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On the use of a friction model in a Volume of Fluid
solver for the simulation of dynamic contact lines
H. Si Hadj Mohand, H. Hoang, Guillaume Galliero, Dominique Legendre
To cite this version:
H. Si Hadj Mohand, H. Hoang, Guillaume Galliero, Dominique Legendre. On the use of a friction
model in a Volume of Fluid solver for the simulation of dynamic contact lines. Journal of
Computa-tional Physics, Elsevier, 2019, 393, pp.29-45. �10.1016/j.jcp.2019.05.005�. �hal-02386087�
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This is an author’s version published in:
http://oatao.univ-toulouse.fr/2
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,
∗
aInstitutdeMécaniquedesFluidesdeToulouse(IMFT),UniversitédeToulouse,CNRS- Toulouse,France
bCNRS/TOTAL/Univ.Pau&PaysAdour/E2SUPPA,LaboratoiredesFluidesComplexesetleursRéservoirs-IPRA,UMR5150,64000,PAU,France cInstituteofFundamentalandAppliedSciences,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) contactangleθ
Y [22].However,ithasbeenputinevidencethatthemicroscopicangleθ
m maydifferfromthestaticangleθ
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
=
2K0λ
sinhγ
(
cosθ
Y−
cosθ
m)
2nkBT (1)where
λ
andK0arethejumplengthandequilibriumfrequency,respectively,kB istheBoltzmannconstant,n isthedensityofadsorption sitesonthesolidsurface usuallyconsidered equalto1
/λ
2 andT isthetemperature. Fromthisrelationthe microscopicmovingcontactangleθ
mcanbeexpressedasafunctionoftheratiobetweenthecontactlinevelocityandthejump velocity K0
λ
of themolecules.Whenthe contactlinevelocityis muchsmallerthan thejump velocity,the relationcanbelinearizedas
cos
θ
Y−
cosθ
m=
ξ
γ
Vcl (2)withtheintroductionofafrictioncoefficient
ξ
atthecontactlinethatcanbeexpressedas[52]:ξ
=
kBT K0λ
3=
μν
Lλ
3 exp−
γ
λ
2(
1+
cosθ
Y)
kBT (3)with
ν
L theflowunitvolumeandμ
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
YnxistheuncompensatedYoungstress,whoseintegrationovertheinterfacesatisfies
−
int
σ
nxYdx=
γ
(
cosθ
m−
cosθ
Y)
(5)Then,RenandE[49] havereformulatedtheGNBCrelationtoproposethefollowingrelationfortheunbalancedYoungstress
cos
θ
m−
cosθ
Y=
β
clγ
Vcl (6)where
β
clisaneffectivefrictioncoefficientatthecontactlineaccountingforboththecontactlinefrictionandtheviscousFig. 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. Numericalset-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 andsameviscosityμ
=
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 forlikemoleculesandδ
= −
1 forunlike-fluidmolecules.Fluid-solid molecular interactionshavebeendescribed usingthesamemodified Lennard-Jonespotential withthe energyscalef s
=
1
.
16,therangeofinteraction
σ
f s=
1.
04σ
andδ
= δ
f s withδ
f sdefinedas:δ
f s=
1 forbothfluidsinthesymmetriccasewhile
δ
f s=
1 for one fluid andδ
f s=
0.
7 for the other one in the asymmetriccase. Inthe following all parameters aremadedimensionlessusingthemassm ofthefluidmolecules,therangeofinteraction
σ
andtheenergyscale.Basedon thisnormalization, thenondimensionalparametersoftheproblemconsideredinthisstudyare H
=
13.
6,ρ
1=
ρ
2=
0.
81,μ
1=
μ
2=
1.
95 andγ
=
5.
5. Weconsider the two differentwetting conditionsstudied in Qianetal.[46]:a symmetriccasecorresponding to the equilibriumcontactangle
θ
Y=
90o and an asymmetric caseobtainedimposingθ
Y=
64o.Thecorresponding 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] deducedthesliplengthsvalues1
=
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+
2i 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. TheNavier-StokesVoFsolver
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
=
Cρ
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+ ∇
TVwithμ
=
Cμ
1
+ (
1−
C)
μ
2. Fγ isthecapillarycontribution,givenby:Fγ
=
γ
∇ ·
nInIδ
I (11)where
γ
is the interfacial tension betweenthe two phases, nI= ∇
C/
∇
C is the unit vector normal to the interfacepointingintofluid1and
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 andvelocityVcloneachwall aredeterminedbyasecondorderlinearinterpolationattheinterfacelocation(C
=
0.
5).Consideringthesituationwheretheinterfaceislocatedbetweenthecell i andi
+
1,i.e.Ci≥
0.
5 andCi+1≤
0.
5,thecontactlinepositionandvelocityrelativetothebottommovingwallarecalculatedas 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 isCˆ
= ˆ
Cm attime(
n+
1/
2)
t where Cˆ
m isobtainedafterm iterations.For2Dsimulationsonaregulargridasconsideredinthiswork:
Cin,fj
=
3 4C nf−1 i,j
+
1 16Cni+f−1,1j
+
Cni−f1,−1j+
Cni,fj−+11+
Cni,fj−−11 (15)withnf
=
1,
...,
m andC0beinginitializedwithCn+1/2.FollowingDupontandLegendre[20],thecurvatureandthenormal
involvedinthecapillaryterm, areobtainedwithtwodifferentvaluesform,mκ
=
12 andmL=
6,respectively.Thecorre-spondingspurious currents intensityhasbeencharacterized andtheir maximum magnitudeevolveas 0
.
004γ
/
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∇
CdV⎞
⎠ ∇
C (16)wherethedivergencetermcorrespondingtothecurvatureisconvertedtoasurfaceintegral.Thistermisthenevaluatedas fluxesoftheinterfacenormalnI
= ∇
C/
∇
Catthesurfaceof
:
Fγ
=
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 controlvolumeusedto calculate thevelocity inthex-direction andcontainingthe contactline(i.e.0
<
C<
1). Thecontributionofthecontactlinetothe momentumbalanceincomesfromthesouthface
South:
Fγ,x,South
=
γ
nI·
nSouth
∇
C·
ex (18)wherethenormalofthesouthfaceisn
= −
ey andthearea isSouth
=
.Theangleθ
m madebetweentheinterfaceandthewallimposes
nI
·
ey=
cosθ
m (19)sothatthecontactlinecontributiontothemomentumin
is
Fγ,x,South
= −
γ
cosθ
m∇
C·
ex (20)Thefrictionmodelproposedforthemovingcontactangle
θ
mispresentedinthenextsection.Timeandgridconvergence4. Thefrictionmodel
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
mmadebetweentheinterface andthewallisreducedcomparedtoitsequilibriumvalue
θ
Y.InadditiontheNaviercondition[43] isappliedonthewalls.ForawallmovingwithvelocityU ,theslipvelocity Vislip offluidi (i
=
1,
2)onthewallwrites Vislip=
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−
cosθ
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 angleθ
m= θ
Y andthe friction is then zero.As a consequence, the only contribution fromthe wall to the momentumbalancecomesfromthecapillarytermcontribution Fγ,x,South
onthesouthface
South givenbyequation (20).Thetotal
contributionofthistermovertheinterfacethickness
δ
n is intFγ,South∼ −
intγ
cosθ
Y∂
C∂
xdx=
γ
cosθ
Y (24)Interestinglythistermisindependentonthenumericalthickness
δ
n oftheinterface.Itisbalancedbythecapillarycontri-butionfromtheotherfacesof
andbythestaticpressureresultingintheLaplacepressurejumpattheinterfaceandan interfaceshapeminimizingthesurfaceenergyandsatisfyingtheimposedangleatthewall[20,39].
Wenowconsiderthecaseofamovingcontactline.Thecontactangleisnow
θ
m.ThenormalvelocityofthesouthfaceSouthbeingzero,theconvectivefluxiszero.Thus,thecontributionofthecontactlinetothemomentumbalanceon
South
mayincludeanadditionalfriction
τ
cl:Fγ,x,South
+
τ
cl=
−
γ
cosθ
m∇
C·
ex+
μ
cl∂
V∂
y(25)
andtheintegralovertheinterfacethicknessbecomes Fγ,x,South
+
τ
cl=
γ
cosθ
m+
μ
cl∂
V∂
yδ
n (26)δ
n beingtypically oftheorderof2-3grid size,a formulationbasedonaviscous likecontribution
τ
cl resultsina griddependentcontributionattheinterface.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 frictionξ
. Therelevantvaluefor
ξ
willbediscussedinsection6.5. Preliminarytests
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 totheimposed contactangle,thesystemstabilizestoformtwo concaveinterfacesofcircularshape withradius R
=
H/
cosθ
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 xi−
xiT H,
E1=
1 N H i xi−
xT Hi(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 beingsecondorder,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 tot
=
1.
5×
10−2.Thegrid ismadewith N=
64.The stabilizedinterface positionattime t=
120 is reportedinFig.4fordifferenttime 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 xit−
xit=5×10−4 (28) TheevolutionofEt1 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.ThemeannormalizederrorE
1 isdefinedusingthesolutionobtainedforthesmallestgridspacing,i.e.
=
H/
256:E1
=
1 N H i xi−
xiH/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.
μ
U/
H∼
γ
/
H .ItresultsacriticalvalueforthecapillarynumberCa=
μ
U/
γ
.Steadysolutions are thenobservedifthecapillarynumberissmallerthanacriticalvalue Ca∗.WhenCa>
Ca∗ therepellingcapillaryforcecan notresisttotheentrainmentofthetwocontactlinesandtheinterfaceiscontinuouslyelongated.ThisisillustratedinFig.7Fig. 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=
90oandanormalizedslip
/
H=
0.
089 forthetwofluids.The critical Capillary number Ca∗ dependson both the ratio
/
H and the imposed contact angleθ
m [34,50,25]. Todeterminethe transitionbetweenstableandunstablesituations,simulations are conductedfordifferentimposedcontact angles
θ
m whileimposing a constant slip/
H=
0.
089.The corresponding valuesof Ca∗ are reportedinFig. 8.Theyarecompared 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-parallelapproximationisverygoodforsmallangles.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. Impactofthefrictionmodel
6.1. Simulationwithastaticcontactangle
Wefirstsimulatethe2DCouetteflowdescribedinFig.1byimposingthestaticcontactangle,sonocontactlinefriction isimposed(
ξ
=
0).Thecapillaryforceonthewallisthuscalculatedwithθ
m= θ
Y (30)The symmetric (
θ
Y=
90o) and asymmetric (θ
Y=
64o) cases are considered. The Navier slip condition is imposed withthe slip lengthsdeduced by Qianet al.[46] from their MD simulations and reportedin Table 1. The grid used forthis comparisonismadeusing N
=
256 andthenormalizedtime stepist
=
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 isTable 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 consideringλ
asthe smallest distance betweentwo wall atoms, i.e.λ
= (
4/
ρ
s)
1/3/
√
2
=
0.
913.Thejumpfrequencyforafluidi displacingitsvaporis[6]K0,i
=
kBTμ
iν
i exp−
γ
λ
2(
1+
cosθ
Y)
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.Whenconsideringaninterfacebetweentwofluids,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 valuesfor the normalized friction coefficients are
ξ
M K T=
32.
8 andξ
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.
42δ
B (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] weconsiderδ
=
6.Accordingly,the frictioncoefficientsforthesymmetricandtheasymmetriccasesareξ
Ren=
3.
024 andξ
Ren=
2.
18,respectively.ThevaluesusedforthefrictioninthefollowingsimulationsaresummarizedinTable2.
ThefrictiongivenbytheMKTpredictionisnotabletoprovideasteadystatefortheinterfaceforboththesymmetricand theasymmetricconfiguration.Theinterfacemovescontinuouslyentrainedbythemovingwallsascommentedinsection5.3 (seeFig.7).Thisindicatesthatthefriction predictedbytheMKTmodelistoolargeandcannotbedirectlyintroduced in aNavier-Stokessolver.However, asteadyinterface isobtainedwhenusingthefriction
ξ
Ren deducedfromMDsimulationsasshowninFig.10.Theagreementappearsverygoodfortheasymmetriccase
θ
Y=
64o whileintheasymmetriccase,thefriction
ξ
Ren overestimatesthefriction requiredtomatchwiththereferencecase. Thisisevidencedby amore importantdisplacementofthecontactlineresultinginamoredeformedinterfacethanobservedinthecaseofreference.
Theseresultsindicatethattheinterfaceshapeseemstobeverysensitivetothefrictionintroducedinthecontactangle modelandthisconfirmstherelevanceofthechoiceofthistestcase.Despiteasmalldifferencebetween
θ
Y andθ
m(aroundFig. 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%.Thesliplengths1 and
2 areunchangedandcorrespondtothevaluesreportedinTable1.The
bestagreementwiththereferencecaseisobtainedfor
ξ
V o F=
1.
7 andξ
V o F=
0.
7 forθ
Y=
90o andθ
Y=
64o,respectively.The corresponding interface shapes are shown in Fig. 11. The frictions
ξ
V o F are one order of magnitude smaller thanpredicted 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 andθ
m= θ
Y=
64o)andwereducethesliplengthatthewalltogetabetteragreementwiththereference.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 theeffectofthegridsizeonthevalueofξ
V o F forthefrictionmodelandontheslipV o F fortheuseofaconstantcontact
angle.Theoptimalvalueisdeterminedforeachgridfollowingtheproceduredescribedabove.Thecorresponding valuesof thefriction andthesliplengthreportedinFig.13convergewiththegridrefinement.Thevaluesare verysensitivetothe gridforsmallresolutions,i.e.N
<
100.Forbetterresolutions,bothξ
V o F andV o F arealmostconstant.
7. Interfaceshape
The objectiveofthissectionisto focusontheinterfaceshape fordifferentcapillarynumbersandratios H
/
.ThegapH 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 modelwiththefrictionξ
V o F given inTable2.The domainsize is L=
4×
H forall thecases.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 fromθ
m=
89.
5oFig. 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=
Hβ
2 x xcl (34)wherexclisthecontactlinedisplacementandtheparameter
β
isafunctionofCa asshownintheinsertofFig.14.Closetothewalltheinterfaceshapedependsonboththecapillarynumberandtheimposedcontactangle.Todescribe theinterface shapeinthevicinityofthewall,weconsidertheevolutionoftheinterface angle
θ (
r)
madeby theinterfacewith 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 tothebottomcontactlineisgivenbyg
(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)
dβ
(36) with f(β,
q)
=
q
(β
2−
sin2β)
[(
π
− β) +
sinβ
cosβ
]+
(
π
− β)
2−
sin2β
(β
−
sinβ
cosβ)
2 sin
β
q2
(β
2−
sin2β)
+
2qβ(
π
− β) +
sin2β
+ ((
π
− β)
2−
sin2β)
When the contactangle satisfies the condition (
θ
≤
3π
/
4) fora liquid displacing a gas (q=
0), the function g can be simplifiedasg
(θ (
r),
q)
≈
θ
3
9 (37)
Fortwoliquidsthisrelationremainsagoodapproximationbutforsmallerangles,typically
θ
≤
π
/
3 forq=
1.Thus,forthe rangeofcontactanglesconsideredherethenextcorrectioninθ
needstobeconsideredintheexpansion[23]g
(θ (
r),
q)
=
θ
3 9−
qθ
4 8π
+
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.Forclaritytheanglevariationθ
m− θ(
r)
isreportedasafunctionofthenormalizeddistancer∗
=
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.
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