this to ✓ p/2. The results = Following CQ, we consider the force balance on apaper, two-dimensional we focus on meniscus the caseof of thisof inviscid calculation motion, agreeKa well1, with and the small results cylinders, of our✏ full numerical1, simulations for unit width (into the page): the vertical force driving the capillary rise is that✓ from85 (see figure 3c). Again, we do not see any indication⌧ that a scaling of the with a fixed contact angle ✓ at= the cylinder wall. To the best of our knowledge, this surface tension, lv cos ✓, while the inertia of the meniscusdoes resists not have rise. a direct Since analogue the massform in (5.8 the) is literature present. on coalescence and breakup of drops. 2 of the meniscus is not constant (m ⇢`hm by the assumption of a constant contact angle), Newton’s second law implies⇠ that cos ✓ d(2.mhPrevious)/dt. In results scaling terms6. thisConclusions lv ⇠ ˙m immediately yields We have presented a fluid-mechanical model for the inertial growth of a meniscus 1/3 2/3 Before developing our own model of the inertial capillary rise around a small hm (lv/⇢`) t . external(2.1) to a circular cylinder. Our potential flow model with a constant contact ⇠ cylinder, we summarize existingangle results is able to for quantitatively this problem. reproduce We begin previously with the published scaling experimental data (CQ) This is the well known capillary–inertia scaling first foundlaws by given Keller by CQ & Miksis for the (1983 evolutionfor a). range of of the cylinder meniscus radii. height We havehm(t) assumed. that inertia dominates viscosity and, In the case of an axisymmetric meniscus rising outside768 R2-2 a circular cylinder,further, CQ that the dynamic contact angle is equal to the static contact angle throughout follow an argument along the same lines. Because of the finite radius of the cylinder,the motion. In reality these assumptions must break down at the very earliest times: 5/3 according to the similarity solution the viscous dissipation uxx/Re t /Re will, in however, the force from surface tension is now lvR cos ✓, while the mass of the 4/3 3 8⇠ fact, dominate the inertial terms ut, uux t for t . Re 10 (corresponding to meniscus is m ⇢`hm hm(R hm). Assuming that hm R and making use of 10 ⇠ ⇠ t . 10 s in dimensional terms). Similarly, the undeformed interface must first react ⇠ ⇥ + / Newton’s second law again, we find to the touch down of the cylinder. After these earliest transients, but with t . ✏3 2, the meniscus height grows like t2/3 – the radius of the cylinder does not play a role 1/4 1/2 hm (lvR/⇢`) t . and(2.2) the problem reduces to the two-dimensional one studied by Keller & Miksis ⇠ (1983). At later times the evolution of the meniscus height with time is governed While the form of the t1/2 scaling in (2.2) may be familiar from the problemby of a family of similarity solutions parametrized by the rescaled cylinder radius 2/3 2 1/3 the surface-tension-driven coalescence of two liquid drops of radius R (see Duchemin,R ✏/t R ⇢`/(lvt ) , provided that t 1 (so that gravity may be neglected). The= introduction= of the parameter R allows us⌧ to collapse previous experimental data Eggers & Josserand 2003, for example), there is no clear analogy between the two ⇥ ⇤ problems. Indeed, in wedge retraction problems, the t2/3 scaling of (2.1) is almostfor the meniscus height with different cylinder radii onto a master curve. Our analysis suggests that the scaling law (2.2) proposed by CQ rests on an implicit generic and so the scaling (2.2) represents a novel twist. However, it is vital to note assumption (that hm R) that is not consistent with the behaviour of menisci growing that a key step in the derivation of (2.2) is the simplifying assumption (madein only the presence of gravity. (We emphasize that this inconsistency occurs only in the 768 768R2-3R2-10 nrilmnsu rise Inertial meniscus meniscusInertial rise rise meniscus InertialInertial meniscus rise

(a) ) c ( (b) (a) ) ) c b ( ( (c) (b) ) ) a ( b ( (c) ) a ( R R R R A WARNING Estimation of mass of meniscus region implicitly assumes that h R m

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(2.2) (2.2) . 2 / 1 t 4 / 1 ) h⇢ / R ( ( R/⇢. h 2 )/ 11 /t 44 t/ 11 /) 2.⇢ / R h ( (h R/⇢ )1/4t1/2. (2.2) (2.2) ` m ⇠v l lv ⇠ m ` ` v l m ⇠⇠ m lv ` a efmla rmtepolmof While problem the the from of form of familiar the problem be t the While1/2 may scaling) from the2.2 ( form in in familiar (2.2 of be ) thescaling may2 may / 1 tt ) 1 be/2 2.2 scaling the ( familiar in of in form from (scaling 2.2 the 2 / )1 thet may problem the While be of familiar of form the fromWhile the problem of the Duchemin, surface-tension-driven(see R radius of Duchemin, the drops coalescence(see surface-tension-drivenR liquid radius two of of of two drops liquid coalescence liquid coalescence drops two of radius of twoR (see liquid coalescence surface-tension-driven Duchemin, dropsthe of radius R surface-tension-driven (seethe Duchemin, o xml) hr sn la nlg ewe h two the Eggers & between Josserand two analogy the 2003 clear Eggers, no for between is example), & there Josserand analogy there clear 2003 example), no is is for , no, for clear there example),2003 analogy example), Josserand there between for & , is no2003 Eggers the clear two analogy Josserand & betweenEggers the two salmost problems. is ) 2.1 ( of Indeed, almost scaling in3 is / 2 wedget ) problems.2.1 ( the of retraction Indeed, problems, scaling 3 problems,/ 2 t in the wedge retraction the retractiont wedge 2/ problems, 3 in scaling problems, Indeed, of retraction (2.1) wedge the isproblems. in almostt2/3 scaling Indeed, of (2.1problems. ) is almost ersnsanvltit oee,i svtlt note to generic vital is and it so note the to However, scaling vital generic twist. is (2.2 it ) novel and represents a so However, the a scaling represents twist. novel) 2.2 ( novel twist. (2.2 a ) represents scaling However, the represents so ) ita2.2 and ( novel is vital twist. scaling generic to the note However, so and it isgeneric vital to note stesmlfigasmto md only that (made a key step assumption only in the (made derivationthat simplifying a the key is of assumption step) (2.22.2 ( in)is of the the simplifying derivation simplifying the is derivation ) the 2.2 of ( in (assumption of 2.2) step isthe key a derivation (made simplifying the that in only step assumption key a that (made only R2-3 768 R2-3 768 768 R2-3 768 R2-3 nrilmnsu rise Inertial meniscus meniscusInertial rise rise meniscus InertialInertial meniscus rise

(a) ) c ( (b) (a) ) ) c b ( ( (c) (b) ) ) a ( b ( (c) ) a ( R R R R A WARNING Estimation of mass of meniscus region implicitly assumes that h R m

3 M h 2 m M Rhm = ⇠ ⇠= ) ) h t1/2 h t2/3 m ⇠ m ⇠ .Hr,a Here, FIGURE). 2002 ( 1. ( a a Quéré ) Images & Here, ). from Clanet F2002 IGURE ( But by experimentsearlier Quéré application1. reported & (a) of Images reportedKeller’s Clanet Theorem by from experiments by gave Clanetexperiments from final reported meniscus & Images height:) Quéréa ( reported experiments 1. (2002 from by).IGURE F Here,Clanet Images ) a a ( &1. Quéré (IGURE 2002F ). Here, a ⇣ R log(` /R) and 0 fibret at of radius hexane and R and 0 air 0.t 225 at fibre mm between touches of hexane radius and interface the air R the interface0.2250 between ⇠ touches mm between mm c touches interface 225 . 0 air the theandR interface hexane touches radius mm of at between225 t. fibre 0 0 andR air and radius hexane of fibre at t 0 and eicsrssaon t Iae oreyo hitpeCae. kthssoigthe = a meniscus showing rises Sketches the around= = Clanet.) showing it.a (Images meniscus Christophe Sketches of courtesy rises Clanet.) around= courtesy of Christophe it. Christophe (Images of it. Clanet.) courtesy= around courtesy Sketches rises of (Images Christophe it. showing meniscus a = = around Clanet.) the rises Sketches meniscus a showing= the are t essential time with ingredientsm h are t height for time determiningessentialwith meniscus m h of ingredients the height scaling scaling the for meniscus of determining of meniscus determining scaling for height the scalinghm ingredients with of determining meniscus time for essential t are height ingredients hm withessential time t are ieotieacircular shown a for outside the rise cases) c ( circular a of and (shownb) wall rise outside for next rise vertical ) thec ( a to cases to a and vertical next of wall (b rise ) wall) riseb ( vertical and a of next to (c) to cases rise next a the vertical outside rise for ) b ( wall of ashown circular and cases (c the ) rise for outsideshown a circular cylinder. interfacial the to The due scaling force laws interfacial the the cylinder. for to hm balancing due (t) by Theare force scaling found found the are by) lawst ( balancingm h balancing for for by hm(t the laws ) found are forceare found) scaling t ( duem h by The to for balancing the interfacial laws cylinder. the scaling force The due tocylinder. the interfacial ihtert fcag fmmnu ftemnsu.Frdtis e text. see tension, details, lv, For with text. the see rate meniscus. tension, the of details, of change For lv, of with momentum momentum meniscus. the of the rate of change of of of the change meniscus. rate momentum of the of momentum with For, v l change details, of of rate thetension, see the meniscus. text. with , v l For details,tension, see text.

(a) ) c ( (b) (a) ) ) c b ( ( (c) (b) ) ) a ( b ( (c) ) a ( R R R R A WARNING Estimation of mass of meniscus region implicitly assumes that h R m

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(a) ) c ( (b) (a) ) ) c b ( ( (c) (b) ) ) a ( b ( (c) ) a ( R R R R A WARNING Estimation of mass of meniscus region implicitly assumes that h R m

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(2.2) (2.2) . 2 / 1 t 4 / 1 ) h⇢ / R ( ( R/⇢. h 2 )/ 11 /t 44 t/ 11 /) 2.⇢ / R h ( (h R/⇢ )1/4t1/2. (2.2) (2.2) ` m ⇠v l lv ⇠ m ` ` v l m ⇠⇠ m lv ` a efmla rmtepolmof While problem the the from of form of familiar the problem be t the While1/2 may scaling) from the2.2 ( form in in familiar (2.2 of be ) thescaling may2 may / 1 tt ) 1 be/2 2.2 scaling the ( familiar in of in form from (scaling 2.2 the 2 / )1 thet may problem the While be of familiar of form the fromWhile the problem of the Duchemin, surface-tension-driven(see R radius of Duchemin, the drops coalescence(see surface-tension-drivenR liquid radius two of of of two drops liquid coalescence liquid coalescence drops two of radius of twoR (see liquid coalescence surface-tension-driven Duchemin, dropsthe of radius R surface-tension-driven (seethe Duchemin, o xml) hr sn la nlg ewe h two the Eggers & between Josserand two analogy the 2003 clear Eggers, no for between is example), & there Josserand analogy there clear 2003 example), no is is for , no, for clear there example),2003 analogy example), Josserand there between for & , is no2003 Eggers the clear two analogy Josserand & betweenEggers the two salmost problems. is ) 2.1 ( of Indeed, almost scaling in3 is / 2 wedget ) problems.2.1 ( the of retraction Indeed, problems, scaling 3 problems,/ 2 t in the wedge retraction the retractiont wedge 2/ problems, 3 in scaling problems, Indeed, of retraction (2.1) wedge the isproblems. in almostt2/3 scaling Indeed, of (2.1problems. ) is almost ersnsanvltit oee,i svtlt note to generic vital is and it so note the to However, scaling vital generic twist. is (2.2 it ) novel and represents a so However, the a scaling represents twist. novel) 2.2 ( novel twist. (2.2 a ) represents scaling However, the represents so ) ita2.2 and ( novel is vital twist. scaling generic to the note However, so and it isgeneric vital to note stesmlfigasmto md only that (made a key step assumption only in the (made derivationthat simplifying a the key is of assumption step) (2.22.2 ( in)is of the the simplifying derivation simplifying the is derivation ) the 2.2 of ( in (assumption of 2.2) step isthe key a derivation (made simplifying the that in only step assumption key a that (made only R2-3 768 R2-3 768 768 R2-3 768 R2-3 WHAT REALLY HAPPENS? To provide more complete data (early times) use numerical simulations Assume: Large Reynolds number (inertia dominated) • D. O’Kiely, J. P. Whiteley, J. M. Oliver and D. Vella • Contact angle is constant and equal to measured equilibrium value

(a)(100 b)

10–2 3 3 –3 c 10 ` –1 / 10 2 m –4 2 h 10

1 –5 10 1 10–2 2 2 10–6 10–3 10–2 10–1 100 101 10–8 10–6 10–4 10–2 100 102 1/2 t/(`c/gtt)

FIGURE 2. Numerical results for the evolution of the meniscus height, h h(✏, t), for m = cylinders of different radius, ✏ R/`c, all with contact angle ✓ 13.(a) Numerical results for ✏ 0.15 (blue curve)= and ✏ 0.037 (red curve) compare= favourably with the experimental= results of CQ with ✏ 0=.153 (blue squares) and ✏ 0.037 (red circles) = 5 4 = 3 respectively. (b) Numerical results for ✏ 10 (red), ✏ 10 (green) and ✏ 10 (blue) = 2/3 = = show that the classic scaling law hm t is observed at early times but not the CQ 1/2 ⇠ scaling hm t . Dashed lines show the equilibrium rise height, hm1, for each value of ✏. (Here, lengths⇠ are non-dimensionalized by ` and times by t (` /g)1/2.) c c = c

(n) (n 1) (n) (n) by, for example, (h (r) h (r))/(tn tn 1). The solution for both h and at each time step was obtained through a coupled approximation of the partial differential equations satisﬁed by these variables. A mesh of quadrilateral elements was used, and this mesh was updated at each time step to take account of the evolution in the computational domain as h(r, t) varies. The mesh was generated with a higher density of nodes in the region around z h(r, t), r ✏. Both the nodal spacing of the mesh and the time step were reduced until= the results= (presented later) were visually indistinguishable from those generated using a ﬁner mesh and a shorter time step.

4. Numerical results To arrive at the model problem given by (3.2)–(3.7) we have made a number of simplifying assumptions. Most notable among these are (i) that the contact angle is constant throughout the motion, (ii) that the meniscus motion as a cylinder touches the interface should be similar to that caused by a cylinder that instantaneously appears in the volume r < ✏, < z < and (iii) that the fluid motion is described by a velocity potential. To1 test whether1 these assumptions are reasonable in a model of the onset of menisci, we now compare the results of our numerical simulations with experimental results (digitized directly from figure 16 of CQ). The only unknown parameter in our model is the equilibrium contact angle ✓.To determine ✓, we fitted the static meniscus shape in images of the original experiments (kindly provided by Christophe Clanet) to numerical solutions of the Laplace–Young equation. We found that ✓ 13 minimizes the residual of this fit and so take ✓ 13 in all comparisons between= theory and experiment reported in this paper. = Figure 2(a) shows a comparison between the numerical solution of our model and the experimental results of CQ for the evolution of the meniscus height hm(t) h(✏, t). (Here, data for the two extremal cylinder radii presented by CQ, subject to ✏ = 1, are used.) We see that the agreement between the model results and experiments⌧ is good, 768 R2-6 WHAT REALLY HAPPENS? To provide more complete data (early times) use numerical simulations Assume: Large Reynolds number (inertia dominated) • D. O’Kiely, J. P. Whiteley, J. M. Oliver and D. Vella • Contact angle is constant and equal to measured equilibrium value

(a)(100 b)

10–2 3 3 –3 c 10 ` –1 / 10 2 m –4 2 h 10

1 –5 10 1 10–2 2 2 10–6 10–3 10–2 10–1 100 101 10–8 10–6 10–4 10–2 100 102 1/2 t/(`c/gtt) Observe: FIGURE 2. Numerical results for the evolution of the meniscus height, h h(✏, t), for m = • (Relatively simple)cylinders model of differentreproduces radius, experiments✏ R/ `quantitativelyc, all with contact angle ✓ 13.(a) Numerical • Apparent p results t behaviour for ✏ seems0.15 to (blue be turn curve) around= and from✏ 0 t. 2037 / 3 to (red flat curve) compare= favourably with the experimental= results of CQ with ✏ 0=.153 (blue squares) and ✏ 0.037 (red circles) = 5 4 = 3 respectively. (b) Numerical results for ✏ 10 (red), ✏ 10 (green) and ✏ 10 (blue) = 2/3 = = show that the classic scaling law hm t is observed at early times but not the CQ 1/2 ⇠ scaling hm t . Dashed lines show the equilibrium rise height, hm1, for each value of ✏. (Here, lengths⇠ are non-dimensionalized by ` and times by t (` /g)1/2.) c c = c

For early times, have seen that vertical scale rise of meniscus meniscus Inertialis Inertial meniscus rise

t2 1/3 = h ) m ⇠ ⇢ (a) ✓) c l( ◆ (b) ) b ( (c) ) a ( Motion self-similar for earlyR times and Laplace equation in ﬂuid (inviscid) R 2 1/3 ⇒ relevant horizontal scale is also: t ⇢ ✓ l ◆

Cylinder radius is a length scale (breaking self-similarity of 2-D problem)

R t2/3

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clnsfrtednmc fieta rise inertial 2.1. of Scalings dynamics for the the for dynamicsScalings of2.1. inertial rise oiae ytereprmna bevtos Qasmdta h otc nl of angle Motivated contact the by that their experimental assumed CQ observations, observations, CQ assumed experimental their that by the contactMotivated angle of ,i prxmtl osatduring the constant liquid on the approximately wetted is ), c , b ( body,1 ✓ figure inin figure✓ 1( body, b,c), is wetted approximately the on liquid constantthe during h oin sn hsasmto,te rsne cln nlssfrteinertia- the the for motion. analyses Using scaling this assumption, presented they they presented assumption, this scaling Using analyses motion. forthe the inertia- oiae yaisi w atclrgoere:()amnsu ett lnrwall planar a dominated to next dynamics meniscus a in (i) two particular geometries: geometries: particular two (i) in a meniscus dynamics next todominated a planar wall n i)amnsu usd etclcylinder (illustrated vertical a schematically outside meniscus a in figure (ii) and ) 1b b1 ) and (ii) figure in a meniscus outside schematically a vertical(illustrated cylinder .I stelte aeta so rmr interest (illustrated primary of is schematically that case latter in the figure is It 1). c ).1 It is figure the in latter case that schematically is of primary(illustrated interest here, but we reasons. consider the pedagogical former for first case case first for former pedagogical the consider reasons. we but here, olwn Q ecnie h oc aac natodmninlmnsu of meniscus Following CQ, two-dimensional a we on consider balance the force force the balance consider on a we two-dimensional CQ, Following meniscus of ntwdh(notepg) h etclfrediigtecplayrs sta from that is unit rise width (into capillary the the page): driving the force vertical vertical force the driving page): the capillary (into width riseunit is that from hl h nri ftemnsu eit ie ic h mass the surface Since rise. tension, resists lv cos ✓ meniscus , while the of the inertia inertia the of the while , meniscus✓ cos v l resists tension, rise. Sincesurface the mass 2 2 yteasmto facntn contact of constant the a meniscus of isnot assumption the constantby m h ` (m⇢ ⇢m ( `hm bythe constant assumption not is of meniscus a the constantof contact )/ ( ✓ ⇠⇠ ✓ ( )/ nsaigtrsthis terms angle), scaling In Newton’s. t d m ˙ h m secondd lawcos v l implies that that implies lv cos law d second mh˙m dt. Newton’s In scalingangle), terms this immediately yields ⇠ ⇠ yields immediately (2.1) . 3 / 2 t 3 / 1 ) ⇢ / h( (h /⇢ )1/3t2/3. (2.1) ` v l m ⇠ ⇠m lv ` ). 1983 ( This Miksis & is the Keller well by known found capillary–inertia first scaling scaling first capillary–inertia found known by well Keller the is & MiksisThis (1983). ntecs fa xsmercmnsu iigotieacrua yidr CQ cylinder, In the circular case a of an outside axisymmetric rising meniscus meniscus rising axisymmetric outside an of a case circular the In cylinder, CQ olwa ruetaogtesm ie.Bcueo h nt aiso h cylinder, the follow of an radius argument finite the along of the Because same lines. lines. same Because the of along the finite argument radius an offollow the cylinder, hl h aso the of mass however, the while the, ✓ forcecos R v froml surface now is tension tension is now surface l from vR cos force ✓, the while thehowever, mass of the n aigueof use meniscus making and R is mm h ⇢`h that m hm(R Assuming . ) hm mh ). AssumingR ( m h m thath ` ⇢ hm m is R and makingmeniscus use of Newton’s second ⇠ law again,⇥ we+ find find + we ⇥ again, law ⇠ second Newton’s

(2.2) . 2 / 1 t 4 / 1 ) ⇢ / R h ( (h R/⇢ )1/4t1/2. (2.2) ` v l m ⇠⇠ m lv ` a efmla rmtepolmof problem the While from the form familiar of be the may t) 1/2 2.2 scaling ( in in (scaling 2.22 / )1 t may the be of familiar form the fromWhile the problem of seDuchemin, the(see surface-tension-drivenR radius of drops coalescence liquid two of of two liquid coalescence drops of radius R surface-tension-driven (seethe Duchemin, o xml) hr sn la nlg ewe h two the Eggers between & Josserand analogy clear 2003 no is , for there example), example), there for , is no2003 clear analogy Josserand & betweenEggers the two salmost is ) problems.2.1 ( of Indeed,scaling 3 / 2 t in the wedge retraction problems, problems, retraction wedge the in t2/3 scaling Indeed, of (2.1problems. ) is almost ersnsanvltit oee,i svtlt note to vital generic is it and so However, the scaling twist. novel (2.2 a ) represents represents ) a2.2 ( novel twist. scaling the However, so and it isgeneric vital to note stesmlfigasmto md only (made that a key assumption step in the simplifying derivation the is ) 2.2 of ( ( of 2.2) is the derivation simplifying the in step assumption key a that (made only R2-3 768 768 R2-3 WHAT CAN WE SAY?

For early times, have seen that vertical scale rise of meniscus meniscus Inertialis Inertial meniscus rise

Cylinder radius is a length scale (breaking self-similarity of 2-D problem)

R t2/3 Important parameter in the problem is then the ratio of these lengths = ✏/t2/3 R .Hr,a Here, ). F2002 IGURE ( Quéré 1. & (a) Images Clanet by from experiments reported reported experiments from by Clanet Images ) a ( &1. Quéré (IGURE 2002F ). Here, a and 0 t at ﬁbre of hexane radius and air R 0.225 between mm touches interface the the interface touches mm between225 . 0 R air and radius hexane of ﬁbre at t 0 and eicsrssaon t Iae oreyo hitpeCae. kthssoigthe = showing a meniscus Sketches rises Clanet.) around= it. Christophe (Images of courtesy courtesy of (Images Christophe it. = around Clanet.) rises Sketches meniscus a showing= the are t time essentialwith m h ingredients height for meniscus determining of scaling the scaling of determining meniscus for height ingredients hm withessential time t are ieotieacircular a shown outside for rise ) thec ( cases and of wall (b) rise vertical a next to to next a vertical rise ) b ( wall of and cases (c the ) rise for outsideshown a circular r on yblnigtefredet h interfacial the cylinder. to due The force scaling the laws balancing for by hm(t) found areare found) t ( m h by for balancing laws the scaling force The due tocylinder. the interfacial ihtert fcag fmmnu ftemnsu.Frdtis e text. see tension, details, For lv, with meniscus. the the rate of of change momentum of of momentum change of of rate the the meniscus. with , v l For details,tension, see text.

(a)(1.0 c) 10–1

t increasing 0.5

0 10–2 1 4 –0.5 0123456 10–2 10–1 100 101 102

(b) 2.0 (d) 1.5 t increasing 1.0 100 0.5 0 –0.5 10–2 –1.0 –1.5 –2.0 10–4 0 10–4 10–3 10–2 10–1 100 101

FIGURE 3. The utility of the family of similarity solutions for understanding numerical results. (a) The interface proﬁle plotted in similarity form for the Keller & Miksis (1983) problem with ✓ 80 (black dashed curve) and the solution of the linearized similarity system (5.3)–(5.6=) with different values of R ✏/t2/3. For the linearized problem, the = displacement may be rescaled arbitrarily; here (3.5) is replaced by ⌘⇢ (0 R) 1. (b) The dependence of on the contact angle ✓ for a plane, R (following; Keller= & Miksis 1983, for example). (c) Full numerical results for a range= 1 of cylinder radii collapse at early times (t 1 corresponds to R ✏) onto the numerically determined solutions of ⌧ the similarity system (5.3)–(5.6) (black dashed curve). Here, ✓ 85. In these rescaled variables, the CQ scaling (2.2) would require that ⌘(0 R) R1/4,= see (5.8). (d) Numerical 3 ; ⇠ results with ✏ 10 and different contact angles ✓ show qualitatively similar behaviour but that the dependence= on ✓ cannot simply be rescaled. Dashed lines show the predicted behaviour when equilibrium is reached, t 1, corresponding to hm hm1. Dash-dotted lines 2/3 = show the value of (✓) hm/t in the corresponding two-dimensional problem, which is recovered in the limit t = 1, R 1. ⌧

2/3 tend to the equilibrium rise height, hm1, the rescaled height hm(t)/t remains on a master curve when plotted as a function of R. We therefore conclude that the family of similarity solutions describes a universal dynamics that takes place after the initial t2/3 behaviour but before the meniscus approaches equilibrium. (We emphasize that precisely the same data set is plotted in figure 4a,b.) The collapse observed in figure 4(a) demonstrates that the cylinder radius R does affect the evolution of the meniscus height and, further, that its effect as the rescaled radius R ✏/t2/3 evolves is captured by the family of similarity solutions with the appropriate= value of ✓. However, the observed collapse of the experimental data can only be consistent with the CQ scaling law (2.2) if ⌘(0 R) R1/4 (5.8) ; ⇠ for some values of R. To search for such a behaviour, we solve the linearized version of the problem (5.3)–(5.6) using a finite difference scheme. (It should be noted that 768 R2-9 UNIVERSAL BEHAVIOUR Introduction of the parameter =gives✏/t universal2/3 dynamics at early times R (Travel through a family of similarity-like solutions, parametrized by )= ✏/t2/3 R In limit = recover (true) similarity solution ofInertial KM83 meniscus rise R 1

(a)(1.0 c) 10–1

t increasing 0.5

0 10–2 1 4 –0.5 0123456 10–2 10–1 100 101 102

This also suggests that at early times the experimental data (with different radii) should rescale(b )onto2.0 a universal curve (for given contact(d) angle) 1.5 t increasing 1.0 100 0.5 0 –0.5 10–2 –1.0 –1.5 –2.0 10–4 0 10–4 10–3 10–2 10–1 100 101

2/3 tend to the equilibrium rise height, hm1, the rescaled height hm(t)/t remains on a master curve when plotted as a function of R. We therefore conclude that the family of similarity solutions describes a universal dynamics that takes place after the initial t2/3 behaviour but before the meniscus approaches equilibrium. (We emphasize that precisely the same data set is plotted in figure 4a,b.) The collapse observed in figure 4(a) demonstrates that the cylinder radius R does affect the evolution of the meniscus height and, further, that its effect as the rescaled radius R ✏/t2/3 evolves is captured by the family of similarity solutions with the appropriate= value of ✓. However, the observed collapse of the experimental data can only be consistent with the CQ scaling law (2.2) if ⌘(0 R) R1/4 (5.8) ; ⇠ for some values of R. To search for such a behaviour, we solve the linearized version of the problem (5.3)–(5.6) using a finite difference scheme. (It should be noted that 768 R2-9 UNIVERSAL BEHAVIOUR Introduction of the parameter =gives✏/t universal2/3 dynamics at early times R (Travel through a family of similarity-like solutions, parametrized by )= ✏/t2/3 Inertial meniscus rise R In limit = recover (true) similarity solution of KM83 R 1 D. O’Kiely, J. P. Whiteley, J. M. Oliver and D. Vella –1 (a)(1.0 c) 0 10(a)(b) 10 100 t increasing t increasing 4 0.5 KM83 1 10–1 0 10–1 10–2 1 1 4 –0.5 3 0123456 1010–2–2 10–1 100 101 10–2102 10–2 10–1 100 10–4 10–2 100 102

(Thisb) 2.0 also suggests that at early times (thed)F experimentalIGURE 4. The data utility (with of the different family of similarity solutions for understanding experimental radii)1.5 should rescale onto a universal curveresults (for (captured given contact from figure angle) 16 of CQ). (a) Experimental results (points) and the results t increasing5 of the0 full numerical simulations with ✏ 10 , ✓ 13 (solid curve) may be collapsed 1.0 10 = = 0.5 onto a master curve by using the similarity rescaling (5.1). (b) A plot of the same data a 0 points as in ( ) but in the rescaling suggested by (2.2) shows good initial collapse that breaks down later on. –0.5 10–2 –1.0 –1.5 to enable us to start with R we use the substitution S 1/R starting with = 1 = –2.0 S10–4 0 and solving the resulting problem using first-order finite differences on a 0 rectangular=10–4 grid10–3 in the10 spatial–2 10 variables–1 10⇢ 0and ⇣10. We1 then step forward to consider S > 0, using second-order differences in S .) The solution of this linearized problem can be rescaled to give the results for any contact angle close to ✓ p/2. The results of this calculation agree well with the results of our full numerical= simulations for FIGURE 3. The utility of the family of similarity solutions for understanding numerical ✓ 85 (see figure 3c). Again, we do not see any indication that a scaling of the results. (a) The interface profile plotted inform= similarity (5.8) is present. form for the Keller & Miksis (1983) problem with ✓ 80 (black dashed curve) and the solution of the linearized similarity system (5.3)–(5.6=) with different values6. of ConclusionsR ✏/t2/3. For the linearized problem, the = ⌘ ( ) b displacement may be rescaled arbitrarily; hereWe ( have3.5) presented is replaced a fluid-mechanical by ⇢ 0 R model1. for ( ) the The inertial growth of a meniscus dependence of on the contact angle ✓ for a plane, R (following; Keller= & Miksis external to a circular= 1 cylinder. Our potential flow model with a constant contact 1983, for example). (c) Full numerical resultsangle is for able a to range quantitatively of cylinder reproduce radii previously collapse published at experimental data (CQ) early times (t 1 corresponds to R ✏for) onto a range the of numerically cylinder radii. determined We have assumed solutions that inertia of dominates viscosity and, ⌧ the similarity system (5.3)–(5.6) (black dashedfurther, that curve). the dynamic Here, contact✓ 85 angle. In is these equal rescaled to the static contact angle throughout the motion. In reality these1/4 = assumptions must break down at the very earliest times: variables, the CQ scaling (2.2) would require that ⌘(0 R) R , see (5.8). (d) Numerical 5/3 3 according to the; similarity⇠ solution the viscous dissipation uxx/Re t /Re will, in results with ✏ 10 and different contact angles ✓ show qualitatively similar4/3 behaviour 3 8⇠ = fact, dominate the inertial terms ut, uux t for t . Re 10 (corresponding to but that the dependence on ✓ cannot simply be10 rescaled. Dashed lines show⇠ the predicted ⇠ t . 10 s in dimensional terms). Similarly, the undeformed interface must first react 3/2 behaviour when equilibrium is reached, t to1, the corresponding touch down of to theh cylinder.m hm1. After Dash-dotted these earliest lines transients, but with t . ✏ , 2/3 = 2/3 show the value of (✓) hm/t in the correspondingthe meniscus height two-dimensional grows like t problem,– the radius which of the is cylinder does not play a role recovered in the limit t = 1, R 1. and the problem reduces to the two-dimensional one studied by Keller & Miksis ⌧ (1983). At later times the evolution of the meniscus height with time is governed by a family of similarity solutions parametrized by the rescaled cylinder radius 2/3 2 1/3 / R ✏/t R ⇢`/(lvt ) , provided2 3 that t 1 (so that gravity may be neglected). tend to the equilibrium rise height, hm1,= the rescaled= height hm(t)/t remains⌧ on a master curve when plotted as a functionThe of introduction. We therefore of the parameter concludeR allows that us the to family collapse previous experimental data for theR meniscus⇥ height with⇤ different cylinder radii onto a master curve. of similarity solutions describes a universalOur dynamics analysis suggests that takesthat the place scaling after law (2.2 the) proposed initial by CQ rests on an implicit 2/3 t behaviour but before the meniscusassumption approaches (that equilibrium.hm R) that is (We not consistent emphasize with the that behaviour of menisci growing precisely the same data set is plotted inin figure the presence4a,b.) of gravity. (We emphasize that this inconsistency occurs only in the The collapse observed in figure 4(a)768 demonstratesR2-10 that the cylinder radius R does affect the evolution of the meniscus height and, further, that its effect as the rescaled radius R ✏/t2/3 evolves is captured by the family of similarity solutions with the appropriate= value of ✓. However, the observed collapse of the experimental data can only be consistent with the CQ scaling law (2.2) if ⌘(0 R) R1/4 (5.8) ; ⇠ for some values of R. To search for such a behaviour, we solve the linearized version of the problem (5.3)–(5.6) using a finite difference scheme. (It should be noted that 768 R2-9 UNIVERSAL BEHAVIOUR Introduction of the parameter =gives✏/t universal2/3 dynamics at early times R (Travel through a family of similarity-like solutions, parametrized by )= ✏/t2/3 Inertial meniscus rise R In limit = recover (true) similarity solution of KM83 R 1 D. O’Kiely, J. P. Whiteley, J. M. Oliver and D. Vella –1 (a)(1.0 c) 0 10(a)(b) 10 100 t increasing t increasing 4 0.5 KM83 1 10–1 0 10–1 10–2 1 1 4 –0.5 3 0123456 1010–2–2 10–1 100 101 10–2102 10–2 10–1 100 10–4 10–2 100 102

(Thisb) 2.0 also suggests that at early times (thed)F experimentalIGURE 4. The data utility (with of the different family of similarity solutions for understanding experimental radii)1.5 should rescale onto a universal curveresults (for (captured given contact from figure angle) 16 of CQ). (a) Experimental results (points) and the results t increasing5 of the0 full numerical simulations with ✏ 10 , ✓ 13 (solid curve) may be collapsed Finally,1.0 do not see any sign of Clanet & Quéré10 diffusive scaling = = 0.5 onto a master curve by using the similarity rescaling (5.1). (b) A plot of the same data a 0 points as in ( ) but in the rescaling suggested by (2.2) shows good initial collapse that breaks down later on. –0.5 10–2 –1.0 –1.5 to enable us to start with R we use the substitution S 1/R starting with = 1 = –2.0 S10–4 0 and solving the resulting problem using first-order finite differences on a 0 rectangular=10–4 grid10–3 in the10 spatial–2 10 variables–1 10⇢ 0and ⇣10. We1 then step forward to consider S > 0, using second-order differences in S .) The solution of this linearized problem can be rescaled to give the results for any contact angle close to ✓ p/2. The results of this calculation agree well with the results of our full numerical= simulations for FIGURE 3. The utility of the family of similarity solutions for understanding numerical ✓ 85 (see figure 3c). Again, we do not see any indication that a scaling of the results. (a) The interface profile plotted inform= similarity (5.8) is present. form for the Keller & Miksis (1983) problem with ✓ 80 (black dashed curve) and the solution of the linearized similarity system (5.3)–(5.6=) with different values6. of ConclusionsR ✏/t2/3. For the linearized problem, the = ⌘ ( ) b displacement may be rescaled arbitrarily; hereWe ( have3.5) presented is replaced a fluid-mechanical by ⇢ 0 R model1. for ( ) the The inertial growth of a meniscus dependence of on the contact angle ✓ for a plane, R (following; Keller= & Miksis external to a circular= 1 cylinder. Our potential flow model with a constant contact 1983, for example). (c) Full numerical resultsangle is for able a to range quantitatively of cylinder reproduce radii previously collapse published at experimental data (CQ) early times (t 1 corresponds to R ✏for) onto a range the of numerically cylinder radii. determined We have assumed solutions that inertia of dominates viscosity and, ⌧ the similarity system (5.3)–(5.6) (black dashedfurther, that curve). the dynamic Here, contact✓ 85 angle. In is these equal rescaled to the static contact angle throughout the motion. In reality these1/4 = assumptions must break down at the very earliest times: variables, the CQ scaling (2.2) would require that ⌘(0 R) R , see (5.8). (d) Numerical 5/3 3 according to the; similarity⇠ solution the viscous dissipation uxx/Re t /Re will, in results with ✏ 10 and different contact angles ✓ show qualitatively similar4/3 behaviour 3 8⇠ = fact, dominate the inertial terms ut, uux t for t . Re 10 (corresponding to but that the dependence on ✓ cannot simply be10 rescaled. Dashed lines show⇠ the predicted ⇠ t . 10 s in dimensional terms). Similarly, the undeformed interface must first react 3/2 behaviour when equilibrium is reached, t to1, the corresponding touch down of to theh cylinder.m hm1. After Dash-dotted these earliest lines transients, but with t . ✏ , 2/3 = 2/3 show the value of (✓) hm/t in the correspondingthe meniscus height two-dimensional grows like t problem,– the radius which of the is cylinder does not play a role recovered in the limit t = 1, R 1. and the problem reduces to the two-dimensional one studied by Keller & Miksis ⌧ (1983). At later times the evolution of the meniscus height with time is governed by a family of similarity solutions parametrized by the rescaled cylinder radius 2/3 2 1/3 / R ✏/t R ⇢`/(lvt ) , provided2 3 that t 1 (so that gravity may be neglected). tend to the equilibrium rise height, hm1,= the rescaled= height hm(t)/t remains⌧ on a master curve when plotted as a functionThe of introduction. We therefore of the parameter concludeR allows that us the to family collapse previous experimental data for theR meniscus⇥ height with⇤ different cylinder radii onto a master curve. of similarity solutions describes a universalOur dynamics analysis suggests that takesthat the place scaling after law (2.2 the) proposed initial by CQ rests on an implicit 2/3 t behaviour but before the meniscusassumption approaches (that equilibrium.hm R) that is (We not consistent emphasize with the that behaviour of menisci growing precisely the same data set is plotted inin figure the presence4a,b.) of gravity. (We emphasize that this inconsistency occurs only in the The collapse observed in figure 4(a)768 demonstratesR2-10 that the cylinder radius R does affect the evolution of the meniscus height and, further, that its effect as the rescaled radius R ✏/t2/3 evolves is captured by the family of similarity solutions with the appropriate= value of ✓. However, the observed collapse of the experimental data can only be consistent with the CQ scaling law (2.2) if ⌘(0 R) R1/4 (5.8) ; ⇠ for some values of R. To search for such a behaviour, we solve the linearized version of the problem (5.3)–(5.6) using a finite difference scheme. (It should be noted that 768 R2-9 CONCLUSIONS FL47CH06-Vella ARI 14 November 2014 9:50 FL47CH06-Vella ARI 14 November 2014 9:50

• Keller’s Theorem shows the equivalence of two ways of thinking about abcforces from surface abctension GAS GAS

γγ γγ

p p

mg mg mg mg

LIQUID LIQUID

Figure 2 Figure 2 (a) Illustration showing Galileo’s understanding that a dense object (CIOS)(a) Illustration may float showing because Galileo’s of the volume understanding of liquid thatdisplaced a dense by object the (CIOS) may float because of the volume of liquid displaced by the meniscus (AICB). Panel a reproduced from Galilei (1663).(Can (b,c) The equivalencemeniscus choose (AICB). demonstrated Panel whichevera byreproduced Keller’s (1998) from Galilei generalization is (1663). most ( ofb,c) The convenient, equivalence demonstrated by Keller’s (1998) generalization of Archimedes’ principle to incorporate surface tension. (b)Therestoringforcesthatcounteractthefloatingbody’sweight,Archimedes’ principle to incorporate surface tension.mg (b,comprise)Therestoringforcesthatcounteractthefloatingbody’sweight,mg,comprise a surface tension force per unit length γ,whichactsparalleltothesurfaceandnormaltothecontactline,andthebuoyancyforceprovideda surface tension that force per you unit length areγ,whichactsparalleltothesurfaceandnormaltothecontactline,andthebuoyancyforce logically consistent!) generated by hydrostatic pressure p acting normal to the wetted surfacegenerated of the body. by hydrostatic It is natural pressure that thep acting resultant normal buoyancy to the force wetted from surface of the body. It is natural that the resultant buoyancy force from hydrostatic pressure is equal in magnitude to the weight of liquid displacedhydrostatic by the pressure wetted bodyis equal ( gray in magnitude shaded area to). ( thec)Keller(1998) weight of liquid displaced by the wetted body ( gray shaded area). (c)Keller(1998) showed that the total restoring force on a floating body (including thatshowed due to that surface the total tension) restoring is the forceweight on of a the floating entire body volume (including of that due to surface tension) is the weight of the entire volume of liquid displaced by that body ( gray shaded area). liquid displaced by that body ( gray shaded area).

Whether to think of the additional force as coming from the weightWhether of liquidto think displaced of the additional by the force as coming from the weight of liquid displaced by the meniscus or from the surface tension force acting on the bodymeniscus is therefore or froma matter the of surface preference. tension force acting on the body is therefore a matter of preference. However, it is important to bear both points of view in mind.However, For example, it is importantif one thinks to onlybear inboth points of view in mind. For example, if one thinks only in terms of the force per unit length provided by surface tension,terms it is tempting of the force to believe per unit that length having provided by surface tension, it is tempting to believe that having a long, convoluted contact line will allow surface tension to providea long, convolutedarbitrarily large contact supporting line will allow surface tension to provide arbitrarily large supporting forces. Thinking instead in terms of the weight of liquid displacedforces. makes Thinking it clear instead that this in terms cannot of the weight of liquid displaced makes it clear that this cannot be the case, for reasons discussed below. be the case, for reasons discussed below. Although impressive for its time, Galileo’s intuitive understandingAlthough that impressive the weight for of its liquid time, Galileo’s intuitive understanding that the weight of liquid displaced by the meniscus must balance the excess weight ofdisplaced a dense ﬂoatingby the meniscus object misses must an balance the excess weight of a dense ﬂoating object misses an important part of the story: Objects cannot lower themselvesimportant arbitrarily part deep of into the the story: liquid Objects and cannot lower themselves arbitrarily deep into the liquid and hence displace an arbitrary amount of liquid in their meniscus.hence Instead, displace the shape an arbitrary of a meniscus, amount of liquid in their meniscus. Instead, the shape of a meniscus, z h(x, y), for example, of a liquid with interfacial tension γz, densityh(x, yρ),, for and example, subject to of a a gravi- liquid with interfacial tension γ , density ρ , and subject to a gravi- = = ℓ ℓ tational acceleration g, must be given by a solution of the Laplace-Youngtational acceleration equation:g, must be given by a solution of the Laplace-Young equation:

n h/ℓ2, (1) n h/ℓ2, (1) −∇· = c −∇· = c where ℓ (γ /ρ g)1/2 is the capillary length, and n ( h, 1)where/(1 ℓ h 2()γ1//2ρisg the)1/2 unitis the (outward capillary length, and n ( h, 1)/(1 h 2)1/2 is the unit (outward c = ℓ = −∇ +|∇c =| ℓ = −∇ +|∇ |

Annu. Rev. Fluid Mech. 2015.47:115-135. Downloaded from www.annualreviews.org pointing) normal to the liquid surfaceAnnu. Rev. Fluid Mech. 2015.47:115-135. Downloaded from www.annualreviews.org (Finn 1986). pointing) normal to the liquid surface (Finn 1986). There are no general results that limit the maximum depth of aThere meniscus; are noindeed, general for results a meniscus that limit the maximum depth of a meniscus; indeed, for a meniscus Access provided by University of Oxford - Bodleian Library on 01/06/15. For personal use only. Access provided by University of Oxford - Bodleian Library on 01/06/15. For personal use only. contained within a sufﬁciently tight corner, the meniscus depthcontained may be within unbounded a sufﬁciently (e.g., tightsee corner, the meniscus depth may be unbounded (e.g., see Finn 1986, p. 116). Nevertheless, in cases of practical interest,Finn the 1986, maximum p. 116). depth Nevertheless, of a static in cases of practical interest, the maximum depth of a static

meniscus is a multiple of either the capillary length, ℓc , or themeniscus typical is radius a multiple of the of object, eitherR the, capillary length, ℓc , or the typical radius of the object, R, whichever is smaller (Finn 1986, Lo 1983). Because the horizontalwhichever length is smaller scale over(Finn which 1986, a Lo 1983). Because the horizontal length scale over which a

meniscus decays is the capillary length ℓc , the typical weightmeniscus of liquid displaced decays is thein the capillary meniscus length ℓc , the typical weight of liquid displaced in the meniscus scales as ρ g min R, ℓ ℓ2. Hence, we expect that a solid objectscales of density as ρ g ρminandR, typicalℓ ℓ2. Hence,radius R we expect that a solid object of density ρ and typical radius R ℓ { c } c ℓ s { c } c s can float only if ρ R3 αρ min R, ℓ ℓ2 for some dimensionlesscan constantfloat onlyα if. ρ R3 αρ min R, ℓ ℓ2 for some dimensionless constant α. s ≤ ℓ { c } c s ≤ ℓ { c } c From this scaling, it is clear that objects with solid densities significantlyFrom this scaling,larger than it is that clear of that the objects with solid densities significantly larger than that of the liquid can only float if R ℓ . In this case, flotation is possibleliquid provided can only that float if R ℓ . In this case, flotation is possible provided that ≪ c ≪ c 2 1 2 1 ρ /ρ α(R/ℓ )− αBo− , (2)ρ /ρ α(R/ℓ )− αBo− , (2) s ℓ ≤ c = s ℓ ≤ c =

118 Vella 118 Vella CONCLUSIONS FL47CH06-Vella ARI 14 November 2014 9:50 FL47CH06-Vella ARI 14 November 2014 9:50

• Keller’s Theorem shows the equivalence of two ways of thinking about abcforces from surface abctension GAS GAS

γγ γγ

p p

mg mg mg mg

LIQUID LIQUID

• CanWhether extend to think ofKeller the additional & force Miksis as coming fromsimilarity the weightWhether of liquidto solutions think displaced of the additional by the to force problems as coming from the with weight of liquidan displaced by the meniscus or from the surface tension force acting on the bodymeniscus is therefore or froma matter the of surface preference. tension force acting2/3 on the body is therefore a matter of preference. intrinsicHowever, it islength important to bearscale, both points though of view in mind. it However, Foris example,hard it is importantif oneto thinks break to onlybear inboth the points oft viewscaling in mind. For example,law if one thinks only in terms of the force per unit length provided by surface tension,terms it is tempting of the force to believe per unit that length having provided by surface tension, it is tempting to believe that having a long, convoluted contact line will allow surface tension to providea long, convolutedarbitrarily large contact supporting line will allow surface tension to provide arbitrarily large supporting forces. Thinking instead in terms of the weight of liquid displacedforces. makes Thinking it clear instead that this in terms cannot of the weight of liquid displaced makes it clear that this cannot be the case, for reasons discussed below. be the case, for reasons discussed below. Although impressive for its time, Galileo’s intuitive understandingAlthough that impressive the weight for of its liquid time, Galileo’s intuitive understanding that the weight of liquid displaced by the meniscus must balance the excess weight ofdisplaced a dense ﬂoatingby the meniscus object misses must an balance the excess weight of a dense ﬂoating object misses an important part of the story: Objects cannot lower themselvesimportant arbitrarily part deep of into the the story: liquid Objects and cannot lower themselves arbitrarily deep into the liquid and hence displace an arbitrary amount of liquid in their meniscus.hence Instead, displace the shape an arbitrary of a meniscus, amount of liquid in their meniscus. Instead, the shape of a meniscus, z h(x, y), for example, of a liquid with interfacial tension γz, densityh(x, yρ),, for and example, subject to of a a gravi- liquid with interfacial tension γ , density ρ , and subject to a gravi- = = ℓ ℓ tational acceleration g, must be given by a solution of the Laplace-Youngtational acceleration equation:g, must be given by a solution of the Laplace-Young equation:

118 Vella 118 Vella