On the use of a friction model in a Volume of Fluid solver for the simulation of dynamic contact lines

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5929

To cite this version:

Si Hadj Mohand, Hacene

and Hoang, H. and Galliero, G. and Legendre,

Dominique

On the use of a friction model in a Volume of Fluid solver for the

simulation of dynamic contact lines.

(2019) Journal of Computational Physics, 393.

29-45. ISSN 0021-9991 .

Official URL:

https://doi.org/10.1016/j.jcp.2019.05.005

On

the

use

of

a

friction

model

in

a

Volume

of

Fluid

solver

for

the

simulation

of

dynamic

contact

lines

H. Si Hadj Mohand

a

,

b

,

H. Hoang

c

,

G. Galliero

b

,

D. Legendre

a

,

∗

a_{InstitutdeMécaniquedesFluidesdeToulouse(IMFT),UniversitédeToulouse,CNRS- Toulouse,France}

b_{CNRS/TOTAL/Univ.Pau&PaysAdour/E2SUPPA,LaboratoiredesFluidesComplexesetleursRéservoirs-IPRA,UMR5150,64000,PAU,France} c_{InstituteofFundamentalandAppliedSciences,DuyTanUniversity,10CTranNhatDuatStreet,District1,HoChiMinhCity700000,VietNam}

a

b

s

t

r

a

c

t

Keywords: Wetting Volumeoffluid Couetteflow Movingcontactangle

WeconsidertheimplementationofafrictioncontactanglemodelinaNavier-Stokes VoF-CSFsolver forthe simulation ofmoving contact lines at thenano-scale. A liquid-liquid interfaceconfined in aCouette flowgenerated bytwo solid wallsmoving atthe same velocityinoppositedirectionsisconsideredtodiscusstherelevanceofthefrictionmodel. The simulationsare comparedwith areferencecase obtainedusing MDsimulationsby Qianetal.[46].WeshowthattheNavierStokessimulationsareabletoreproducetheMD simulationsforboththeinterfaceshapeandthevelocityfield.Theappropriatecontactline frictionisfoundtobegridconvergentandofthesameorderasthefrictionmeasuredin MDsimulations.Adetailedinvestigationoftheinterfaceshapehasrevealedanauto-similar linearprofileinthecenterofthechannel.Closetothewalltheinterfaceshapefollowsthe classicalLogevolutiongivenbytheCoxrelationdespitethewallconfinement.

1. Introduction

Dynamic wetting is encountered in various industrial processes and natural systems [51,3], such as coating [16,24], agrochemicaltechnology[61],printingtechnology[69],roadapplications[64] andwelding[26].However,thesimulationof moving contactlinesisstill challengingfornumericalsimulation.Thisismainly duetotheintrinsicmulti-scalenatureof thecontactlineconnectingthethreephases. Inparticular,theclassicalfluiddynamicstheory,i.e.Navier-Stokesequations coupled withno slipboundary conditions,is not consistent to describe the fluidsdynamic in the contactlineregion. It predicts an infinite viscous stress andpressure atthe contactline, known asthecontactline singularity [32,21].During thelastdecades,alotofworkhasbeencarriedouttoremovethissingularityandtoaccuratelymodelthemovingcontact line [27,7,10,57]. Both molecular and hydrodynamic approaches have been considered. To tackle the multi-scale aspect of the moving contact line problem, i.e. to be able to interpret experiments [37] or to perform numerical simulations [2,20,59,39,41] of drop size ofinterest for applications, the interface at large scale is connected to molecular effects at the contactline. The apparent contactanglemeasured at a hydrodynamic or macroscopicdistance LM fromthe contact lineislinked tothecontactlinevelocityandthemicroscopic angle

θ

m measured atamicroscopicdistance Lm (Cox[14], Voinov[65]).

θ

m isusually supposed to beconstant andiscommonlyset equalto theequilibrium(Young) contactangle

θ

Y [22].However,ithasbeenputinevidencethatthemicroscopicangle

θ

m maydifferfromthestaticangle

θ

Y whenthe

*

Correspondingauthor.

contactline ismoving. Toaddressthisproblem, the MolecularKineticTheory(MKT)gives themicroscopic contactangle fora movingcontactlineconsidering surfacedisplacements orjumpoffluidmoleculesfromadsorption siteslocalizedat the wall [8]. Inthismodelbased onthe theory ofrateprocesses[29], thecontactline fluctuatesmicroscopically around its meanpositionandfluidmoleculeslocatedatthewallandsufficientlyclosetotheinterfacejumpfromadsorptionsites localizedateachsideofthecontactline.Whenthecontactlineisstaticatitsequilibriumposition,themeanrateofjumps is the same in both forward andbackward directions. However, when the contact lineis moving, the mean jump rates becomeasymmetricandyieldsanexcessofjumpenergyinthemovingdirection.Thisexcessofenergyiscompensatedby thestressassociatedtothedifferencebetweenthestaticandthemicroscopiccontactangles(thesocalleduncompensated Youngstress)andprovidestheexpressionforthecontactlinevelocityrelativetothewallas:

Vcl

=

2K0

λsinh



γ

(cosθ

Y

−

cos

θ

m

)

2nkBT



(1)

where

λ

andK0arethejumplengthandequilibriumfrequency,respectively,kB istheBoltzmannconstant,n isthedensity ofadsorption sitesonthesolidsurface usuallyconsidered equalto1

/λ

2 andT isthetemperature. Fromthisrelationthe microscopicmovingcontactangle

θ

mcanbeexpressedasafunctionoftheratiobetweenthecontactlinevelocityandthe jump velocity K0

λ

of themolecules.Whenthe contactlinevelocityis muchsmallerthan thejump velocity,the relation canbelinearizedas

cos

θ

Y

−

cos

θ

m

=

ξ

γ

Vcl (2)

withtheintroductionofafrictioncoefficient

ξ

atthecontactlinethatcanbeexpressedas[52]:

ξ

=

kBT K0

λ

3

=

μν

L

λ

3 exp



−

γ

λ

2

(1

+

cos

θ

Y

)

kBT



(3)

with

ν

L theflowunitvolumeand

μ

thedynamicviscosityofthefluid.

AsignificantnumberofexperimentshasbeenconductedtoverifythepredictionsoftheMKTapproaches,andtheir vari-ants,todescribemoving contactlines[31,60,33,17,30].However,classicalopticaltechniquesprovideonlyglobalquantities suchasthemacroscopiccontactangle,thecontactlinevelocityandtheradiusofspreadingdroplets[55,48,66].Thus,access to localquantitiesclosetothecontactline(suchasthemicroscopiccontactangle)bythesetechniquesislimitedandthe validationoftheapproachesissoindirect.Recently,theAtomicForceMicroscopy(AFM)hasbeenappliedtoaddress mov-ingcontactlineproblemsandthemicroscopiccontactangleandtheinterfaceshapehavebeenmeasuredatthemolecular scaleinwellcontrolledconditions[13].Interestingly,theseAFMmeasurementsrevealedthatthemicroscopiccontactangle iscloselyrelatedtothecontactlinevelocityasexpectedfromtheMKT[13,18].

Acomplementaryroutetohaveaccesstolocalinformationregardingthemovingcontactlineprobleminmodelsystems is to useMolecular Dynamics(MD) simulations [35,62,46,15,6,54,40]. Thesemolecular scale simulations are nevertheless restricted tothe descriptionof nano-systems.Theyhave shownagood abilityto capturethecomplex interfacial and in-termolecular effectsoccurringwithin thecontactline region.Theyare usefultotest anddevelop modelsto describe the moving contactline behavior, e.g. MDsimulations havebeen used to improve the MKTmodel [15,6,54]. Similarly, Qian andco-workers [46,47],on thebasis ofMDsimulations forliquid-liquidinterfaces incontactwithan idealsolidsurface, developed theGeneralized NavierBoundary Condition(GNBC) to describe the(slipping)boundary conditionsatthe con-tactlineinacontinuum approach. Theydemonstratedthat theuncompensatedYoungstressinapartiallywetting caseis counterbalanced byviscous andfriction dissipations,the latterbeingexpressedasthecontactlinevelocity multipliedby a frictioncoefficient. IntheGNBCmodel,thecontactlinevelocity relativetothe wallisrelatedtotheunbalancedYoung stressthroughtherelation[46,47]:

β

Vcl

= −

μ

∂

V

∂

n

−

σ

Y

nx (4)

where

β

isaslipcoefficientand

−

σ

Y

nxistheuncompensatedYoungstress,whoseintegrationovertheinterfacesatisfies

−



int

σ

_nxYdx

=

γ

(cos

θ

m

−

cos

θ

Y

)

(5)

Then,RenandE[49] havereformulatedtheGNBCrelationtoproposethefollowingrelationfortheunbalancedYoungstress

cos

θ

m

−

cos

θ

Y

=

β

cl

γ

Vcl (6)

where

β

clisaneffectivefrictioncoefficientatthecontactlineaccountingforboththecontactlinefrictionandtheviscous shear. Note that under the presentform the GNBC modelappears similar to the MKT relation (see Eq. (2)) butstrictly

Fig. 1. Couetteflowconfigurationconsideredhere.ThetwowallsaremovingwithoppositevelocitiesofmagnitudeU .Thereportedshapeoftheinterface correspondstoasteadystablesolution.

speakingtheMKTonlyconsidersthecontactlinefrictionwhiletheGNBCconsidersthecontactlinefrictionandtheviscous shearfriction.TheGNBCconditionhasbeensuccessfullyimplementedasboundaryconditionincontinuumsimulationsby usingthediffuseinterface(phasefield)methodbasedontheCahnHilliardmodel[47,19,5,56,4,42,70],byusingtheArbitrary Lagrangian-Eulerianmethodcombinedwitha finiteelementmethod[28] andbyusingthefront trackingmethod[67,68]. Tothebestofourknowledgeit hasnotyetbeenconsideredina VolumeofFluid(VoF)- Continuum SurfaceForce (CSF) formulation.

The aimofthiswork isto considertheuse ofafriction modelforthe simulationofnano-scale moving contactlines usinga VoFapproach.Forthispurposewe haveselectedthesteadynano two-phase Couetteflows consideredby [46,49] becauseaccurateMDresultsareavailable foradirectcomparison.The maininterestofthisflowconfigurationisthatthe slipatthecontactlineissteadyandperfectlycontrolled.Indeed,therelativecontactlinevelocity Vcl betweenthecontact lineandthemovingsolidwallisequaltothedrivingvelocityU ofthesolidwall.

Thepaperisorganizedasfollows.Thenumericalset-upisdescribedinsection2.TheVoF-CSFNavier-Stokessolverused forthisstudyispresentedinsection3.ThefrictioncontactanglemodelanditsimplementationinsideaVoF-CSFapproach arediscussedinsection4.Aseriesofadditionalvalidationsofthecodeisproposedinsection5.Thenumericalsimulations ofthenano Couette flowsconsidered inthisstudyaredetailedinsection 6andthe descriptionoftheinterface shapeis reportedinsection7.

2. Numerical set-up

A VoF-CSF solver with no interface reconstruction is used for the resolution of the Navier-Stokes equations and the models described insection 4 are consideredas boundarycondition forthe contactline. We aimto (i) demonstratethe abilityofaVoF-CSFmethodtosimulateflowsatthenano-scaleand(ii)todiscusstheperformancesofthedifferentmodels when used for predictive simulation of moving contact lines. To make relevant the comparison andthe discussion, we consider the two-phase Couette flow shown in Fig. 1. A cartesian systemof coordinates (ex, ey) is used. The walls are paralleltothex-directionandseparatedbythedistance H .TheyaremovingwithoppositeconstantspeedofmagnitudeU

alongthex-direction.Twoimmiscibleliquidsofsamedensity

ρ

=

ρ

1

=

ρ

2 andsameviscosity

μ

=

μ

1

=

μ

2 areconsidered. Theinterfacebetweenthetwofluidsisdescribed withaconstantsurface tension

γ

.Aplane interfaceisinitiallyimposed perpendiculartothewallsmakinganinitialcontactangleof90o.Thisflowisverydifficulttocarryoutexperimentally[63] butithasthegreatadvantagetoprovidea2Dsteadyinterfaceshapewithmovingcontactlinesataconstantvelocity.The dynamiccontactanglewiththewalldiffersslightlyfromthestaticcontactangle,asshownby MDsimulations[46,49].In addition,thestabilityofthisflowhasbeenstudied[34,50,25] andthecorrespondingtransitioncriteriawillbeusedforthe validationofthesimulations.

TheMDresultsofQianetal.[46] willbeusedasthereferenceforthecomparisonwiththeVoF-CSFsimulations.Inthe MDsimulationsofQianetal.[46],interactionsbetweenfluidmoleculesofmassm aremodeledbyamodifiedLennard-Jones potential of the form 4





(

σ

/

r

)

12

− δ(

σ

/

r

)

6



, where r is the distance between two molecules,



and

σ

are the energy scaleandtherangeofinteraction, respectively.

δ

=

1 forlikemoleculesand

δ

= −

1 forunlike-fluidmolecules.Fluid-solid molecular interactionshavebeendescribed usingthesamemodified Lennard-Jonespotential withthe energyscale



f s

=

1

.

16



,therangeofinteraction

σ

f s

=

1

.

04

σ

and

δ

= δ

f s with

δ

f sdefinedas:

δ

f s

=

1 forbothfluidsinthesymmetriccase while

δ

f s

=

1 for one fluid and

δ

f s

=

0

.

7 for the other one in the asymmetriccase. Inthe following all parameters are madedimensionlessusingthemassm ofthefluidmolecules,therangeofinteraction

σ

andtheenergyscale



.Basedon thisnormalization, thenondimensionalparametersoftheproblemconsideredinthisstudyare H

=

13

.

6,

ρ

1

=

ρ

2

=

0

.

81,

μ

1

=

μ

2

=

1

.

95 and

γ

=

5

.

5. Weconsider the two differentwetting conditionsstudied in Qianetal.[46]:a symmetric casecorresponding to the equilibriumcontactangle

θ

Y

=

90o and an asymmetric caseobtainedimposing

θ

Y

=

64o.The corresponding imposed velocitieson the wallare U

=

0

.

25 and U

=

0

.

2, corresponding to capillarynumbers Ca

=

0

.

089 andCa

=

0

.

071,respectively.From theirMDsimulations,Qianetal.[46] deducedthesliplengthsvalues

1

=

2

=

1

.

625, and

1

=

1

.

625,

2

=

3

.

67,forthesymmetricandasymmetricconfigurations,respectively.ThevaluesarereportedinTable1. InQianetal.[46],thedomainoffluid2isinitiallyintroducedbetweentwodomainsoffluid1tomakepossibletheuse ofperiodicalconditions.Itresultstheexistenceoftwointerfacesinthecomputationaldomain.Here onlyoneinterfaceis simulatedandtheexactonephaseCouetteflowsolution

Table 1

Valuesofthesliplength consideredinthiswork.

Name Symmetric case (θY=90o) Asymmetric case (θY=64o) Reference

1− 2 1.625−1.625 1.625−3.67 Qian et al. [46] 1,V o F− 2,V o F 1.22−1.22 1.42−3.21 Adapted value Vi

(

y

)

=

2 y H

+

2 i U (7)

is imposedforthe twofluidsasboundarycondition atthe inletandoutletofthedomainlocated farfromtheinterface. As aconsequencethedomainlength L has beenreducedcompared toQianetal.[46].The effectofthedomainlength L

(i.e.theimposedboundarycondition)hasbeencheckedbycomparingL

=

136 andL

=

68,showingamaximumdifference betweenthetwosolutionslessthan0

.

5%.ThedomainlengthconsideredinthefollowingisL

=

68.

ThecomputationaldomainofsizeL

×

H isdescribedusingaregulargrid4N

×

N whereN isthenumberofcellsinthe directionnormaltothemovingwalls.DifferentgridsmadewithN

=

32,64,128and256willbeusedinthefollowing.For allconsideredgrids,theslipzonewillbefullyresolved.Themaximumgridsizeconsidered,



=

0

.

425 forN

=

32,isthree timessmallerthantheminimumsliplengththatwillbeconsideredhere.Forthemoreresolvedsimulations(N

=

256),the slipzonewillbedescribedbymorethan20cells.

3. The Navier-Stokes VoF solver

3.1. 1-Fluidsystemofequation

TheNavier-StokessimulationsareperformedusingtheincompressibleVolumeofFluid(VoF)solverimplementedinthe JADIMcode[20,39].TheVoFfunctionC (C

=

1 influid1andC

=

0 influid2)istransportedby

∂

C

∂

t

+

V

· ∇

C

=

0 (8)

where V

=

C V1

+ (

1

−

C

)

V2 representstheone-fluidvelocityfield.TheinterfaceisnotreconstructedinourVoFapproach. An accuratetransportalgorithm[11] basedonFCT(Flux-Corrected-Transport)schemes[71] is usedtokeepthenumerical interfacethickness

δ

n ofabout2-3gridcells.

UndertheassumptionofNewtonianincompressiblefluidswithinisothermalconditionsandwithoutphasechange,the conservationequationsofmassandmomentumtakethefollowingforms:

∇ ·

V

=

0 (9)

ρ



∂

V

∂

t

+

V

· ∇

V



= −∇

P

+ ∇ ·  +

g

+

Fγ (10)

where

ρ

=

C

ρ

1

+ (

1

−

C

)

ρ

2 is the fluid density, P

=

C P1

+ (

1

−

C

)

P2 is the pressure andthe viscous stress tensor is



=

μ



∇

V

+ ∇

T_V



_with

_μ

₌

_Cμ

1

+ (

1

−

C

)

μ

2. Fγ isthecapillarycontribution,givenby:

Fγ

=

γ

∇ ·

nInI

δ

I (11)

where

γ

is the interfacial tension betweenthe two phases, nI

= ∇

C

/

 ∇

C



is the unit vector normal to the interface pointingintofluid1and

δ

I denotesthedeltadistributionfunctionoftheinterface.

The system of equations (8)-(10) is discretized using the second order finite volume method. A staggered mesh is used consistingin locating theVoF function C andthe pressure P inthe centerof thecontrol volume while the veloc-itycomponentsarelocatednormaltothefacesofthecontrolvolume.Timeadvancementisachievedthroughathirdorder Runge-KuttamethodfortheadvectiveandsourcetermsandtheCrank-Nicolsonmethodisusedfortheviscousstress.The incompressibilityissatisfiedattheendofeachtimestepthroughaprojectionmethod.

BoundaryconditionsfortheVoFfunctionC ,arerequiredontheboundariesofthedomainwithnonzeronormal veloc-ities. Thevalue ofC isimposed attheinletandoutletofthedomainto C

=

1 (fluid1) andC

=

0 (fluid2), respectively. In thefollowing,theliquid-liquidinterface isdefinedby C

=

0

.

5.The correspondingcontactlinepositionxcl andvelocity

Vcloneachwall aredeterminedbyasecondorderlinearinterpolationattheinterfacelocation(C

=

0

.

5).Consideringthe situationwheretheinterfaceislocatedbetweenthecell i andi

+

1,i.e.Ci

≥

0

.

5 andCi+1

≤

0

.

5,thecontactlineposition andvelocityrelativetothebottommovingwallarecalculatedas

xcl

=

(0.5

−

Ci+1

)

xi

+ (

Ci

−

0.5)xi+1 Ci

−

Ci+1

,

(12) Vcl

=

U

+

(0.5

−

Ci+1

)

Vi

+ (

Ci

−

0.5)Vi+1 Ci

−

Ci+1 (13)

where Vi isherethetangentialfluidvelocity atthewall.Thevalue ofVcl isdeterminedateach timestepandisusedto calculatetheinstantaneousvalue ofthecontactanglewithoneofthemodelsconsidered inthisstudy.The contactangle betweenthe interface and thewall isimposed through the calculation of theCapillary contribution asdescribed in the followingsection.

3.2. Capillarycontribution

The capillarycontributioninthemomentum equation issolved usingtheclassical ContinuumSurface Force (CSF) ap-proach[12]: Fγ

=

γ

∇ ·



∇

C

|∇

C

|



∇

C (14)

Awellknownproblemofthisformulationisthegenerationofspuriouscurrents[36,45,1].Inordertodecreasespurious currentsintensity,aclassicalsolutionconsistsincalculatingthesurface curvatureandthenormalfromasmoothed distri-bution C [

ˆ

12]. Thesmootheddistribution isC

ˆ

= ˆ

Cm _attime

₍

_n

₊

₁

_/

₂

_)

_{t where C}

ˆ

m _{isobtainedafterm iterations.For2D} simulationsonaregulargridasconsideredinthiswork:

C_in_,f_j

=

3 4

C nf−1 i,j

+

1 16

Cn_i+f−_1,1_j

+

Cn_i−f₁−_,1_j

+

Cn_i,f_j−₊₁1

+

Cn_i,f_j−₋1₁



(15)

withnf

=

1

,

...,

m and

C0beinginitializedwithCn+1/2.FollowingDupontandLegendre[20],thecurvatureandthenormal involvedinthecapillaryterm, areobtainedwithtwodifferentvaluesform,mκ

=

12 andmL

=

6,respectively.The corre-spondingspurious currents intensityhasbeencharacterized andtheir maximum magnitudeevolveas 0

.

004

γ

/

μ

[20], in agreementwithothercodesusingtheBrackbill’sformulation.Thecorrespondingspuriouscurrentcapillarynumberismuch smallerthan thecapillarynumbersconsidered inthiswork, indicatingthat spurious currentsare notexpectedto induce anyperturbationattheinterfaceandonthevelocityfield.

Considering the finite volume method used for the discretization of the equations, the surface tension contribution integratedinacontrolvolume



isexpressedas



 Fγd



=

γ

⎛

⎝



V div

∇

C

 ∇

C



dV

⎞

⎠ ∇

C (16)

wherethedivergencetermcorrespondingtothecurvatureisconvertedtoasurfaceintegral.Thistermisthenevaluatedas fluxesoftheinterfacenormalnI

= ∇

C

/

 ∇

C



atthesurface



of



:

F_γ



=

γ

⎛

⎝



 nI

·

nd



⎞

⎠ ∇

C (17)

Inrelation(17) theterminbracketsisthecurvatureand

∇

C indicate thedirectionandlocationoftheimposed capillary contribution.Thecontactanglemadeby theinterfacewiththewallisused asaboundary conditioninthecalculationof thecapillaryterm(14) inthe momentumequation. Toillustratethis, let’s considera controlvolume



usedto calculate thevelocity inthex-direction andcontainingthe contactline(i.e.0

<

C

<

1). Thecontributionofthecontactlinetothe momentumbalancein



comesfromthesouthface



South:

F_γ,x,South



=

γ

nI

·

n



South

∇

C

·

ex (18)

wherethenormalofthesouthfaceisn

= −

ey andthearea is



South

= 

.Theangle

θ

m madebetweentheinterfaceand thewallimposes

nI

·

ey

=

cos

θ

m (19)

sothatthecontactlinecontributiontothemomentumin



is

Fγ,x,South



= −

γ

cos

θ

m



∇

C

·

ex (20)

Thefrictionmodelproposedforthemovingcontactangle

θ

mispresentedinthenextsection.Timeandgridconvergence andadditionalvalidationsarereportedinsection5.

4. The friction model

Astatedbefore,theobjectiveofthisworkistointroduceafrictionmodelwithinaVoF-CSFsolverforsolvingnanoscale moving contactlines. As presentedinthe introductionthe uncompensatedYoung stress needs tobe compensatedby an additionalfriction inducedbythe motionofthecontactline.Thisfriction hastwo effectsthatneedto becorrectly intro-ducedinthesimulations:(i)theshearisenhancedatthecontactlineonthewalland(ii)theangle

θ

mmadebetweenthe interface andthewallisreducedcomparedtoitsequilibriumvalue

θ

Y.InadditiontheNaviercondition[43] isappliedon thewalls.ForawallmovingwithvelocityU ,theslipvelocity V_islip offluidi (i

=

1

,

2)onthewallwrites

V_islip

=

Vi

−

U

=

i

∂

Vi

∂

y (21)

where

i is thewall sliplengthoffluid i, Vi is thevelocity offluid i paralleltothe walland y isthe coordinateinthe directionnormaltothewall.

As discussedinthe introductionsome frictionneeds tobe introducedatthe contactline. Twopossiblewaysare now examinedadoptinganumericalpointofview:

- BytheuseofaMKT- GNBClikeformulation

cos

θ

Y

−

cos

θ

m

=

ξ

γ

Vcl (22)

- Bytheuseofanadditionalviscousfriction

τ

cl atthecontactline:

τ

cl

=

μ

cl

∂

V

∂

y (23)

In orderto selecttheappropriate wayto introduce thefriction modelin aVoF-CSF formulation, let usconsidertheir respectivecontributions tothemomentumbalanceatthecontactline.Tosimplifythediscussionwefocusonthebottom wall (south boundary condition). When the wall has no motion (U

=

0), the angle made by the interface is the static angle

θ

m

= θ

Y andthe friction is then zero.As a consequence, the only contribution fromthe wall to the momentum balancecomesfromthecapillarytermcontribution Fγ,x,South



onthesouthface



South givenbyequation (20).Thetotal contributionofthistermovertheinterfacethickness

δ

n is



intFγ,South



∼ −



int

γ

cos

θ

Y

∂

C

∂

xdx

=

γ

cos

θ

Y (24)

Interestinglythistermisindependentonthenumericalthickness

δ

n oftheinterface.Itisbalancedbythecapillary contri-butionfromtheotherfacesof



andbythestaticpressureresultingintheLaplacepressurejumpattheinterfaceandan interfaceshapeminimizingthesurfaceenergyandsatisfyingtheimposedangleatthewall[20,39].

Wenowconsiderthecaseofamovingcontactline.Thecontactangleisnow

θ

m.Thenormalvelocityofthesouthface



Southbeingzero,theconvectivefluxiszero.Thus,thecontributionofthecontactlinetothemomentumbalanceon



South mayincludeanadditionalfriction

τ

cl:

Fγ,x,South



+

τ

cl



=



−

γ

cos

θ

m

∇

C

·

ex

+

μ

cl

∂

V

∂

y





(25)

andtheintegralovertheinterfacethicknessbecomes

F_γ,x,South



+

τ

cl



=

γ

cos

θ

m

+

μ

cl

∂

V

∂

y

δ

n (26)

δ

n beingtypically oftheorderof2-3grid size



,a formulationbasedonaviscous likecontribution

τ

cl resultsina grid dependentcontributionattheinterface.However,aformulationthroughcos

θ

musingaMKT-GNBClikeexpression(Eq.(22)) provides acontributiontothemomentumequation independentofthegridsize.Asaconsequence,theMKT- GNBClike formulationgivenbyEq. (22) combinedwiththeNavierslipcondition(Eq.(21)),willbeintroducedintheVoF-CSFmethod. The use ofthis modelrequires thevalue ofthe staticcontactangle

θ

Y andthe value of thecontactline friction

ξ

. The relevantvaluefor

ξ

willbediscussedinsection6.

5. Preliminary tests

The Navier-Stokes solver of JADIM hasbeen intensively validatedfor both 2D and 3D simulations. The VoF approach associated withthe simulation ofmoving contactlines hasbeen detailedin[20,39] where numerous validations canbe found relatedtostaticshapeofdropsonsurface aswellasspreadingorslidingdrops.We reportherethegrid andtime convergenceofthesimulationusingtheproposedfrictionmodelandtwoadditionaltestcasesrelevantforthepresentstudy.

Fig. 2. Liquid drop confined between two static walls. (Left) initial condition. (Right) Stabilized shape (t=100).

Fig. 3. Liquiddropconfinedbetweentwostaticwalls.(Left)Comparisonbetweenthenumericalsimulationsatt=100 andtheexactshape:∗N=32, ◦N=64,×N=128 and N=256.(Right)EvolutionoftheerrorsE1(◦)andE∞(×)asafunctionofthegridrefinement.

We firstconsider thestaticshape ofa dropconfinedbetweentwo walls. Then, we studythegrid andtimeconvergence forthe code usingfriction modelfor thesimulation ofthe Couette flow considered inthis study.Finally, we investigate thetransitionbetweenstableandunstableinterfaceshape fortheconfigurationconsideredinthisworkanddescribedin section2.

5.1. Shapeofadropconfinedbetweentwostaticwalls

Thegeometryconfigurationconsidered forthistestisclosetotheonepresentedinFig.1.Avolumeoffluid2initially delimitedbytwoverticalinterfaceswithfluid1isintroducedinthemiddleofthechannelasshowninFig.2(left)forming initial contactangles of90o with thewalls. A constantcontactangle

θ

m

=

80o isimposed on thetwo walls. Due tothe imposed contactangle,thesystemstabilizestoformtwo concaveinterfacesofcircularshape withradius R

=

H

/

cos

θ

as showninFig.2(right).

Thesimulations performedwiththegrids N

=

32,64,128and256arecomparedinFig. 3(left)attimet

=

100 with theexact circularshape.Theerror betweenthe numericalpositionx andthe exactposition xT H is measuredusingthe normalizedmaximumdifferenceE_∞andthenormalizedmeandifference E1definedas:

E_∞

=

1 Hmaxi





x_i

−

x_iT H



_{ ,}

E1

=

1 N H



i





x_i

−

xT H_i



_

(27)

Theerrors E_∞ and E1 arereportedinFig.3(right)asfunctionofthegrid refinement



.Thefigure clearlyshowsagrid convergence of order



4/3 between



and



2 forboth E_∞ and E1. The same order of convergenceis observed when varying the imposed contact angle

θ

Y. The numerical discretization schemefor the Navier-Stokes solver in JADIM being secondorder,thereductionoftheorderofthegridconvergenceisattributedtothesmoothingprocedureappliedto C for

thecalculationofthecapillarycontributionFγ . 5.2. Timeandgridconvergenceforthefrictionmodel

We report in this section time and grid convergence testswhen using the friction model.The convergencetests are performedforthe simulationof the2D Couette flow considered inthisstudy(see Fig. 1). Forclarity onlythe tests per-formedwiththefrictioncoefficient

ξ

=

1

.

7 areshown. Verysimilarresultswereobservedwhenconsideringotherfriction coefficients.

We firstconsiderthe effectofthe time steponthe solution.The normalizedtime stepis variedfrom



t

=

5

×

10−4 to



t

=

1

.

5

×

10−2_{.Thegrid ismadewith N}

₌

_{64.The stabilizedinterface positionattime t}

₌

_{120 is reportedinFig.4} fordifferenttime steps. Aclearconvergenceis observed.We introducethe error Et

Fig. 4. TimeconvergenceperformedforthefrictionmodelforN=64 withthefrictioncoefficientξcl=1.7.(Left)Interfacepositionforthedifferenttime

stepsatt=120.(Right)NormalizederrorEt

1 asfunctionofthetimestept.

Fig. 5. Gridconvergenceperformedforthefrictionmodelwithξ=1.7 witht=2×10−3_{.(Left)Interfaceshapeforthedifferentgrids.(Right)Evolution}

ofthemeannormalizederrorE

1.

betweenthe solutionobtainedfor



t andthe solutionofreferenceobtainedforthe smallesttime stepconsidered



t

=

5

×

10−4: E₁t

=

1 N H



i





x_it

−

x_it=5×10−4





(28) TheevolutionofEt

1 with



t reportedinFig.4showsthattheoverallnumericalmodelissecondorderconvergentintime. The grid convergenceisnow considered.The numberof cells N is varied from32to256. Forallthe simulations the time stepissetto



t

=

2

×

10−3 andthesimulationtime ist

=

120.Thistimestepensures timeconvergenceforall the meshesconsidered.AsshowninFig.5,thesimulationconvergeswhendecreasingthegridsize.Themeannormalizederror

E

1 isdefinedusingthesolutionobtainedforthesmallestgridspacing,i.e.



=

H

/

256:

E₁

=

1 N H



i





x_i

−

x_iH/256



_

(29) TheevolutionofE

1 isreportedasafunctionofthegridspacing



inFig.5.Thetimeevolutionofthecapillarynumberand theviscousshearrateprofileatthewallarealsoshowninFig.6forthedifferentgridstested.Theseplotsclearly indicate thatthegridconvergenceisensuredforallthereportedquantities.Theinterfaceshapeappearstobemoresensitivetothe meshrefinementandtheconvergencerateisthen



4/3_{.Thispointwillbefurtherdiscussedinsection5.1.}

5.3. Transitionfromstabletounstable2Dtwo-phaseCouetteflow

Thislasttestcaseisrelatedtothestabilityofthe2Dtwo-phaseCouetteflowconsideredhere.Thisflowconfigurationis knowntopresentsteadyorunsteadysolutionsdependingonthevelocityimposedonthewalls.Asteadyinterfaceposition can beobserved aslongastheentrainmentoftheinterface bythe shearflowcan bebalanced bythe capillaryrepelling force [34,50,63,25],i.e.

μU

/

H

∼

γ

/

H .ItresultsacriticalvalueforthecapillarynumberCa

=

μU

/

γ

.Steadysolutions are thenobservedifthecapillarynumberissmallerthanacriticalvalue Ca∗.WhenCa

>

Ca∗ therepellingcapillaryforcecan notresisttotheentrainmentofthetwocontactlinesandtheinterfaceiscontinuouslyelongated.ThisisillustratedinFig.7

Fig. 6. Gridconvergenceperformedforthefrictionmodelwithξ=1.7 witht=2×10−3_{.(Left)Timeevolutionofthecontactlinecapillarynumber}

Ca=μVcl/γforthedifferentgrids.(Right)Velocitygradientatthewall.

Fig. 7. Successiveinterfaceshapesforatypicalunsteadysituation.Fromtoptobottomt=0,2,4,6 and8.Theimposedcontactangleisconstantθm=90o

andthenormalizedslipare 1/H= 2/H=0.089.

wheretheinterfaceshapeisshownatdifferenttimestepswhenimposingafixedcontactangle

θ

m

=

90oandanormalized slip

/

H

=

0

.

089 forthetwofluids.

The critical Capillary number Ca∗ dependson both the ratio

/

H and the imposed contact angle

θ

m [34,50,25]. To determinethe transitionbetweenstableandunstablesituations,simulations are conductedfordifferentimposedcontact angles

θ

m whileimposing a constant slip

/

H

=

0

.

089.The corresponding valuesof Ca∗ are reportedinFig. 8.Theyare compared tothe criticalcapillary numberobtainedwiththe quasi-parallel approximation proposed by Jacqmin[34]. We havesolvedthesystemofequations(Eq.2.7b-2.9ofJacqmin[34])usingafirstordercenterEulerscheme.Theassumption of aquasi-parallel flow induces asolution valid forplane interfaces makingthe solution a priorivalid forsmall contact angle

θ

m withthe wall. As showninFig. 8 theagreement betweenthe Navier-Stokes simulations andthe quasi-parallel approximationisverygoodforsmallangles.Then,thedifferenceincreasesandthequasi-parallelapproximationisshownto overpredictthetransitioninagreementwiththecomparisonmadeinJacqmin[34] withphasefieldsimulations.However, thepredictiongivenbythequasi-parallelapproximationremainsinreasonableagreementevenatlargecontactangle.

Fig. 8. Criticalcapillarynumberasfunctionoftheimposedcontactangleθmwith 1/H= 2/H=0.089.◦Numericalresults,(continuousline)theparallel

flowsolutionfromJacqmin[34].

Fig. 9. Interfaceshapewhenusingthestaticcontactanglemodelfor(left)thesymmetriccaseθY=90owith 1= 2=1.625 and(right)fortheasymmetric

caseθY=64owith 1=1.625 and 2=3.67.MDsimulationsarereportedusingcircles.

6. Impact of the friction model

6.1. Simulationwithastaticcontactangle

Wefirstsimulatethe2DCouetteflowdescribedinFig.1byimposingthestaticcontactangle,sonocontactlinefriction isimposed(

ξ

=

0).Thecapillaryforceonthewallisthuscalculatedwith

θ

m

= θ

Y (30)

The symmetric (

θ

Y

=

90o) and asymmetric (

θ

Y

=

64o) cases are considered. The Navier slip condition is imposed with the slip lengthsdeduced by Qianet al.[46] from their MD simulations and reportedin Table 1. The grid used forthis comparisonismadeusing N

=

256 andthenormalizedtime stepis



t

=

2

×

10−3.The simulationsarecompared tothe MDsimulationofQianetal.[46] inFig.9.

Thestaticmodelisabletocaptureasteadyinterfaceshapecomparabletotheexpectedonebuttheslipatthecontactis toolargetofittheMDresults.ThedifferenceintheCLpositionisnoticeablyunderestimated(about30%)forthesymmetric casewhileabetteragreementisobservedfortheasymmetriccase:thedifferenceisabout20%andthebottomwallwhilea goodagreementisobservedontheupperwall.Thissuggeststhatthestaticmodelisnotadaptedandthatanextrafriction needsbeaddedatthecontactlineasdiscussedinthefollowingsection.

6.2. Simulationwiththefrictionmodel

Thefrictioncoefficient canbe determinedusingtheMKTformulationgivenbyrelation(3).Alltherequiredparameters are givenin Qianet al.[46]. In their MDsimulations the temperature was controlled at kBT

=

2

.

8. The jump length is linkedtotheadsorptionsitesdensitylocalizedatthewallandcanberelatedtothedensityofthesolid

ρ

s as

λ

=

ρ

s−1/3

=

Table 2

Valuesofthefrictioncoefficientξconsideredinthiswork.

Name Symmetric case (θY=90o) Asymmetric case (θY=64o) From Eq. Reference

ξRen 3.02 2.18 (33) Ren and E [49]

ξM K T 32.8 57.9 (32) Ramiasaetal.[48]

Sevenoetal.[53]

ξV o F 1.7 0.7 Adapted value This work

Fig. 10. (left)the symmetriccaseθY=90o with thefrictioncoefficientξRen=3.024 and for(right) theasymmetriccaseθY=64o with thefriction

coefficientξRen=2.18.MDsimulationsarereportedusingredcircles.

0

.

813.Note that a very close value is obtained by considering

λ

asthe smallest distance betweentwo wall atoms, i.e.

λ

= (

4

/

ρ

s

)

1/3

/

√

2

=

0

.

913.Thejumpfrequencyforafluidi displacingitsvaporis[6]

K0,i

=

kBT

μ

i

ν

i exp



−

γ

λ

2

(1

+

cosθY

)

kBT



(31)

wheretheunitflowvolumeofthefluidistakenas

ν

i

=

1

/

ρ

i.Thecorrespondingvalueofthejumpfrequenciesare K0,1

=

K0,2

=

0

.

32 and K0,1

=

K0,2

=

0

.

18 forthesymmetric(

θ

Y

=

90o)andtheasymmetriccases(

θ

Y

=

64o),respectively.When consideringaninterfacebetweentwofluids,asinthecaseconsideredhere,theeffectivefrictionisthen[48,53]

ξ

= ξ

1

+ ξ

2 (32)

where

ξ

i is thefriction of fluid i given by Eq. (3) withthe jumpfrequency givenby Eq. (31). Thecorresponding values for the normalized friction coefficients are

ξ

M K T

=

32

.

8 and

ξ

M K T

=

57

.

9 for the symmetric andthe asymmetric cases, respectively.

The contactline friction can alsobe determined with theMD simulations ofRen andE [49]. They considerthat the friction at the contact line is composed of viscous and frictional parts. Here, the Navier slip condition, i.e. the viscous contribution,isimposedontheentirewall,includingthecontactlineregion.HenceonlythefrictionalpartreportedbyRen andE[49] needstobeconsidered.Itisgivenby

ξ

Ren

=

0.42δB (33)

where

δ

is the contactline thicknessand B isthe fluid friction coefficient. In the asymmetric case, the fluid friction is different inthe two fluidsand the average value B

= (

B1

+

B2

)/

2 is considered. Based on Renand E [49] weconsider

δ

=

6.Accordingly,the frictioncoefficientsforthesymmetricandtheasymmetriccasesare

ξ

Ren

=

3

.

024 and

ξ

Ren

=

2

.

18, respectively.ThevaluesusedforthefrictioninthefollowingsimulationsaresummarizedinTable2.

ThefrictiongivenbytheMKTpredictionisnotabletoprovideasteadystatefortheinterfaceforboththesymmetricand theasymmetricconfiguration.Theinterfacemovescontinuouslyentrainedbythemovingwallsascommentedinsection5.3 (seeFig.7).Thisindicatesthatthefriction predictedbytheMKTmodelistoolargeandcannotbedirectlyintroduced in aNavier-Stokessolver.However, asteadyinterface isobtainedwhenusingthefriction

ξ

Ren deducedfromMDsimulations asshowninFig.10.Theagreementappearsverygoodfortheasymmetriccase

θ

Y

=

64o whileintheasymmetriccase,the friction

ξ

Ren overestimatesthefriction requiredtomatchwiththereferencecase. Thisisevidencedby amore important displacementofthecontactlineresultinginamoredeformedinterfacethanobservedinthecaseofreference.

Theseresultsindicatethattheinterfaceshapeseemstobeverysensitivetothefrictionintroducedinthecontactangle modelandthisconfirmstherelevanceofthechoiceofthistestcase.Despiteasmalldifferencebetween

θ

Y and

θ

m(around 3o_{)thepositionofthecontactangleisnoticeablyimpacted.Thesepointsarefurtherdiscussedinthenextsection.}

Fig. 11. Steadystateinterfaceshapefor(left)thesymmetriccaseθY=90oand(right)theasymmetriccaseθY=64ofor(blueline)thefrictionmodel

(adjustedfriction)and(redline)thestaticmodel(adjustedsliplength)(seethetext).MDsimulationsarereportedusingredcircles.

6.3. AdaptedfrictionfortheVoF-CSFsolver

TheobjectiveisnowtofindthecorrectfrictionthatneedstobeusedinaVoF-CSFsolvertorecovertheinterfaceshape obtainedwiththe MDsimulation.Forthat purposewe adaptthe valueofthe friction coefficient

ξ

in ordertominimize thedifferencewiththeinterfaceshapegivenbytheMDsimulationofreference.Thereportedvaluescorrespondtoamean difference lessthan0.5%.Thesliplengths

1 and

2 areunchangedandcorrespondtothevaluesreportedinTable1.The bestagreementwiththereferencecaseisobtainedfor

ξ

V o F

=

1

.

7 and

ξ

V o F

=

0

.

7 for

θ

Y

=

90o and

θ

Y

=

64o,respectively. The corresponding interface shapes are shown in Fig. 11. The frictions

ξ

V o F are one order of magnitude smaller than predicted by theMKTmodel andofthe sameorder ofmagnitudeasthose deduced fromMDsimulations (see Table 2). Some velocity profiles parallel tothe wallsare reported inFig. 12atdifferent vertical positions andcompared withthe correspondingprofilesfromtheMDsimulations.Asshown,theagreementforbothcasesisfoundverysatisfactory.

Note that, a similar approachcan be conductedconsidering the staticcontactangle model.Now the contactangleis keptfixedtothestaticcontactangle(

θ

m

= θ

Y

=

90o and

θ

m

= θ

Y

=

64o)andwereducethesliplengthatthewalltogeta betteragreementwiththereference.The“best”sliplengthsarecomparedinTable1withthevaluesdeducedfromtheMD simulation. Boththe interface shape(Fig. 11)andthe velocityprofile(Fig. 12) arenow inverygood agreementwiththe MDsimulations.

Theseresultsclearlyindicatethat,oncecalibrated,both thefriction model(adjustedfriction)andthestaticmodel (ad-justed sliplength) are ableto provide goodresults onboth the interface shape (Fig. 11) andthe velocity field (Fig. 12). Oneimportantconclusionisthat Navier-Stokes/VoF/CSFsimulationsareabletoreproduceflowsandcontactlinedynamics atthenano-scale.Thisisinlinewithpreviousfindings[49,9,38,57,44] whichhaveshownthathydrodynamicsisincredibly consistenttodescribeliquidflowsandinterfacebehavioruptothenano-scale.Wealsoshowthatitisnecessarytoconsider an adaptedboundarycondition atthecontactline. Thetwo approachesconsideredhere(frictionversus slip)seemsto be able to beused asboundary conditionsinan equivalent way.However the changein thedynamic contactangleis here relatively small(around3o).We canexpectthat forlargervariations ofthedynamiccontactanglecomparedtothestatic angle,anadjustmentofthesliplengthisnotrelevantbecausethecorrectangleatthewallisnotimposed.

6.4. Influenceofthegridsizeonthefrictioncoefficient

The modeladjustmentreportedabove wereperformedfora givengrid correspondingto N

=

256.Wenowinvestigate theeffectofthegridsizeonthevalueof

ξ

V o F forthefrictionmodelandontheslip

V o F fortheuseofaconstantcontact angle.Theoptimalvalueisdeterminedforeachgridfollowingtheproceduredescribedabove.Thecorresponding valuesof thefriction andthesliplengthreportedinFig.13convergewiththegridrefinement.Thevaluesare verysensitivetothe gridforsmallresolutions,i.e.N

<

100.Forbetterresolutions,both

ξ

V o F and

V o F arealmostconstant.

7. Interface shape

The objectiveofthissectionisto focusontheinterfaceshape fordifferentcapillarynumbersandratios H

/

.Thegap

H of thechannel isincreased whilekeeping unchangedthe sliplength andthe physicalproperties (see section 2). The values H

=

13

.

6, 450 and2250 are consideredcorresponding to H

/

=

8

.

37,277 and1385,respectively. Thesimulations are performedusingthe friction modelwiththefriction

ξ

V o F given inTable2.The domainsize is L

=

4

×

H forall the cases.Thegridsize ischosen toensurethattheslipzoneisresolved andthegridconvergencehasalsobeencheckedfor thenewcasesconsidered.Thewallvelocityisalsovarieduptothelimitofstabilityoftheinterface.Theinterfaceshapefor velocityrangingfromU

=

0

.

03 toU

=

0

.

08 arereportedinFig.14forH

=

450 andN

=

256.Thiscorrespondstocapillary numbers rangingfrom Ca

=

0

.

106 toCa

=

0

.

284 andcontactanglesgivenby thefrictionmodelvarying from

θ

m

=

89

.

5o to

θ

m

=

88

.

6o.ForthiscaseweobservedthattheonsetoftheunsteadyregimeisbetweenU

=

0

.

08 and U

=

0

.

09,i.e.for

Fig. 12. Tangentialvelocityprofilesfor(top)thesymmetriccaseθY=90oand(bottom)theasymmetriccaseθY=64o.Comparisonbetweentheadjusted

staticmodel(dashedlines)andfrictionmodels(continuouslines)andtheMDresults(symbols)fromQianetal.[46]:(top)y= −6.375 (◦);y= −4.675 ();y= −2.975 ( );y= −1.275 ().(bottom):y= −6.375 (+);y= −3.825 (×);y= −1.275 ();y=1.275 ( );y=3.825 ();y=6.375 (◦).

Fig. 13. InfluenceofthegridsizeonthefrictioncoefficientξV o F and◦thesliplength V o F forthestaticmodel.N isthenumberofcellsalongthe

Fig. 14. (top) Interface shape for H=450 and for wall velocity U=0.03 (◦), U=0.04 (), U=0.05 (), U=0.06 (), U=0.07 ( ) and U=0.08 ().ContinuouslineEq. (34).(bottom)Normalized interfaceshapeshowingthelinear evolutioninthechannelcenter. (insert)β versusCa for(◦)H=13.6,(◦) H=450 and(◦) H=2250.

capillarynumberbetweenCa

=

0

.

284 andCa

=

0

.

319.As expected,when thecapillarynumberisincreasedtheinterface deformationisenhanced.

A detailed inspection shows that the interface shape exhibits an auto-similar behavior in the channel center when varying the wall velocity asreportedinFig. 14.When reporting y normalized by H

/

2 as a function ofx normalized by theinterfaceabsciseat y

=

H

/

2 (notedx

(

H

/

2

)

)theinterface shapesalmostcollapseandfollowalinearshapethatcanbe describedwith y

=

H

β

2 x xcl (34)

wherexclisthecontactlinedisplacementandtheparameter

β

isafunctionofCa asshownintheinsertofFig.14. Closetothewalltheinterfaceshapedependsonboththecapillarynumberandtheimposedcontactangle.Todescribe theinterface shapeinthevicinityofthewall,weconsidertheevolutionoftheinterface angle

θ (

r

)

madeby theinterface with the x-axis parallel to the walls. In the reference frame attached to the wall the contactline is a receding (resp. advancing) contactlineatthebottom (resp.top)wallforFluid2.According tothe hydrodynamicmodel ofCox[14], the interfaceangle

θ (

r

)

atthedistancer tothebottomcontactlineisgivenby

g

(θ (

r

),

q

)

=

g

(θ

m

,

q

)

−

Ca Ln

(

r

/ )

(35)

Fig. 15. VariationoftheInterfaceangleθm− θ(r)asafunctionofthenormalizeddistancer∗=r/ tothecontactlineforH=450 andCapillarynumbers

Ca=0.106 (top)andCa=0.284 (bottom).Continuousline:Coxrelation(35) withg(θ (r),q)givenby(36),Dottedline:Coxrelation(35) withg(θ (r),q) givenby(37),Dashedline:Coxrelation(35) withg(θ (r),q)givenby(38).

g

(θ,

q

)

=

θ



0 f

(β,

q

)

d

β

(36) with f

(β,

q

)

=

q

(β

2

−

sin2

β)

[(

π

− β) +

sin

β

cos

β]

+



(

π

− β)

2

₋

_sin2

_β



_(β

₋

_sin

_β

_cos

_β)

2 sin

β



q2

_(β

2

₋

_sin2

_β)

₊

_2q

_β(

_π

_{− β) +}

_sin2

_β



_{+ ((}

_π

_{− β)}

2

₋

_sin2

_β)



When the contactangle satisfies the condition (

θ

≤

3

π

/

4) fora liquid displacing a gas (q

=

0), the function g can be simplifiedas

g

(θ (

r

),

q

)

≈

θ

3

9 (37)

Fortwoliquidsthisrelationremainsagoodapproximationbutforsmallerangles,typically

θ

≤

π

/

3 forq

=

1.Thus,forthe rangeofcontactanglesconsideredherethenextcorrectionin

θ

needstobeconsideredintheexpansion[23]

g

(θ (

r

),

q

)

=

θ

3 9

−

q

θ

4 8

π

+

O

(θ

5

₎

₍₃₈₎

Theserelations are reported inFig. 15forq

=

1 and compared tothe numericalsimulations for H

=

450. Twocapillary numbersCa

=

0

.

106 and Ca

=

0

.

284 havebeenselected.Forclaritytheanglevariation

θ

m

− θ(

r

)

isreportedasafunction

ofthenormalizeddistancer∗

=

r

/

tothecontactline.As clearlyshown,relation(35) with g

(θ (

r

),

q

)

givenbyexpression (36) isabletodescribetheevolutionoftheangleinthevicinityofthewalldespitetheconfinementimposedbythetwo wallsmovinginoppositedirections.Itisremarkablethatrelation(35) obtainedwithaderivationbasedonmatchinginner andouterregionsthroughanintermediateregionseemstostillapplyinsuchaflowconfiguration.Theagreementbetween numericalresultsandtheCox relationisimprovedwhenthe capillarynumberisdecreased. Notethatrelation(37) often used for thedescription of dynamic contactlines is farto describe the evolution ofthe interface shape for two liquids. However,relation(38) appearstobeaveryinterestingapproximationtothefunction g

(θ (

r

),

q

)

.Theseresultsindicatethat numericalstrategies[20,59,39,58] usingcontactanglemodelsbasedontheuseoftheCox hydrodynamicmodel(Eq.(35)) remainsrelevantforsuchconfinedflows.

8. Conclusion

Wehavepresentedtheimplementationofadynamiccontactanglemodelbasedonthecontactlinefrictionina Navier-Stokes VoF-CSF solver for the simulation of moving contact lines at the nano-scale. The dynamic contact angle model requires the value of the staticcontactangle andthe value of thecontact linefriction, while the sliplength is used to imposeaNavierboundarycondition.Aliquid-liquidinterfaceconfinedinaCouetteflowgeneratedbytwowallsmovingat the samevelocity inopposite directionsisconsidered to discussthe relevance ofthefriction model.The simulationsare comparedwithareferencecaseobtainedbyMDsimulations[46].WeshowthattheNavierStokessimulationsareableto reproduce theMD simulationsforboth theinterface shape andthevelocity field.The appropriate contactlinefriction is found tobe gridconvergentandofthesameorderasthefrictionmeasuredinMDsimulationswhilethefrictiondeduced from the MKTmodel seems not able to providean appropriate friction fora Navier-Stokes solver asconsidered here. A detailedinvestigationoftheinterface shapehasrevealedan auto-similarlinearprofileinthecenterofthechannel.Close tothewalltheinterface shapeisobservedtofollowtheclassicalLogevolutiongivenby theCoxrelationdespitethe con-finementimposedbythewalls.AdditionalMDsimulationsandappropriate experimentsarerequiredtofurtherinvestigate the friction at a moving contact line. These results will be of importance for the implementation of friction models in Navier-Stokessolversforthesimulationsofmovingcontactlines.

Acknowledgements

TheauthorsacknowledgetheCarnotISIFORprogramsupportforthepost-doctoralgrantofHSHMandCALMIPfor pro-viding computational resources throughthe projectP1519. Wewouldlike tothankAnnaïgPedrono forthehelp andthe supportwithJADIM.

References

[1]T.Abadie,J.Aubin,D.Legendre,Onthecombinedeffectsofsurfaceforcecalculationandinterfaceadvectiononspuriouscurrentswithinvolumeof fluidandlevelsetframeworks,Comput.Fluids297(2013)611–636.

[2]S.Afkhami,S.Zaleski,M.Bussmann,Amesh-dependentmodelforapplyingdynamiccontactanglestovofsimulations,J.Comput.Phys.228(2009) 5370–5389.

[3] G.Ahmed,O.ArjmandiTash,J.Cook,A.Trybala,V.Starov,Biologicalapplicationsofkineticsofwettingandspreading,Adv.ColloidInterfaceSci.249 (2017)17–36,https://doi.org/10.1016/j.cis.2017.08.004.

[4]F.Bai,X.He,X.Yang,R.Zhou,C.Wang,Afiniteelementmethodforthenumericalsolutionofthecoupledcahn-hilliardandnavier-stokessystemfor movingcontactlineproblems,Int.J.Multiph.Flow93(2017)130–141.

[5]K.Bao,Y.Shi,S.Sun,X.P.Wang,Afiniteelementmethodforthenumericalsolutionofthecoupledcahn-hilliardandnavier-stokessystemformoving contactlineproblems,J.Comput.Phys.231(2012)8083–8099.

[6]E.Bertrand,T.D.Blake,J.DeConinck,Influenceofsolid-liquidinteractionsondynamicwetting:amoleculardynamicsstudy,J.Phys.Condens.Matter 21 (46)(2009)464124.

[7]T.Blake,Thephysicsofmovingwettinglines,J.ColloidInterfaceSci.299(2006)1–13.

[8]T.Blake,J.Haynes,Kineticsofliquid/liquiddisplacement,J.ColloidInterfaceSci.30(1969)421–423.

[9]L.Bocquet,E.Charlaix,Nanofluidics,frombulktointerfaces,Chem.Soc.Rev.39(2010)1073–1095.

[10]D.Bonn,J.Eggers,J.Indekeu,J.Meunier,E.Rolley,Wettingandspreading,Rev.Mod.Phys.81(2009)739–805.

[11]T.Bonometti,J.Magnaudet,Aninterfacecapturingmethodforincompressibletwo-phaseflows.Validationandapplicationtobubbledynamics,Int.J. Multiph.Flow33(2007)109–133.

[12]J.Brackbill,D.Kothe,C.Zemach,Acontinuummethodformodelingsurfacetension,J.Comput.Phys.100(1992)335–354.

[13]L.Chen, J.Yu, H. Wang, Convexnanobendingat amoving contact line:the missing mesoscopiclinkin dynamicwetting, ACSNano 8(2014) 11493–11498.

[14]R.Cox,Thedynamicsofthespreadingofliquidsonasolidsurfaces.part1:Viscousflow,J.FluidMech.168(1986)169–194.

[15]J.DeConinck,T.Blake,Wettingandmoleculardynamicssimulationsofsimpleliquids,Annu.Rev.Mater.Res.38(2008)1–22.

[16]A.Deblais,R.Harich,A.Colin,H.Kellay,Tamingcontactlineinstabilityforpatternformation,Nat.Commun.7(2016)12458.

[17]G.Delon,M.Fermigier,J.Snoeijer,B.Andreotti,Relaxationofadewettingcontactline.part2.experiments,J.FluidMech.604(2008)55–75.

[18]Y.Deng,L.Chen,Q.Liu,J.Yu,H.Wang,Nanoscaleviewofdewettingandcoatingonpartiallywettedsolids,J.Phys.Chem.Lett.7(2016)1763–1768.

[19]Y.Di,X.P.Wang,Precursorsimulationsinspreadingusingamulti-meshadaptivefiniteelementmethod,J.Comput.Phys.228(2009)1380–1390.

[20]J.Dupont,D.Legendre,Numericalsimulationsofstaticandslidingdropwithcontactanglehysteresis,J.Comput.Phys.229(2010)2453–2478.

[21]V.E.Dussan,S.Davis,Onthemotionofafluid-fluidinterfacealongasolidsurface,J.FluidMech.65(1974)71–95.

[22]J.Eggers,Hydrodynamictheoryofforceddewetting,Phys.Rev.Lett.93(2004)094502.

[23]R.T.Foister,Thekineticsofdisplacementwettinginliquid/liquid/solidsystems,J.ColloidInterfaceSci.136 (1)(1990)266–282.

[25]P.Gao,X.Y.Lu,Onthewettingdynamicsinacouetteflow,J.FluidMech.724(2013)R1.

[26]M.Gatzen,T.Radel,C.Thomy,F.Vollertsen,Wettingandsolidificationcharacteristicsofaluminiumonzinccoatedsteelinlaserweldingandbrazing, J. Mater.Process.Technol.238(2016)352–360.

[27]P.G.deGennes,Wetting:staticsanddynamics,Rev.Mod.Phys.57(1985)827–863.

[28]J.F.Gerbeau,T.Lelièvre,Generalizednavierboundaryconditionandgeometricconservationlawforsurfacetension,Comput.MethodsAppl.Mech.Eng. 198(2009)644–656.

[29]S.Glasstone,K.Laidler,H.Eyring,TheTheoryofRateProcesses:TheKineticsofChemicalReactions,Viscosity,DiffusionandElectrochemical Phenom-ena,McGraw-Hill,NewYork,1941.

[30]F.Henrich,D.Fell,D.Truszkowska,M.Weirich,M.Anyfantakis,T.H.Nguyen,M.Wagner,G.Auernhammer,H.J.Butt,Influenceofsurfactantsinforced dynamicdewetting,SoftMatter12(2016)7782–7791.

[31]R.Hoffman,Astudyoftheadvancinginterface.i.interfaceshapeinliquid-gassystems,J.ColloidInterfaceSci.50(1975)228–241.

[32]C.Hu,L.Scriven,Hydrodynamicmodelofsteadymovementofasolid/liquid/fluidcontactline,J.ColloidInterfaceSci.35(1971)85–101.

[33]H.Huppert,Thepropagationoftwo-dimensionalandaxisymmetricviscousgravitycurrentsoverarigidhorizontalsurface,J.FluidMech.121(1982) 43–58.

[34]D.Jacqmin,Onsetofwettingfailureinliquid-liquidsystems,J.FluidMech.517(2004)209–228.

[35]J.Koplik,J.Banavar,J.Willemsen,Moleculardynamicsofpoiseuilleflowandmovingcontactlines,Phys.Rev.Lett.60(1988)1282–1285.

[36]B.Lafaurie,C.Nardone,R.Scardovelli,S.Zaleski,G.Zanetti,Modellingmergingandfragmentationinmultiphaseflowswithsurfer,J.Comput.Phys. 113(1994)134–147.

[37]N.LeGrand,A.Daerr,L.Limat,Shapeandmotionofdropsslidingdownaninclinedplate,J.FluidMech.541(2005)293–315.

[38]R.Ledesma-Alonso,D.Legendre,Ph.Tordjeman,Nanoscaledeformationofaliquidsurface,Phys.Rev.Lett.108(2012)106104.

[39]D.Legendre,M.Maglio,Comparisonbetweennumericalmodelsforthesimulationofmovingcontactlines,Comput.Fluids113(2015)2–13.

[40]A.Lukyanov,A.Likhtman,Dynamiccontactangleatthenanoscale:aunifiedview,ACSNano10(2016)6045–6053.

[41]J.Luo,X.Hu,N.Adams,Curvatureboundaryconditionforamovingcontactline,J.Comput.Phys.310(2016)329–341.

[42]J.Luo,X.P.Wang,X.C.Cai,Anefficientfiniteelementmethodforsimulationofdropletspreadingonatopologicallyroughsurface,J.Comput.Phys.349 (2017)233–252.

[43]C.Navier,Mémoiresurlesloisdumouvementdesfluides,Mém.Acad.R.Sci.Inst.Fr.6(1823)389–440.

[44]P.Poesio,A.Damone,O.Matar,Slipatliquid-liquidinterfaces,Phys.Rev.Fluids2(2017)044004.

[45]S.Popinet,S.Zaleski,Afront-trackingalgorithmforaccuraterepresentationofsurfacetension,Int.J.Numer.MethodsFluids30(1999)775–793.

[46]T.Qian,X.P.Wang,P.Sheng,Molecularscalecontactlinehydrodynamicsofimmiscibleflows,Phys.Rev.E68(2003)016306.

[47]T.Qian,X.P.Wang,P.Sheng,Avariationalapproachtomovingcontactlinehydrodynamics,J.FluidMech.564(2006)333–360.

[48] M.Ramiasa,J.Ralston,R.Fetzer,R.Sedev,Contactlinefrictioninliquid-liquiddisplacementonhydrophobicsurfaces,J.Phys.Chem.C115 (20)(2011) 24975–24986,https://doi.org/10.1021/jp209140a.

[49]W.Ren,W.E,Boundaryconditionsforthemovingcontactlineproblem,Phys.Fluids19(2007)022101.

[50]M.Sbragaglia,K.Sugiyama,L.Biferale,Wettingfailureandcontactlinedynamicsinacouetteflow,J.FluidMech.614(2008)471–493.

[51]K.Sefiane,M.Shanahan,M.Antoni,Wettingandphasechange:opportunitiesandchallenges,Curr.Opin.ColloidInterfaceSci.16(2011)317–325.

[52]D.Seveno,T.D.Blake,S.Goossens,J.DeConinck,Predictingthewettingdynamicsofatwo-liquidsystem,Langmuir27(2011)14958–14967.

[53]D.Seveno,T.D.Blake,S.Goossens,J.DeConinck,Correctionto’predictingthewettingdynamicsofatwo-liquidsystem’,Langmuir34(2018)5160–5161.

[54]D.Seveno,N.Dinter,J.DeConinck,Wettingdynamicsofdropspreading.newevidenceforthemicroscopicvalidityofthemolecular-kinetictheory, Langmuir26(2011)14642–14647.

[55]D.Seveno,A.Vaillant,R.Rioboo,H.Adão,J.Conti,J.DeConinck,Dynamicsofwettingrevisited,Langmuir25 (22)(2009)13034–13044.

[56]S.Shao,T.Qian,Avariationalmodelfortwo-phaseimmiscibleelectroosmoticflowatsolidsurfaces,Commun.Comput.Phys.11(2012)831–862.

[57]J.Snoeijer,B.Andreotti,Movingcontactlines:scales,regimes,anddynamicaltransitions,Annu.Rev.FluidMech.45(2013)269–292.

[58]Z.Solomenko,P.Spelt,P.Alix,Alevel-setmethodforlarge-scalesimulationsofthree-dimensionalflowswithmovingcontactlines,J.Comput.Phys. 348(2017)151–170.

[59]Y.Sui,P.Spelt,Anefficientcomputationalmodelformacroscalesimulationsofmovingcontactlines,J.Comput.Phys.242(2013)37–52.

[60]L.Tanner,Thespreadingofsiliconeoildropsonhorizontalsurfaces,J.Phys.D,Appl.Phys.12(1979)1473–1484.

[61]P.Taylor,Thewettingofleafsurfaces,Curr.Opin.ColloidInterfaceSci.16(2011)326–334.

[62]P.Thompson,M.Robbins,Simulationsofcontact-linemotion:slipandthedynamiccontactangle,Phys.Rev.Lett.63(1989)766–769.

[63]E.Vandre,M.S.Carvalho,S.Kumar,Delayingtheonsetofdynamicwettingfailurethroughmeniscusconfinement,J.FluidMech.707(2012)496–520.

[64]S.Vassaux,V.Gaudefroy,L.Boulangé,A.Pévère,V.Mouillet,V.Barragan-Montero,Towardsabetterunderstandingofwettingregimesattheinterface asphalt/aggregateduringwarm-mixprocessofasphaltmixtures,Constr.Build.Mater.133(2017)182–195.

[65]O.Voinov,Hydrodynamicsofwetting,J.FluidDyn.11(1976)714–721.

[66]K.Winkels,J.Weijs,A.Eddi,J.Snoeijer,Initialspreadingoflow-viscositydropsonpartiallywettingsurfaces,Phys.Rev.E85(2012)055301(R).

[67]Y.Yamamoto,T.Ito,T.Wakimoto,K.Katoh,Numericalsimulationsofspontaneouscapillaryriseswithverylowcapillarynumbersusingafront-tracking methodcombinedwithgeneralizednavierboundarycondition,Int.J.Multiph.Flow51(2013)22–32.

[68]Y.Yamamoto,K.Tokieda,T.Wakimoto,T.Ito,K.Katoh,Modelingofthedynamicwettingbehaviorinacapillarytubeconsideringthe macroscopic?mi-croscopiccontactanglerelationandgeneralizednavierboundarycondition,Int.J.Multiph.Flow59(2014)106–112.

[69]H.S.Yoon,H.T.Lee,E.S.Kim,S.H.Ahn,Directprintingofanisotropicwettingpatternsusingaerodynamicallyfocusednanoparticle(afn)printing,Appl. Surf.Sci.396(2017)1450–1457.

[70]H.Yu,X.Yang,Numericalapproximationsforaphase-fieldmovingcontactlinemodelwithvariabledensitiesandviscosities,J.Comput.Phys.334 (2017)665–686.