Ph St

FINGE R ING INSTABILITY OF A

GRAVITATIONALLYDRIVENCONTACT LINE

BY

@J O HN MARK JERR ET TB.Sc.(HollS.}

Athes issub m itt edto theSchoolofGra duate Studiesin par tial fulfilm entofthe

requirem entsfer the degree of Mast erof Scie nce

De pa rtm en t ofPhysics Memorial Unive rsity ofNewfoun d land

April 199£

St.John's Newfoun dlan d

1.1

National library olCanacla

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ABSTRACT

Presented hereare the results oflUIexperimenta l studyoftheflowof a\'i~cou, fluidsheetdown a dryinclined plane.Thethree-phase ContAct line atth....front ofthe flow is initiallystrai ghtbutbecomes unstabletoaroughl)' perjodic \'Ariation in its downslope positionwhenthe sheet becomes thinenough.Frommeasurcmcnls ofthe contactline position as a functionof timefor angles0intherange0"<n<!;,')".the flow isanalyzed both before anda.ftertheinsta bilityoccurs,and thedevelopment or the fingerpatternisparameter ized,These resultsarecompared wilhthose round in previousexperimentsandthose predictedby theory.

ACKNOWL EDGEMENTS

I wouldliketo thankmysupervisor,Dr. JohndeBruyn,forhelpfuladvice,

consuhe tlon,financialsupportandmany proof readings ofthisthesis,the physics depart mentforteachingassista ntshipsthrougho utmymastersprogram andMr.Mike Ryan fortheconstructi onofmyexperimentalapparatus.

iii

ABSTRACT..

ACKNOWL[;DGE~IENTS TABLEOFCO NTENTS LIST OF FIG URES..

TA B LE OFCON T E N T S

. ii

... .. ... . .. . ... iii

... .. •.•... .. ... ... .. ..iv ....•. ... ... ..vi

LISTOFTABLES ix

Cha pterIINTR ODUCTI ON

1.1Introduction 1.2 PreviousWork l.3Purposeend Scope

Chapter2THEORY 2.1 Intr oduct ion.

2.2 Wetting Phenomen on...

2.3ContactAngleHyeteresia... ..,.... .. •.

2.4 Contac tLine Singularity . 2.5Precur sorFilm ..•

. . . . ... . .... ... . . ...I

... ,

.... .. . .. ... ... ... 8

.. 10

. lO

.1I

.. ....13 . ..14 2.6Lubricat ion App roximations

2.7Theoret icalwork of refer ence I.• ... . ... ... ...•.

2.8Theoreticalworkorreference 13...

..16 ... ....18 .2l

2.9Theoreticalworkorreference12 . "2~

ChApter3,\P PARATUS,\:-;0 EXPEliHIE:-;!AL TECH:"IQUE

.... •28

. 29

... •..31 :1.1Apparall15. .... . ... . ... .. . . ... .... . ... •. .. •.._ :1.2 Fluids....• ....•...

:1.3 Experiment...lTechnique.. .. .

3.4RawData Gathering...•.. . ... ...•..•... .... .. . .. . ...33

Cll<l.pter~RESULTS 4.1Gf:neral Observaticns. 4.2 BeforetheInstability .

~.3AneetheInsta.bility .

.. . ...35 ... •. . . ...•...35 . ....••...•.••.. •.••...•..38

Chapter5DISCUSSION

5.1Beforethe Instabilit y 11

5.2Aftt'TtheInstab ilit y... .. ... ..•.. .. ...•.. •.... . ....•.•.;2

Ch"'pler6CONCLUSIONS

6.1Conclusions .... .•. . •.. .... .. ... ... . .•. .•.. ...•... ..•••.. . ...•.. . . . .83 6.2 Future Con siderations... .. .. .. .•... ....•...•••.... . .• .•.... .••...83

References . ... ...•.. .... ...•.. ...86

Lis t of Fig ur es

Figu reI-I. Avolu meofRuidRo...ing downadry,solidinclined~llrfA({'.

Figu re1-2.Rlvuler (paralldsidedfing~ )pauem

Figu re 2-1.Three phase solid·liquid·gascont actline...• . . . ... . . .

•••:1

••:1

. 11

Figure2·2 . The general formof the conte cr angle()Il:lafunct ionof contactline velocity.12 Figu re2-3.Advancing andrecedingcontactanglesIl:la function ofsurfaceroughness. From

referenceI? . • ...13

Figure2-4.Two possible method,ofadvanceof a. fluidfilm.Rolling motion[a]andadva nce

over aprecursorfilm (b) I·j

Figure2-5. Unperturbed80wprofile calculatedfromEq.(2-25)forthreevalues ofbincluding anarbitrar y shift alongtheedirectionsothat themaximalineup.Fromreference 13.•..• •. • •••. .••.•••••• •• •• • ••.. •. •. • •••• •• •.. •• • ••••• • • •••• • ...23

Figure2-6.

P

asa funct ionofq,where positivevaluesoftJindic ateunstablemodes.from

reference13 24

Figure2-7. Calculatedflow profilesfora contactangie of 70°, incliniltionangle45° and capillarynumbers0.03,0.10 and 0.144for profile, a,bandc,respec tively,From reference12.. •.. •• . . .. ••. . .. •.... .. ... ... . . ...,...27 Figure4-1.Posit ionv.!.timeforHMOon plexiglas,ata

=

3°.Tbeposit ion of theuniform frontbefore the inst abilit y (opensquares]advances faste r than11/3atearlylimes and appearsto approac h1\11/3growth (dashedline) atlatertimes. Thesolid squaresand,tanshow thefingerlipand trough posit ions,respec tively,afte r

vi

thein.'lta bility. and thedott edlineshowsthepower 111...growt hofthe finger tip

position.. . ....37

Figure 4-2. The downslopeposition oftwo pointsontheconta ct line ofa 250an3sample or liMO on plexiglsssatOr=8°.The squarescorrespond toapointwhich ieatthe tiporafinger alterthe instabilityoccurs,andthe starsto oneina trough.The lines arefits to Eqs. (4- 2) and(4- 3)ferthe fingerandtrough,respectively...39 Figure 4· 3. Thesolidsymbols show the exponent(Jof Eq.(4- 2) asa.function ofsinOrfor glycerin onplexiglass (a),HMO on plexiglass(b),glycerinon glass(c)andH~IO on glass(d).The dashedlinesshowthe averagefor each case. Thedatapoints indicate themeanvalues andtheerror barsthe rangeIromfitsto several fingers at eachangle.The open symbolsshow the exponent/30for the flowbeforethe instability,determinedfrom Eq. (4-1). Forclarity,errorbars {or these pointsare not shown;they aresimilarinsizetothose for theothe.:data... ..,.41 Figure4-4. The solidsymbolsshow the amplitudeAofEq. (4-2) for glycerinon plexiglasa (a),HMO on plexigleaa(b), glycerinon glass(c) andHMOon glass (d).The lines arefits to Eq.(4- 4). Theopensymbolsshow the amplitudeAOfor the flow beforetheinstability,determinedfrom Eq. (4-1) 46 Figure 4-5. The exponentialslowingtimer ofEq.(4-3 )as afunction of sin c for glycerin onplexigleee (a),HMO onplexiglass(b), glycerin onglass(c) and HMO on glass

(d ). The linesare powerlawfitstoEq.(4-5) 51

Figure 4-6. The amplitudeB of Eq.(4-3)as a functionofsin o: forglycerin on plexiglass(a), HMOon plexiglass(b), glycerinon glass (c)andHMO onglass (d ). The dashed

linesshow the averageoreach at high anglC3 56

Figu r e4-1.The asymptoticvelocityCof Eq. (4-3)^11.9a function of sifl."{or glycerinon vii

ple:cigldM(a),HMOonplexiglass (b). gl)'cerinonglASS(e)and liMOon~I,,!l.' (d)... .•. . ...••.. . . •. . .. .. . . . ... . . .. . ... . .. •... .. . 60

Figure4-~.TheaveragewA\'elengthIofthe finger pattern' forthe plf.xigla.•"1II1rf"ct'(,,) And the glass surface[b],- -is a fito( Eq.(-1-6)todata(orliMOwith b).

=

OAO±.02 forthe plexiglass surfaceand_bA

=

0.39±.OSforthe gl,," lIurfllcc.

- - - isasimilarfit(orglycerinwithb).

=

OAt±.03(or the plexiglll.'5 lIl\T(an"

andb"=OA 9±.02 for the glasssurface is a lineof-1/3slope for

comparison.. . .•. . . ... . . . 66

Figure4·9.Thefinger width0asa functionof sin0for the ple:dglass surface (a)and thegla~~

surface(b).- - is afitof Eq.(-1-7)tc dara(or liMOwith 66

=

0.06 ±.0<1for the plexiglas! surfaceandb6=O.59 ± .04 forthe glass surface.--- - isasimilar litforglycerinwith66

=

0.53±.03for the plexigla..s surface and 66=0.51±.03

forthe glass surface 6I:l

Figure5-1. The ratioIII(rom myexperim~lalresults(orthe plexigllLUsurface(a)"ndtIlC~

glasssurface(b)along wit h thetheoreticalprediction o( referenee 13. - -

HMO;- - - glyo.:rin; theoretical result 76

Figure 5-2.Thegeneral (armof the contactangle 9 attheliquid-sclid-vepcr contactline,ua funct iono( contactline velocity.Here asystem withcontact engle 91 isadvancing

witbvelocityVI 79

Figure5-3.The contact lineshortlyafterthe onsetoftheinsta bilityis sketched as asolid line. 9 isgreaterthan ,lessthanandequal to9Aat the pointsX,YandZ respectively.Thedotte d linesshowschematicallythegrowthor a rivulet.. . ..80

viii

List of Tabl es

Table1.Propertiesornuids st udied 30

TableII. PMamelers d.aracterizingfinll:ertipposit ion...• •...•.. ...50 TableIll.Parameterscharacterizingtroughposition fH TableIV.Paramet erschar acterizing wavelengt handlingerwidth .... ... .. • •;0 TableV.Summaryornata from Tab lesII,IIIandIV... 85

Chapter1 Intr-oduction 1.1Introducti on

The flowof a viscous fluidsheet downa dry inclinedsurfacedisplaysacommon yetinterestinginstability.Itis seen in paint flowingdown a walland rain running downawindow pane.Thisinstabilityisneitherwell understoodtheoreticallynor thoroughly characterized experimentally.

Consider avolume of fluiduniformly distributed behinda gateatthe upper end of adryinclinedsurface.Asthegate isopened, the fluid beginstonow down the slope withthe contact line atthefrontofthe flowbeinginitiallystraight,as illustr ated in Fig. \.1.Whenthesheetthins toacertain fluid-dependentthickness, thecontact line becomesunsta bleto a roughlyperiodic cross slopevariationin its downslopeposition.1 Forcertai n fluid-solid combinations,l.2suchas glycerinon a plexiglasssurface,thisvariat iongrows intoa series of rivulets orfingers offairly uniformwidth(parallelsidedfingers). Thesefingers continueto flowdownhillwhile the upslope troughsbetween thefingersstopshortlyaftertheinst ability develops.

Thusthe surfacebetweenthe fingers remainsdryandthis pat tern, shownin Fig.l- 2, does notcomplet ely wet thesurface.

Forotherfluid-solid combinations,I such as silicone oil on glass, thevariation grows into afairlyperiodic patter noftriangularshaped fingers (sawtooth pattern).

Inthis caseboththe fingersand the troughs continuetoflowdownhill afterthe instabilitydevelops,althoughat different rates.This flow, showninrig. 1-3,will wettheentiresurface.

Thesephenomenaare importanttomany industrial processeswhere weltingby a thin filmof fluidisrequired.Two examplesarethespincoatingof a magnetic

Fig. I-I.A volume of fluid flowing down ... dry, solid surface inclined at^aDangleQto the horizontal.Initially the contact line is straightas shown, buteventually it becomes unstable,developing fingen.

Fig.1-2.Rivulet [parallel sided finger)pattern.

ww

Fig.1-3.Sawtootb pattern .

storage disk byafluidinitiallyplacedat its axisofrota tionand thelubricantcoating on enginebearin gs.Ifthefluids used intheseexamplesdid not completelywetthe surfaceundesirab le'dry patches'couldform.For theabovecases this ...ouldresultin poorquality disksandengines which needfrequent bearing replacement.In addition welt ing phenomena andthephysicsbehindthemare oflongstandinginterestboth experimenta llyandtheoretically4.5and have been well studied in both areassince the pioneering wor k of Young(1805)6,Reynolds(1886)7and Rayleigh (1890).8Even with overonehundred yearsof study thesephenomenaare stillnotthoroughlyunderstood.

1.2 PreviousWork

Thefingeringinstabilityofa movingcontact line hasbeen studied experimentally byHuppert !andSilviandDussan V.2A relatedinst ability,the fingeringinstability inthe spreadingof arotatingliquiddrop,basbeenst ud ied byMeloe!al.3

Huppert stud ied thisinst ability usinga perspexsurfaceandthreefluids,namely glycerinandtwo silicone oils.Heobservedtherivu letpa tte rn withglycerin,and the sawtoot hpattern with bothoils.He derivedanexpression for theposition,Xn,ofthe stablecontactline,neglecting surfacetension,andfound

(I - I ) whereAis theinitialcross sect ionalarea,JJis theviscosityandt is the time after releaseofthe fluid. He also derived,includingsurfacetension,an expressionforthe wavelength ofthe instab ility,

(1-2) whereuis the surfacetension andpis the fluiddensity.Hefoundthat bothEqs.

(1-1)and(1-2)agreed wellwithhis experimenta lobservati ons.

SilviandDUMan V2studied the sameinstability usingglycerinonbot hglAM and plexiglasssurfaces. TheyoMerved therivuletpAtter nforglycerinonpklligla.u.;u

didHuppert,Ibutfou ndthe sawtoot h pattern forglycerin on,!Ilass. Theymeasured theadvancingcontact Angle tobeiO"and18" on plexi,!lIA" andglassrespectively, Thustheyconcludedthatthe size of thea.dvancing contactangleisanimport ant factorindeterminingwhetherthe rivu letortheuwtooth patternwill emerge for a givenliquid-solid combinlLtioD. Theyalsofound tha ttheirexperimentalobeeevauons werein agreement withEq.(1- 2)derivedbyHuppert.

Meloetal.3placedadrop ofsilicone oila:t theaxisofrot atio nofa silicon wlI.fer, Theliquidwett ed thesubst rate,i.e.,theliquid-solidcontact anglewas zero. As the waferwasrotated,the drop spreadoutwith an initia.lly circular conta.ctline.

Whenthe thicknessofthe fluid wassmall enoughthecontact line became unstable and fingers form ed.Whileintbiscasethecontactlinewasdriven bycerllrifugal ratherthan gravitationalforce,thebasicphenomenaobservedweresimilartothose describedin theprevioussection.

Inadditionto thetheo~tkalworkof Huppert¹mentionedbriefly wove,lheord- lealwork on thisinstab ilitybubeencarried out by Schwartz,9Hocking.IOGoodwin andHomsyl2and TroinetaI.^Il

Schwa.rtz9carried out numericalsimulationsusing eqUAtions derivedinthelu- hricationapproximationandincluding surface tension.Theadvancingcontact angle was takento he zero,correspondingto a liquid whichstronglywetthe surface, and a no-slipboundary condition was appliedat the edgeortheflowcorresponding to theexperimenta l constraintof a wall.He foundthat theno-slipconditioncaused the contact linetobe retarded at theedgeof the flow.This perturbationiniti atedfinger-

ing and thedisturba nce propagatedinwardalongthecontactline. Hefound thatthe longest finger iswedge shaped whichis in agreementwith experimentalobservations of wellingflows.

Fromsca ling argumentsand dimensional considerationshe found thatthe wave- length should go like

,\'"(sina)- 1/4 (1 - 3)

incont rasttotheexperimental and theoreticalfindings of Huppert! andtheexperi- mentalresultsof Silviand DussanV2that >'""'" (sina)-! /3.

Without theno-slipboundarycondition,small periodicperturbationsimposed on thestraight front eventually grew into fingers.

Neglecting surface tensioncaused thefingering phenomenon to disappear, both withand withoutthe no-slipboundary condition.This seems to confirm Huppert's suggestion1 thatsurfacetension providesthe destabilizing force.

Hocking^lOperformed a linear stabilityanalysisof a fluidridgea.sopposedto a fluidsheet. One reason for selecting a fluid ridgecomes fromthe experimental observation3•11of a bulge inthe free surface of the fluid near the contact linewhen the fluid sheet is sufficiently elongated and the suggestionthatthe observedinstability ofthe contact. line is a result of the dynamicsin this ridge.Anotherreason is, of course,tosimplify the fluid sheetproblem. Thusin studying a fluid ridge, Hocking hoped to simplify the problem while retaining the importantdynamical processes. He found the ridge to be linearly unstable but not to fingers.Rather,thefluid tended to collecttoone side of thechannel and Row down the planethere. Tha.t is,he found the lengthsca le or the instability to be dependenton the channel width in contrast to experimental observations.!,2

Hocking also consideredthe nonlineardevelopmentofthe insta bilityand pre' sented preliminarynumericalresults thatsuggesttheformat ionof afinger of l1uill moving down the plane withawidth that isindependent ofthe chan nel width .He foundno indicationofthesawtoothpatte rnobserved with somel1uid -solid comhina- tiona and proposedthat thesawtoothpatternis only atransient phenomenonwhich would eventuallyevolve intothe rivulet patte rnifthe surfacewas longenough.

GoodwinandHom~yl2haverecently lnveetigatedthe base stalewhich develops priortotheinstabilit y using acombination of analysisandnumericalsolution. They showed thatit isnot possibletomodel the flow nearthecontactline inthelubrice- tionapproximatio n ifanon-zeroconta ctangle is imposed asaboundar y conditio n, without requiringinfinite velocitiesattheco ntactline.They derived andsolvednu- merlcallyStokesflow equationsfor theregion nearthecontact linewithacontact angle bound arycond ition. Theyfoundthepresenceof a hump orbulge in theIiulds free surface nearthecontact line as observedexperimentally.3,11Themagnitudeof thisbulge increased for increasing contact angle, increasinginclination angleandlJC- creasingcapillary number.The capillarynumberCa

=

p.Ule expresses the relative magnitude of viscousandsurfacetension forces;Uis acharacteristic velocity.They alsoobserveda secondar ybulge at large anglesofinclination.Theydemonstrated thatthe bulge near the contactline arisesfromkinematicconsideratio ns andnotfrom the contactlinesingularity. From theirresults theyconcludedthat , exceptpossibly atvery small contact angles andsmallCa, thelubricationapproximat ionsare not validnear the contact line.

Trciantt al.13 studiedthe lubricationequations withsurfacetension,Theyde- rived a form for the flow profile priortothe instability.This profileis a combination of an"outer" region,givenby Hupperts' solut ion,Eq.1-1,which ends abruptly at

%=%,.,andan "inner"region nearthe contact linewhichhasabulge,anawhich is smoothed bysurface tension.Usingscalingarguments,the inner andouterregions are matched,resultingin a flow profilesimilarto that observed experimentally.Then they matchedthis solutionto athin precursor wet tingfilm. Thiswas donetore- move thecontact line singular ity,discussed furt her inchapter2.Alinear stability analysis was thenperformedon the resultlngprofile. They analysedthe growthof imposedperiodic perturbationsoverarange of waven umber! and found thefrontto he unstableto wavenumbersq:is.91withamaximum growthrate for awavelengthof

>.

=

^141.Here,I

=

^HjI3Ca)I/3is thecharacterist ic lengt h overwhich surfacetension

competeswithgravityandHis thefilm thickness.

1.3 Purposeand Scope

In thisthesis

r

presentmeasurements ofthe behaviourofthe cont actline bot h before and after theinstability.Theposition ofthecontact lineasa function of time was measured for three fluids ontwosurfaceswithindin ationanglesin therange

00<a<55⁰•The flowwas thenanalysedin terms of empiricalfitting functions.

Theaverage wavelengthofthe instability and the width ofthe fingers were also studied.

This workwas carr iedoutinorderto addto the experimental knowledgeofthe movingcontact line.Measurementsweremade sc aa to cherecteriacthe motionof the contactline and thelength scaleoftheinstability which developed. My results, forthemost part,confirmsome theoret ical predictionsofthemotion ofthecontac t lineand thelength scaleoftheinstability,whileatthesame timesupplyingnew information about thefinger widthfor which I have seen noexperimental results or theoreticalpredictions. Most ofmy experimentalobservat ionsagree withthose

o r

others whileone,the pattern producedby glycerinonglass,doesnot.

In Chapte r2 I willdiscusssome of the basictheoryandterminologyrelevant to thestudyof dynamic contact lines. includingwelting, contactanglee,conracc angle hysteresis,thecontact linesingularity, precursorfilms andthelubricationapproxi- mations.Iwill alsodiscuss someof the previoustheore ticaltreatmentaof thecontact lineinstabilityin moredetail.Chapte r 3willconta indescriptionsofthe cxpen men- talapparatusused,the fluids andtheir relevent physicalproperties,the experimental procedu reandthe data gatheringprocedure.In Cha pte r-1the experimentalobser- vationsandresults of fitsofthe datato f'mpiricaleqllRtioO!\will hepresented, while in Chapt er5theseresults willbe discussedandsometheoreticalexplanations for theexperimental ob5l"TVations will bepresented.Chapter6 will includetheconclu- sions, and somepossibilitiesforfuture experimentsusingthe same apparat uswillbe presented.

Chapter 2 Theory

2.1Int rodu ct ion

The dynamics of moving contact linesarepoorly understood.The usualthee- reticalapproximationsoffluidmechanicsused to describefluidtll)w break down at acontactline.hhasbeenshown14 that fora Newtonian,incompresibleAuid with a no-slip boundarycondition ,unbounded forcesresult at thecont act line. In the remainderof this~hesisIwillrefer to the unboundedforcesat thecontactline as the

"contactlinesingularity."Anotherprobleminvolvesthecontact angle the fluid makes withthe surfaceof thesolid.Ithas been showntheoreticallySandexperimentally l5 lhat the observedorapparentcontact angle may not heequa l tothe actualcontact angle.The apparentcontact anglemayhave a rangeof valuesfor whichthecontact linedoes notmove (contactangle hysteresis).A thin precursorfilm maypreceedthe macroscopically observablecontactline resultinginthe observableflow moving over apreccated, as opposed to dry,surface.The precursorfilm also causesconfusionas tothe positioningofthe contact line in theoreticalcalculations.

2.2 WettingPhenomen on

Young'sequation,llwhich expressesthebalance of horizontalforcesatthestat ic three-phase contactline of asolid-liquid-gassystem, asshown in Fig.2·1,statesthat (2-1)

where "{=

" 'g

is the interfacialtensionbetween theliquid and gas, ,." between the liquid and solid and"{,gbetween the solid and gas, andDeisthe liquid-solidcontact angle.Thespreading parameterSis defined inthe following way: anareaofsolid- gasinterface has asurfacefree energy of"{.gwhile thesame area coveredby a thick

10

coat ingof liq uidhas a surface freeenergy ofl'+1~1'5is justthedifference between these two surface free ener gies.S is written

s

⁼1'9-1,/-1 (2-2)

12 - 3)

WhenS<0,i.e.,

"9

< ...,+1d. the liquiddoes notspread;thiscorrespondstopart ial

welting.IfS ?:0 thereis no balance of horizont alforces and completewett ing occurs.

5

=

0 corresponds to()e=O.

g..

liquid

~Iid

/

contact line

Fig.2· 1.Three phase contact line sbowing interfacialtensions"'Y. "'Y"

and"'Y" Andstaticcon tactAngle9t•

Inequilibrium verticalforces must also balance; these vertical forces canarisedue to,forexample,capillary forcesand l1uidweight.

2.3ContactAngle Hysteresis

Figure2-2 isa graphofcontactangle versus contact linevelocityfora typical fluid-solid system.

11

-

Fig.'· 2, The generaJform of the contact lineangle 8 atthesolid- liquidcontact line, asa. function ofcontact line velocity.

For contact anglesOR<IJ<0Athecontac tline doesnot move,whileitadvances forIJ>0Aandrecedes {or0< OR' This phenomenon,wherebythecontactline doesnot move eventhoughthe contactangleis varied Ircm its equilibrium sta tic valueisknownas conta ct anglehyst eresis, andisverycommon.Oneexperimentally verifiedcauseofhyste resisis themicroscopic roughnessof thesolidsurCace.16,17By coating a smoot hsolidsurfacewit han organic monolayer,Zisman16foun ditunusual forSAandORto differwhereas Dett reandJohnson,17 usingsurfaces ofincreasing roughness,found that 8A " (JR.Fig.2-3. {or allsurfaces.

Other possiblecausesofconta ctanglehysteresis are chemicalcontamin ants or inhomogeneitiesinthesolidsu rfaceandsolutes intheliquid whichmay deposit afilm onthe surface. The surface condition playsanimportant rolein wett ingphenomena.

as it can affect the sizeoftheliquid-solidcontactaugle,16,17 and inthe caseof my 12

:lOoo• • , I , 4 • t • 0

''''~''C(Nour;~NUI _

Fig. 2-3. Advancingand receding o:oohcla.ngles,SA MId SR,esa luncti onofsurfaceroughness fromreference 17.

experiments ca n therebyaffectthe patte rnwhich developsfromtheinstability.2

2.4 ContactLineSing u la rity

I£a fluidmoving along a solidsurfaceisassumedtobe Newto nia n, incompressible and to obey the no-slipboundary cond ition.thenunbound edforces will beproduced at the contactline.14 This singularityshowsupin thelubricat ion approximations, to be discussedin section2.6, by requiring a 90° contact anglebetweentheliq uid andsolid.Sincethelubricat ion approximationsare usedprimarilyforthespreading of thin films wherethe velocityvectoris approximately parallel to thesolidsurface, obtainin ga 90° contact angle implies thatthe lubricationapproximationsarcnot valid nearthe contactline.Thesingulari lycan be ignoredifit. isknown Lhat the fluidncar the contact line doesnot affect the dynamicsof theflow in theregionof the

13

contact line and itcanberemoved byrelaxing theno-slipboundary condition . The no-slip boundaryconditionis usedsimp lybecauseitremoves the singularitybut docs net comefrom anyphysical understandingofrbefluid flownear thecontactline.IS 2.5Precur sorFilm

Afluid mayadvance ona dry subst ratein twoways: 1JByarolling motion,as shownin F'ig. 2-4(a). 2JBymeans ofathin precurso r wettingfilmwhich advances ahea doftheobservablecontactlineasshownin Fig .2-4(b).Inthefirst case,the observablecontactline isbetween the fluid and the dry surface, whileinthe second caseitisbetweenthe observable fluid andanalreadywetted surface.

/ I I I solid

g.. (b)

g..

pre c u rs or

,

film

/ // / '/ " , / ' "

,

solid

Experiments indicat e14 that a drop of honey moving downan inclined glasssur- facetend s to roll andnotslide. Therollingmotion was studiedbyplacinga drop of dye onthesurface of thehoney andfollowingthe dye' s motion as the honeymoved.

In myexperiments the fluidalsoappear edtorolldown the slope.

14

Experiments byHardy 19 withdrops of acetic acid.ona horizontalglass surface (it was important that theairbe dry) Indicatedthata precursor film waspn~"llt beyondthe observablecontactline,eventhough the drops did not undergoany visihle changein shape.He detected the film's presenceby measuring a significantdcctensc in thevalue ofthe staticfriction ofthe surface.Drops of castoroiland paraffinon the samesurfacedid not emitanymeasurableprecursorfilm.19

Experimentswere performedby Bascomfl aJ.2t1usingnonpolar fluids onclean, smoothmetalsurfacesin thepresenceof both saturatedandunsatu rated air. All theliquids used werechosen becausetheyamacroscopicallyobservablecontact angle of zerodegrees. Theyfoundthat a precursorfilm was alwayspresent regardless ofwhetheror not theair was saturated,thesurfaceroughened or the llquids nltra purified,although these variablesdidaffectthe speedofadvance of the film and whether ornot the macroscopicallyobservable body offluidwould spreadover the precursor film. For example,squalaneon stainlesssteel exhibitedaprecursor film approx imately 20 Ain thicknessas measuredwith anellipsometer,anda leadingedge which.moved with.speedsintherange 0.03 to 1.0jjrn!3.

The precursor filmmay causesome confusionlUIto which leading edge, precur- sor or macroscopically observed,thecontactline shouldbe associatedwith.^fiIf one choosestheprecur sor's leadingedge,thentheproblemof thecontactline singular- itywill be encoun teredandcalculat ionsinvolvea detailedanalysis ofa fil:.i\.1with an L"known,anisotr opic st ress tensor.(Theanisotropyisdueentirelytothe fluid's motion. Ahuid atrest has an isotropi c stresstensor.)Analternativechoice would be to placethe contact line at its observedpositionand model theprecursorfilmas partof the surface.

15

2.6Lu bri cationAppr ox im ation s

InIn'IIlYtheoreticaltrea tments of the contact line instability, thefluid sheet callhi:modelledas athin filmand the lubrication approxima t ions areused. These approximation s allowa considerablesimplificationoftheNavier-Stck csmomentum eqnurlons. Thisdiscussion ofthelubr icationapproximat ionsfollowsthatof Ref. (21).

TileNavier-Stokea momentumequation foranincompressibleNewtonianfluidin whichthevelocityis a continuousfunction ofspatialcoordinatesis21

where

?:-

⁼tJ[+(jj,V)v,

(2 -4)

(2- 5) the material derivative, accountsforspatialaswellas time partial derivatives,pis thefluiddensity,pistheto talpressure,IJis theshearviscosity andjj

=

iu

+

jv

+

kw isthe velocity. Thepressure,p,could,asin thesystemstudiedinthis thesis,include hyd rosta tic terms. We take thefluidsheettoliein theX-IIplane.

TheRey nolds number,Re

=

pULIIJ' expresses th erelativemagnitudeofinertial andviscous forces. HereUis thesurface velocityandListhe filmbreadth. For smPlllRe, thetime derivatives(oraccelerations]aresm allcompared withthe viscous termsinvolvingV2ii,andsimilarly forsmallRethe inertialtermsp(v .V)ii arealso negligiblecomparedto the viscousterms.Applyingtheabovesimplificationsto Eq.

(2-4)gives the Stokesflowequations

(2-6) Furthersimplificati onofEq. (2- 6 )can bemade afterputt ing it into dimensionless formbynormalizing in-plane filmvelocities withrespec tto surfa ce velocityV,vertical

16

velocitybyUh/

c..

randII dista nces withresp ect tofilm breadt hL.:di~tan«"5with respectto filmthickness hand pressure withrespectto pU2•lV,the veloci tylu the : direct ion, is oforderh(Lsma llerthanuend vandunbe 1I1'gIectcd .'tIThellonnll1i1A"l1 r,jandkcomponents orEq. (2-6)thenbecome

and

8p'

ifl

=0,

(2- 711)

(2-7b)

(2-7c)

whereprimesindicatenorm alized quantities . SinceIi«:Land termst12/{);/l, iP/8y,2and

a

^{2/Oi 2areo ft he}same order,then termsinvolving82/81 2alLtI[il/ oy''}.

in Eqs.(2-7 )are much smallertha n termsinvolving82/82,2.Applyin g thl'!ICaim- plificat;ons toEqe,(2-7)andredimen sionalizingresu lts in the equation susedinthe lubricat ionapprox im ation,nal1"lCly

(2-Sa)

or, writingeach componentofEq.(2-8a)eltplicitly,

(2-81)

fortheidirectionand

8p 82"

8Y

=JJ"l);f 17

(2 -&)

forthejrllrectlonwith p::::>pix,y),If

=

II(X,y,z)and u

=

v(z,y,e].

2.7Theoretic al wor k of Refer ence1

Huppert! used thelubricat ion approximat ions to analyzetheflowof a thin, vis-

COilSfluid filmbefore the contact lineinsta bilityoccured. Inthis sectionrrederive Ids results. lt shouldbenotedhere thatthe lubricat ion approximationis valid only in the regionaway fromthecontact linesince ncar the contactlinefilm thickness aurlother quan tit ies varysignificantlyover distances~h. Thus the approximat ions made in thelast sectionarenotvalidnearthe contact line.

Assumingthefree surfaceto be Aat so thatsurfacetensioneffectsare negligible, andignoringcontact line effects, the pressure at a distancezdownthe slopedue to thefluidin the filmlying upslopeofxisp

=

-pgxsin a. Thusop/ox

=

-pgsinQ

whereQis the inclinationangleand 9 isthe accelerationdue to gravity. Using the lubricat ionapproximation,Eq.(2-8a),andthe aboveexpression for ap/ax,we find that the y·independent downslopemomentu m equat ionis

O= pg sina +I' O.z2

a'u

· (2- 9) Here,xis downslope coord inate,yis cross-slopecoordinateand z is thecoordinate normaltothe surface. FollowingHuppert, thecontac tline effectscanbeneglected.if theeffectofsurface tension issma llcomparedto gravityeffects,or morespecifically, ifthe Bond number B=pgL2/q:>I,whereois the surface tension andLisa characte rist iclengthscale

or

thecurrent.Inthe regionwherethe free surfaceis not stronglycurved.

r.e.

awayfromthecontact line, thecontributionto thepressure in thefluidduetosurfacetensionis negligiblecomparedtc thehydrostatic contribution.

However , ncarthecontact line the surface is significantlycurved andthe surface tension contribution isnolonger negligible.

18

UsingEq. (2-9)andthe boundaryconditio ns1I=0 at : =0 (theno-slip condition) and{Juj8z

=

0 atz

=

Ii(ta ngentialstress at the freesurfacemust he zero), the fluid velocity as a functionof height:: inthefluidsheet is

u

=

(pg~inojJl)

( h

-~)z, Theheight averag edfluidvelocity(1l)is given by

(2-10)

(2-11)

Theequatio nof continuity,aplal+'V.(pv)

=

0, whenaveragedoverthehd ght of thefluidsheet,becomes

(2-12) for a fluidsheet ofheighth(l:,t ).Substit ut ingEq.(2-11) into Eq.(2-12)gives the par tialdifferentialequation forthe unknown free sur faceh(x,!)tohe

(2- tl)

FromEq.(2-13) we seethathis constantalongcharacteristicsgivenby

(2-11)

IfweletIt

=

f(x },for exam ple, integrationofEq.(2-14) produ cesanequationfor the char act eristi cs

(2-15)

whereXoistheinitia l valueofthecha racteristic.Thereforethesolutionof Eq.(2-13) atlongtimesis

h

=

[P(z-xO)/ pg sino] I/2Cl/ 2

~(p / pg sin a) I/'lxl/2j - l/ 2 x;$:1"0, 19

(2-16,)

(2-16b)

whereh in Eq.(2-16b)is independentof initia lcondit ions.The equation expr essing conservationof massis

{'.II I

10 h(x,tjdx=A, (2-17)

whereXIIisthe valu e ofxat thefrontof thecurrent andAis the initial cross-sect ional PromEqs .(2-16b)and(2-17)we find thatsometimeafterthe release ofthe fluid

(2-(8)

ThusHupp er tpredict s that thelengthof thefilm growslike xn "'"tl/ 3.By substit ut ion orXnfromEq.(2-18)intoEq.(2-16b)wefind that the thickness of the fluid atthe frontof thefilmis

(2-19)

Since, fromEq.(2-10), thefluidvelocityincreaseswithheight,thesolut ionof Eq.(2-13)willdevelopinto ashock atlarget.Thisunphysicalresultis due to the neglectofsurfacetension, whichwilltend to smoot hthe free surfaceprofilenea r the now front .

Huppert also deri vesanexpression for thelengthsca leof the contactlineinetabil- ity.lie first finds the for m of the quasi-st eady two-dimensio nalfluid front byincludi ng surfacetensio nand matchin gthe tip onto the mainRow given byEq. (2-18).The additiontoEq.(2-1 3)ofthe termsduetosurface ten sionlead s to

(2 - 20)

Ncar thefluidfrontthedominant balanceisbetween the gra.vitationaland sur face tens ionterms inEq.(2-20), andthus

pg

sinQ ~:::::

^{(lj3)u h}

~~.

20

(2-21)

This givt'9

~/~

^...(length)3....(ol,)/(pgaiun], (2-221

Therefore a"typical"lcngthscaleis - (tTh/ pg5in(\')1/3.Tal-iliA:h

=

^hnattlwconti\("t linegivesalengthKillefor thecontactlineof (tThn/pgsinQ)1/ 3.Sinretheim~tability resultsfroma.competitionbet....eengravityamlsurfacetellsion.itisreasonableto identifythislengthscalewith the wavelengthof the instability,and thuslIUPIK-rt predictsthat thewavelength ofthe instabilityvarieslike(sina)-1/3.

2.8Th eore t icalwor kofReference13

Troianela1.¹³useHuppert's solution,Eq. (2-tGb),whichendsabruptly:,t Z..Xn,Eq. (2-18),forthefluldprofilefarfromthecontnctline (outer region).

Thenusingthelubricationapproximatio nswit hsurfacetension,the heightprofile, h(z,",),is obtain edfrom the solut ion of the height·avcrAgctlcontinuityequAtion

~+ V . h(O) = O.

Here,(11)=(u)i+{v)jwhere(1.1)and(u)areheighteveregedvelocities intheiand jdirections,respectively. They findasolution (ortheshapeof thesurfacencarthe contact line andmatch thi,to Huppert's solution.Inthelubrica tionapproximation thevelocityis givenby

p(V)=(h2/3)[pgsino!

+tTV xl .

(2-2') where the curvatureX isgivenbyX~(ffl h/8z 2

+

82h/ihi ) intheirapproximation.

Inordertoremove thecontactline singularity(seesection 2.4),theymatch the resultantfluidprofileto athin precursorfilm.Theythenperformalinearstability enelyeie cfthissolutio nandfindit tobe unsta bletospoatiallyperiodicdieturbancce.

I will now discusstheirprocedurein moredeta il.

21

For the unperturbed flow awayfrom thecontact line, the flow profile is given byHuppert 'scquancns,Eq.(2- 16b), which ends abruptly at.I'n given by Eq. (2- 18).Near the contac tline(the"inner" region) the flow profile will besmoothed by surfacetension.Troianet al.work in a reference frame movingwith the contactline with velocityVa

=

dx"ldt.Tofind the unperturbedprofile in the inner region they write the profileash(x,y,l)

=

hll(t)ll({,t )and requireH-0I as~

=

^.I'll--+co in orllerto match the solution (h_ hnlin the outer region. Thedimensionless length (=x/' ,wherexis distance alongimeasured fromthe contact lineand 1= h/ (3 Ca)I /Jisthecharacteristic length over whichsurfacetension competes with gravity,whereit is assumed that the capillarynumberCo :;: p.UO/u

«

1.Assuming small/le,a lime-independentsolution of Eqs. (2- 23)and(2- 24)determinesthe functionJJ{(,/)

=

IlO(O.

The boundarycondit ions are first that for~-000,the innerand outersolutions must match.i.e.,allderivatives ofH withrespect toxmust vanish andNO __ I, and second that near the contectiine the dynamicsmust take into account the singularity due to theno-slipboundarycondition(ii

=

0 ath

=

0). Troianer al.13 remove this singularity by matching their flow profile to a thin precursorfilm of thicknessbhll ,

whereb<:I.The equationthey find for the flow profileis

H'(1_83H,)~(1-b3) _(1 b).!-

o 8{3 (I-b)

+

H' (2-25)

Thesolutionof Eq.(2-25) givesthe profile shown in Fig.2·5 for three valuesof b.

They then performa linear stability analysis of theuniformprofile to small per- turbationsin the j-directicn ,neglectingterms of orderb

«

1.They define(

=

y/I and lookat perturbations withdimensionless wavevectcr q

=

Q/l.The positionof

22

;;.

Fig. 2-5. Unperturbed flow profilecalculatedfromEq.(2-25)for thr ee valuesofb.ua{unct ionof6{, which includes anarbitra.rysbift along the{ direction50tbat themaxima line up.Fromreference13.

the boundaryis displacedfrom

e=

0 to {=(B,where

(S((,I)=-A(() B(I ). (2 - 26)

IfA«()

=

COlI(q(),thentheregion-~/2<(q()<w/2is a sectionof the boundary perturbedin the forwarde-directicn,i.e.,a finger.The tlme-dep..ndent amplitude

or

the perturbationis assumedto be oftbeformB(t)

=

BO'f!JlrwhereT

=

J[Uo(t)/~dt is proportionaltothe distancetravelled.IfP>0, thenaB/o!>0and thefingerwill grow.

Solving numericallythe linearized continuityequation withthe appropriatehound- aryconditions, Troian dal.13_{findthe growthrate}fJto dependon the dimensionless wavenumberq==Qlland precursor film thicknessb as shown in Fig.2-6.Positive

23

valueso(Pindicate unstable modes. The profile isunstable(orq~0.9and even though thereis amaximumgrowth ratefor).

=

^{141,a wholerange}of wavenumben areunstable. Thisimpliesthatthedominantwavelength fortheinstab ility will be ).=14/,but perturbat ionswith ot herwavelengths will also grow.Thismay explain whythe experimentally observedinst abilityisnot perfectlyperiodic.I.2

Fig. 2·6{Jas a (undionofq,wherepositivevaluesoffJindicate unstablemodes. From reference13.

2.9 Theoreticalwork of Reference12

Goodwin aDd Hormy12 solve (orthe two-dimensionalflow field and Iree surface shape in the vicinityof the cont act line. They a.!Isumea slow moving, visCOUINew- tonianfluid advancingon a dry,inclinedsolidplane under the influence of gravity.

They fint recap Huppert 's)work with the lubricationapproximation neglecting sur- face tensionand find that thisresults in ashock-type solution.In ordertoresolve this shocksolution theyincludethe first ordereffects ofsurface tensionin the lubrication theory but obtain a different unphys ical result .Then Goodwinand Homsyusea

24

differentscalingin ordertoprescribeitcontact angleboundary conditionbut filld thisresults in an unrealist iccontactline velocity.Finally the)'formulatea Stokes flowproblem whichpermitssatisfaction of acontactangle boundary condition.,\1·

lowing slip nearthe contactline removesthecontactlinesingular ity.TIll'Yshowthat the innerregion is governedby Stokes flow while inthe outer regiontilt'nowiswell describedby lubricationtheory.

Following the procedureofIluppcrt,1 Goodwin andIIoll1sy l2ca lculate, intht' lubricationapproximation without surfacetension.that thelocation of the leading edge priortothe instabilitygoeslikexn(t ) :::::(3/2)2/3ll/3 ,wit horscalcdlike z....A/li' whereh^tis thecharacteristi c lengthnorma l to theslope.and the thicknessofthe fluid atXflishn{t)~(3/ 2) 1/3t- l/ 3. withhscaled by h^t•Theseequationsall:similar to Eqs. (2-16b) and (2-18)above derivedbyHuppert. Sincesurfacetension was neglected. a shock-typesolutionisobtainedwiththe front locat edat.r'I'

Next they rescale the problem to includethefirstorder effectsorsllrracetension whileretainingthe lubri cat ion approximations.Theyworkin a referenceframemov- ingwiththe average velocity ofthe contactline.They findthepositio n or therr(~

surface to be describedby the different ial equation

(2-27) subject to the bounda rycondi tionsh

=

0 atx :::0,hr...-00asx...0andh ...I asx_-00 .in thelimith...0, the rate ofcha nge of curvature ,(j1h/8x3,must be unbou ndedinorderto satisfyEq.(2- 27). Sinceit is alsorequircd5that the slope or the freesurface beunbounded at the contactline. a solution is not easilyfound.

Goodwinand Homsyfound that a contactanglebounda rycondit ioncouldnot besatisfiedusingeitherof theabove cases, and so tried a new set of scalinge which

25

would allowthemtospecifysuch a condition,Theywere also interested in how the

1I{J·~lipconditionaffcctlod their ability to modelthe flow near the conta ctlineso a

gt!nl'talizedslipboundary co nditio n was introduced,Goodwin andHomsy again used the lubricationapproximationincluding the firstordereffect sof surface tensionin a referenceframemoving withtheaverage velocityofthecontac tline.

Theno-slip boundaryconditionwas replacedbyone in which theslipvelocityis proportionalto the product ofthe velocitygradient at the wall anda funct ion,S(h), ortill!fluidthickness.Followingthesameprocedu re asfor the preceedingcase, they findthedifferentialequationfortheshape of the freesurface to be

~= ~-l

8x' S(h)·h+h' . (2-28)

Thesingularity at theorigin can beremoved by specifyingS(h ) ,." O(h- I)ash--tO.

However, sinceS(h)isproportionaltothe slip velocity, thisint rod uces asingularity intheslip velocity.

Theabove threeattemptsto obtaina solutionatthe contactlineusingthe lubrl- cation approximationsallcontaina singularity.Thesingularityiseither inthera te or change orcurvat ure orthefree surface with respectto positionatthe contactline for boththeslipand no-slipmodelsorinthe slipvelocityif aslip modelischosen. Thus theyconcludethat thelub ricat ion approximationsareof limitedvalue in modelling the nowin the region or the contactline,

Finally, Goodwin andHomsy deriveand solvenumericallyStokes 001'.'equations Ior the region near the contactline with a contactlineboundarycondition. The Stokes flow equations,

(2-29) 28

are simplificationsofthe NuK-r·Stokes equations fora_\'i~usduid....ith\"t'rysma.ll Reynoldsnumber.The pressurepinEq,(2-29)indudnsrav;tytermswhileGoodwin andHomsywritethesravitalionaltermsseparately.The Sto\.;"1\0...f'quationsdilfer fromthelubrin tion equationsintbt'Vp=&p/8ri+iJp/8yj+8p/iJ:t and'V'io'=

ifliJ/8z'+ffliil8y'+8'iJ/lJ:'rerStokesflow,while inthe lubricationappro:timation 'ilp=8p18zi+8pllJlIja.ndV'iiisappro:timat.edby;Pula:'.Inthisappro:t;mation an inn" solutioncanbeobtained.GoodwinandHonu )'lindsolutionsIcrthefree surface profileover a.ra.ngeof capillu ynumber,contactangleandindinationangle of 0.01 ::;Ca:50.164,W:s:~::;^{';00and 1.5°:5Q}:5135°.Asanexemplc,Fig.

2-1illustrates the calculatedfreesurfaceshape for a contactilngleofiO°,inclination angleo{ .j5° and Ca =0.03,0.10andO,IHfcrccrvesa.b andcrespecnvely,

Goodwin andHomsylindtha.tnearthecontactlinea humporbulge or Ruid ec- curs,the sizeof wbichinereeseewithdecreasingcapillarynumber,increasingcontacl angleand increasinginclinationangle.They demonstrate tha.tthebulgeisnotdue tothecontactline singularitybutistheresult or an ;nten.etiunbetweeninterfaci&1 forces and andthestressfieldinsidethe fluid.

27

Ch ap te r 3 ApparatusandExperimentalTechnique

3.1Apparatus

The experimentalapparatusis conceptua llysimilar to thatused byHuppert ' andSilviandDussan V.'!A sheet ofplexiglass orglass122cmin thedownslope (xl direction,91cm inthecrossslope(y)direct iona.nd 1.3cm thick wasclamped ina rigid aluminum frame.The angle of inclinati onQwas adjustable;themeasurements here were takenover a range of0⁰<Q<55°.Graphpaperwitha.grid oftwoinchsquares markedon itwas placedunder the experimentalsurfaceto facilitatemeasurementof the contactline position.Analuminumgatespanningthe fullwidth ofthesurface was placed ncar thetopoftheslope.Foamrubberweathe rstrippingwas attached alongthebottomofthegate toprovide a seal to preventleakage of thefluidpriorto openingthegate. Theseal alsoprevented the gate fromscratchingthesurface. The gate was hingedat itsupper endso that its axis of rotationwasin the ydirect ion and approximately IOemabove thesurface.Thus when thegatewasopenedit swung out and upallowing thefluidto flowdown theslope.Analternate methodofopeningthe gatewasto pullitdirectly upfrom the surface, butthisusually resulted injanuning ofthegalewhichinturncauseduneven release or the fluid.

Theexperiments wererecorded using a BurleIndustr iesCCOvideocameracon- nectedto avideocassette recorder.Thecamerawas mountedapproximately IOOem to125cmaboveandperpendicularto the surface.A12.5mmfocallengthlens gave thecamera a fieldofview wide enough to include thefullwidt hof theexperimental surface. A BurleIndustriesvideo monitor wasusedtoaccuratelypositionthe camera , towatch each experimentfrom thecamera'sviewpointandtoanalyze therecorded experiments. ALaser 286j2Lpersonal computerwas employed fordoingfitstothe

28

dataasdiscussedbelow.

3.2Flu ids

Experimentswere carriedoutusingthreefluids:glycerin22,'heavy'mineraloil (HMO)23 and'light ' mineraloil(LMO).24Thedensity,viscosity,surfacetensionand stat ic contact angleof eachfluid weremeasuredusingthe methodsdiscussedbelow andare shown in 'fableI.

The nrstthreeofthese are thephysicalproperties of the fluidswhich enterinto the theoretical treatmentsdiscussed in theprevious chapter,andthe fourth,as argued by Silvi andDussanV2,isimport antin determiningthepattern tha tdevelopsafter the instability.Thefluid propertiesof HMO andLMO arethe sameexcept for the viscosity, so a comparisonofthe flowdevelopmentforthese twoHuldawill showthe effect,if any,of viscosity.

Thedensity ofeachfluidwas easily determinedusing a MettlerAE'260electronic balancebaving aresolution of.0001g.The mass of a known volumeof tluiddivided by its volumegave thedensity.

A GilmontInstruments size 3falling ball viscosimeterwas used alongwitha stop- watch to measure eachfluid's viscosity.Theviscosimeterwasfilled witha Iluidand the time of descentthrough a givenheightofa ball ofknown density was measured.

The equationp=K(Pb- PJ)tgave the viscosity in(s/em·^s)·10- 2 (centipoise).

where uisthe viscosity,Kis a constant equalto O.63em2/s'lforthisviscosimeter,1'6 andPJ are the densitiesins/em'!>of theball and fluidrespectivelyandtis the time ofdescentin seconds.

Thesurface tension was measured using a Cenco 70545torsion wire tensiometer.

Thisinstrumentconsistsofa torsion arm damped to themiddle ofa torsionwire.A

ss

TableI.Properties of fluids studied

static contact stat iccontact surfacetension viscosity density angle [plcxiglesa] angle[glass]

(1(dyn/cm) I.l(g/cms) p(g/ cm 3)

o

[degrees] 8(degrees)

Glycerin 59± I ll.l±.l 1.26±.01 6O±2 5O±2

HMO 34± 1 1.5±.1 O.81± .Ol 14 ± 2 16±2

LM O 32±1 O.S ±.1 0.85±.01 14±2 16±2

30

platl num-Iridlum wirering of5.992em circumferencewas connected to the otherend ofthetorsion.arm. Afterthe instrumentwascalibratedwithknown masses,the ring was immersedin thefluidbeing measured.Thetorsionon thewirewas increasedand the fluid sur face lowered simult aneously so that the torsion arm remainedat its zero posrtion. The scale readingwastakenwhenthe ring brokefree ofthefluidsurface.

Fro mthecalibrat ioncurveandacorrection based onthe fluiddensitythesurface tens ionwascalculat ed.

The staticconta ctangle was measuredbyplacingadropof fluid onahorizonta l piece of glassor plexiglass. The videocamerawas then positioned so that its optic axis wasapproxima telyalignedwit htheglassorplexiglaeasurface andthe middleofthe drop .Then a framegrabber ,which was used to controlthe ca mera by computer, took apictu re. Fromthe printout ofthe picture thestaticcontact angle was measured.

An interesting observationconcerning thestat iccontact angle isthatover a period ofseveralhou rs it decreased.Thisdecreasewas smallfor HMOon both glassana plexlgleas,forwhichitchanged Ircm>-17°to '" 14° over2hours,but for glycerin onglass[plexiglass giving similarresults)itwentfrome-75°to '"50°in5minutes, to -- 43°in 10 minutes andto'"20° in 2.5hours. Thislarge changein the static cont act angle for glycerin is due to theabsorption by theglycerinof moisturefrom the air.Duringexperiments, the fluidWMleft behind the gate forapproximately 5 minutes and for this reason the valuecfthe staticcontact angle forglycerin given in TableIis the valueafter 5 minutesexposed tothe air.

3.3 Expe r imentalTech niq ue

All experime ntswereperformed atroomtempera tu re which was not especially controlled and variedbetween 21°Cand 27°Coverthecourseofthlework.

31

Beforeeach experimentthesolidsurface was carefullycleanedtoensureuniform

~urfaceconditions.The cleaningprocedurewas as follows.After most ofthe fluid from a previousrun had been removed,thesurface waswashed twice usingdetergent in water and rinsedeach timewit h cleanwater. Thena commercial glasscleaner, Windex,wasapplied to the surface which was then wiped cleanwit hpapertowel or achamois.Followingthis thesurfacedriedquickly. Wipingthe dry surfacewithdry papertowel crea ted astaticchargewhichhad a dramaticandadverse effectonthe now;carewastaken toavoidthis.

Following thecleaningprocedure, the surfacewas inclined to theappropriat e angle anda.levelwasused to makesureit waslevel intheydirection.Thevideo camera wassetperpendicular to the surfaceand positionedso that thegatewas at the upperendofthevideomonitorscreen.Thena known volumeof fluid(250cm3 fortheexperimentsreportedhere)waspouredbehind thegateand left standing fora sufficientlylong timethatit was evenlydistributedand stationary.Priorto opening the gate , the releventparametersof the experiment such as theexperimenta lsurface, angleof inclinati on,typeand amountof fluid and room temperat ure were recorded on the videotape.Thecamera was focussed.on thesurface,thenthe gate wasopened.

and thefluid floweddowntheslope with aninitiallystraight contactline. Carewas takingopening thegatesoastokeepitstraight and not allowonesideto open beforethe other whichwould resultin an uneven releaseofthe fluid.Itwas also important not to open the gatetoofastor the gate wouldthrow drops ofthe fluid several centimetersdown the deansurface.Since the fluidsused were dear, only a shadow of thecontact linecould beseen.Thecurved surfaceofthe fluid nearthe contact line caused thelightpassingthroughitto be refractedinsuch a way as to produceashadow ofthe conta ctlineonthegridbelow.Inorde rtohaveonlyone

32

cont~ctline shadow,onlyone setof ceilinglightswas lefton in the lab during each run.The lightswere sufficientlyfar away so as to producejUltone shadow.In afew laterexperimentswith other fluids,asingle light was placedin (rontof the slope.

This producedaverydistinctshadowofthe contactline althoughfingersnearthe sides ofthe slope hadtheir shadows displacedslightlygiving the impression ofaluger distancebet weenthe fingers.Theactualfingerscouldbe5t'f1\due to reflectionsfrom the finger tips,so this latt erproblemwaseasily cvereome.

ExperimentsusingHMOwere performedover theranges2° :SQ:S 21°and 2° ::;^Q :S20°for the plexiglassandglass surfaces respectively whilethe ranges for glycerin were4°::;Q ::;30° and4°:5Q::; 54° fortheplexiglassand glass surfaces respectively.For inclinationangleslowerthan thelower limitforeach the contact linedeveloped into poorfingerswhichtendedtoflow inthecrossslopeas wellasthe downslop e directionsand satisfact orymeasurements wer enotposaible.For indi nati onangles larger than the upperlimit foreach the fluidwouldsplashdown overthesurfacemuchlikeawave breakingon abeach ,resulting in a nonuniform flow front .

3.04RawnataGa th ering

The raw data wu obtainedby direct measurement ofthe contactlineposition fromthe videomonitor duringreplayofarecordedexperimentalrun.Theposition asa fundionof limeofthreeorfourfingersandtroughsWIl.Smeasuredforeech run.

The VC Rused here incorporated a real-time counte rsothatthevideotapecould be moved ahead anappropriate numberofseconds thenstopped totake measu rements.

The zero timeof each runwutaken asthe time at whichthe contact line shadowfirst appeared from under theopened gate. Thefingerand trough measurements started

33

atthe pointat whichthecontact line began to show the instability.Also measured werethe distancedownslope,z", at whichthecontactline becameunsta ble, tbefin&er width, i,ta.ken as theCullwidth a.t half the fingerlen!th and the a.veragewavelen&tb, I,oCthefingerpallern.The fluid at the edges of theflow was held back by the"'ath al.each edgeoCthesurface causingthe contactlinetobecurved inthatarea. For thisreasonthelingers closesttoeachrotewere not includedinthe calculationof the average wavelength.

34

Chapter 4 Results 4.1General Observations

Experimentswith allthree fluidswere performed onthe plexiglaas surface.while only glycerin andHMO were usedon theglaeasurface.In allcasestherivulet(parallel sided finger) patte rn wasobserved.Since I took substa ntially moredata for liMO thanfor LMO,I will quote onlydatafor HMOexcept incases weretheLMOdata is of importancealthou.;:hdat a for LMOis shown onsome ofthe figuresbelow.In the case of glycerin,thetroughsappearedtostop shortlyafterthe contactline became unst ablewhile for HMO,thetroughs continued tomove veryslowlydownhill.The observedfingeringpatte rn wasnot perfectly period ic, nor werethefinger lengths perfectly uniform.Several preliminary experiment al runsweretriedwith siliconeoil onglass,for whichtheunstable contact line developed into thesawtooth patternin whichboththe fingersandtroughscontinuedtomove downhill,but at differentrates.

In thiscase the fingering patt ern was fairlyperiodic.The fingerlengthswere more uniformthanforthe rivulet pattern butst ill not perfectlyuniform .

The fluid at the edgesofthe flowwas heldbackbythewallsat theedgesor the surface. Thisperturbation of the contactline resulted in fingers formingat the edges.

AlthoughSchwartz9 found numerically that thisedge perturbationcausedfingering as it propagatedinward along thecontactllne,all experimentshere indicate thatthe entire contactline becomes unstable simultaneously. Duetothesurface'slarge widt h (91cm),any edge effects are expectedtobe small in themiddle regionof thesurface.

4.2 Before the Instability

Huppert! predicted that ,prior to the instabilit y,the positionofthe uniform contactlineshould advanceliket1/3; his measurementsagreedwit h thisprediction.

35

Dueto this,Ianalyzedmy datafrombefore theinstabilityby fittingtheposition to a powerlaw in time,i.e.,

(4-1)

Fitsto Eq. (4-1)were performedonlyforCr:514⁰forglycerin and0':58⁰forHMO.

At higheranglesthetime before theinstabilitywastoo shortfor enough position measurementstobe made and mea ningful fitsto Eq.(4-1)werenot possible. The dat a waswell described,forthemostpart , by Eq.(4-1 )buttheexponenttJowas substa ntia lly largerthanthevalueof1/3predictedbyHuppert .Infact the exponents, shownas opensymbolsinFigs. 4·3,are inreasonablygood agreement withthe exponentscharacteriz ingthe growthofthefingerswhichdevelopaftertheinsta bility.

similarly,the valuesofthe amplit udeAoofEq. (4-1),shown asopensymbolsin Figs.

4.4, agreewithinerrorwiththecorrespondingvalue forthe flowafterthe instab ility.

Fits todata from two differentruns havingthe sameexperime ntal parameters producedresults which wereequal withinexperimenta l error,thusindicat ingthat theflow isreproducible.

The discrepency between myresultsand the11/3behaviourfoundbyHuppert canbe understoodfrom Fig. 4-1,whichshows the front positionas afunction of time forHMOon plexiglasswith0'

=

8⁰,bothbeforeand aft er the instabilit y.The pre-insta bilitydata,shown asopen symbols,seemsto approachat1/ 3behaviour,but only afteratransientperiodin whichthe front advancesmorequickly.

This behaviour is!lCCIlonlyat the smallest angles. Atlarger angles,a~50 for HMO and0'~8⁰for glycerin ,theinstab ilityOCCUT9beforethetransient hasrelaxed and theapproachtot^l/ 3behaviourisnot observed.

Thefilmthickness at theonsetoftheinstability was notmeasuredquantit atively.

36

oo

'

.

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,

.

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31

HoweverroughestimatesweremadebasedonHup~rt"result thatthe fluidthicknes, atthe (rontofthe,hocksolution,Eq.(2-18),IesivenbyEq.(2-1 9).My data(or

Znsuggestthatthecontactline becomesunstablewhen thefluidthins toacertain fluid-dependent thickness ontheorderofthreemillimeters.Thls criticalthicknes,i, Itn~O.33an (or glycerinandItn ::::::0.27em (orHMO.

4.3Aft ertheIn s t a b ili t y

Since I amunawareofanytheoret icalpredict ions{orthe growt hof thedeveloping fingersbeyond theregimewherelinearstability analysis is valid,25I analyzedmydata interms ofempiricalfittingfunctions. The downslopetipposition after the instability,

%d,was welldescribed,within experimentalerror,forallfluidson both substratesat allangles a bya powerlawintime:

(4-2)

wheretheamplitud eA,the exponentPand the time origintowereall used asIree paramete".AtypicalfittoEq.(4-2) isshowDinFig.4·2for HMO on plexiglass at S·,

Thetrough position,Z'I>wasfitt edtothefunct ion

(4-3)

withB, C.Dand., as free parameters.Atypicalfitof thisfuncti onto experimental dalaisalsoshownin Fig. 4-2.

Otherfittingfunctionsweretried butgaveless satisfacto ry descriptionsofthe data. In particular,powerlawfibtothe trough dataweretried,buttheygave satisfactoryfitsonly whenthetroughscont inuedto moved withsufficieIltlyhigh

38

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speeds,e.g.,the power lawfit would not workforglycerin troughsbutdid work for some HMOtroughs.

Figs. 4-3(a) to (d) show the tip growth exponentf3as a function ofsin o for glycerinand both oilson plexiglass, (a) and (b),and for glycerin and HMOon glus, (c)and(d).

Each point on these plots (andsimilarplotsbelow)representsthe averageof resultsfrom fits to a number(three for verylow anglesandfour for otherangles) of fingers(ortroughs) froma particularrun..The errorbarsrepresentthe rangeof valuesobtained from all of the 6tsto thatdata set.The range of valuesobtainedfor different runs atthe same angle is rough lythesame indicatingthat any variabilityin surfacecondition betweenrunsis no greater than thevariabilityacrossthe slopefor asingle run.

For both fluids,glycerinand HMO , onbothsubstrates,f3isessentiallyinde- pendentof0withinexperimentaluncertainties.The mean valueofIJfor the pled- glasssurfaceis 0.65±.04 for glycerinand 0.52±.05 forHMO; forthe glasssurface fJ =O.55 ±.05 forglycerinand0,48 ±.OIforHMO. (Errorshereand elsewhereinthis thesis are statisticalstandard deviationa.]These valuesoff3are alsoshownin Table II.My vaJue for glycerinfingerson plexiglass isin agreement,withinexperimental error, withtheexponent01'.6reportedby Huppertl.fJdoes showa slighttendency to decreasewith increasing0forHMO on bothsurfaces andfor glycerinonglass but it is difficulttotell jfthis trendisreal from my data.The values ofIJand

130,

where valuesoff30are re presented as open symbols in Figs. 4-3,appeartobe equa lwithin experimentaluncertaintyin mostcases.

Figs.4·4showthedependence offinger growthamplitudeAonQforplexiglaee, 40

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Fig.4~3(a).The solidsymbolsshowthe exponent{JofEq. (4-2) as a.lunctian ofjinQ(orglycerin onplexigla.ss. The dashedlineashowtheavera«efor each case.Thedatapointsindicatethemeanva.luesandtheerrorbarsthe r<lOgefromfilato aeveral tinge",AteKb a.cgle.The ope nsymbolsshow the exponent{1ijforthe flowbeforethe iostability,determinedfrom Eq.(4-1).Forc1uity, error ban forthesepointsare notshown;theyIHCsimiliouin size to those (ortheotherdatL

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