lubrication approximation for fluids with shear-dependent viscosity bruno m.m. pereira参考-匠人.pdf

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Received:30April2019;Accepted:20May2019;Published:28May2019
Abstract:WepresentdimensionallyreducedReynoldstypeequationsforsteadylubricating?owspressiblenon-Newtonian?uidswithshear-,dependingonthestrengthofthepower-lawcharacterofthe?uid,thenovelequationcaneitherpresentitselfasahigher-:the?:power-lawfluid;shear-dependentviscosity;Reynoldsequation;’lubricationapproximation(Reynolds[1])oftheNavier-Stokesequationisacornerstoneofclassical?,manyofthelubricatingoilsthatarecurrentlyinusecannotbeappropriatelydescribedbytheNavier-,theyshearthin,displaystress-relaxation,instantaneouselasticity,non-linearcreep,thresholdforthestrainratebeforetheystartto?ow,thixotropy,,rateandintegraltypehavebeendevelopedtodescribethenon-Newtonianbehaviorexhibitedbysuch?uids,andlubricatingapproximationshavebeenderivedforavarietyof?uid?,weareonlyinterestedinaveryspecialsub-classofnon-Newtonian?uids,namelyweareinterestedindevelopingalubricationapproximationforshearthinning?-Newtoniancharacteristics,[2],seealso[3–5].Earlystudiesconcerningthelubricationapproximationforthe?owofpower-lawconstitutiverelationswereprimarilyconcernedwithpurelyonedimensional?owswhereinertialeffectsdonotmanifestthemselves(seeforinstanceShuklaetal.[6]);othersconcernnon-inertial?owsFluids2019,4,98;doi:/?uids4020098rnal/?uids:..Fluids2019,4,982of17(seeforexampleParkandKwon[7]),andyetothersconcernthelubricationapproximationforapowerlaw?uid,underin?nitewidegapapproximation(seethestudybyJohnsonandMangkoesoebroto[8]).Bourgin[9],seealsoKacouetal.[10],developedlubricationapproximationfor?uidsofthedifferentialtypeandHarnoy[11]studiedthelubricationapproximationforanelastic-viscous?[12]studiedthe?owofaviscoelastic?uidinajournalbearingusingaperturbationanalysiswhichwasshowntobeincorrectbySanAndresandSzeri[13].Caletal.[14]developedalubricationapproximationforviscoelastic?uidsandshowedthatviscoelasticitycanhavepronouncedeffectforcertainvaluesofthe?lmthicknessandinthecaseofthejournalbearing,,in?owsinvolvinghighpressuresasencounteredinelastohydrodynamics,ount(seeBarus[15],Bair[16]).Lubricationapproximationhasbeendevelopedinthecaseof?uidswithpressuredependentviscositybyRajagopalandSzeri[17]al.[18].Finally,Fusietal.[19]havestudiedthelubricationountinertialeffects;?uidsliketheBingham?uidthathaveathresholdinthestressfor?owtotakeplacearebestdescribedbyconstitutiverelationswhereinthekinematicsisdescribedasafunctionofthestressratherthanexpressingthestressintermsofthekinematicalvariableinthetraditionalmanner(seeRajagopal[20,21]foradiscussionofsuch?uidsaswellasmoregeneral?uidsthataredescribedbyimplicitconstitutiverelations).Asthereisathresholdforthestressbeyondwhichthe?uidstartsto?ow,thegoverningequationsarequitedifferentfromthelubricationapproximationobtainedinthecaseoftheotherstudiesthatemploy?uidmodelsthatdonothavesuchathresholdforthe???uencethe?owcharacteristicsattheordersofapproximationconsideredinthiswork(seeNazarovandVideman[22]fortheinertialcorrectiontotheReynoldslubricationapproximation).Thepower-law?uidmodelunderconsiderationhastwoconstantsthatdetermineitsviscosity,aconstantpower-lawexponentnandanotherconstanta0thatdeterminesthedepartureoftheviscosityfromtheNewtonianviscositywhenthepower-lawexponentis2(seeEquation(7)).Aformalperturbationanalysisiscarriedout,assumingtwodifferentpossibilitiesforthematerialparametera0,namelythatitisoftheorderO(e3)andO(e2),,wesimplyobtainahigher-odercorrectiontotheclassical(linear)Reynoldsequationbutifa0isofO(e2)esfullynonlinearandmustbesolvedtogetherwithanODEforthemainpartofthe?,twoproblemsaresolved,the?rstbeingthe?uid?owbetweenarollingrigidcylinderandarigidplane,?owofpressible,homogeneouspower-law?uid?vr+[rv]v+rp=rf+divS,(1)?tdivv=0,(2)wherer>0istheconstantdensityofthe?uid,v=(u,v)isthevelocity?eld,Sisthedeviatoricstresstensor(T=pI+SistheCauchystresstensor),pisascalarvariable,oftenreferredtoasthemechanicalpressure,pressibilityconstraint(2),andfisanexternalforceactingonthe?,pisnotthemeanvalueofthestressbutforthemodelconsideredinthispaperitis(seeRajagopal[23]foradetaileddiscussionofthenotionof:..Fluids2019,4,983of17pressure,itsuse,misuseandabuse).Thedeviatoricstresstensorfora?uidwithshear-dependentviscositywhichweshallstudyisrelatedtothesymmetricpartofthevelocitygradientasfollows:n222S=2m01+a0jDjD,(3)wheren1isthepower-lawexponentanda0ascalarcoef?cientrelatedtothepower-lawcharacter21T
ofthe?uid,withdimensionsoftimesquared(dim[a0]=T).Moreover,D(v)=2rv+(rv)denotesthesymmetricpartofthevelocitygradient(rv)=?vi,andm>?-dimensionalthin?owswithoutexternalforces(f=0).Wewillnowintroducethefollowingdimensionless(starred)quantities(x,y)=L1(x,y),v=U1v,p=P1p,S=S1S,a=U2L2a,(4)00whereLandUrepresenttypicallengthandvelocityscalesandwherethecharacteristicpressurePandthecharacteristicstressSaretakentobeP=S=mUL1.(5)0heusualReynoldsnumberRethroughRe=rULm1,(6)0andmakethefollowingassumptionsthatareappropriateforaclassoflubricationproblems:the?owtakesplacebetweentwoalmostparallelsurfacessituatedaty=0andy=H(x);thelubricating?lmisthin,thatis,H(x)=eh(x),wheree1denotesasmallnon-dimensionalparameter;the?owisslowenoughortheviscosityhighenoughsothatRe=O(e);thepower-lawparameterwillhavetobesuchthata0=O(e2)ora0=O(e3).Next,wewilldropthestarsandintroducethefastvariabley!e1y(thestretchednormalcoordinate).en222Re([rv]v)+rp=2div1+a0jDjD,(7)divv=0.(8)Theadherenceboundaryconditionsonrigid,impermeableboundariesreadasv=U0,U0
e=0,U0
e=U0,aty=0,(9)yxv=Uh,Uh
e=0,U0
e=Uh,aty=h(x),(10)ntwhereU0andUhdenotegivenconstantvelocitiesoftheboundariesandenandetstandfortheunitnormalandtangentvectorsaty=h(x).Thesevectorsarerelatedtotheunitvectorsexandeythroughtheformulae?(eh(x)y)ex+?(eh(x)y)ey0r(eh(x)y)?x?yeh(x)exeyen==p=p,(11)kr(eh(x)y)k1+e2(h0(x))21+e2(h0(x))2ex+eh0(x)eyet=p.(12)1+e2(h0(x))2:..Fluids2019,4,=O(e3)Wewillassumethefollowingansatzv(e,x)=e0v(1)(x)+e1v(2)(x)+e2v(3)(x)+...,(13)p(e,x)=e2p(0)(s)+e1p(1)(x)+e0p(2)(x)+e1p(3)(x)+...,(14)wherex=(x,y)andthefunctionsv(j),p(j)andareofO(1).AccordingtotheassumptionsRe=O(e)anda0=O(e3),andthusonsettingRe=eR,a=e3a,(15)(13)and(14)intoEquations(7)and(8),weobtainatO(e3)in(1)thatp(0)=p(0)(x),.,independentofy;atO(e2)in(1),atO(e1)in(2)andatO(e0)in(9)and(10)dp(0)?2u(1)=2,(16)dx?y?p(1)=0,(17)?y?v(1)=0,(18)?yu(1)=U0,v(1)=0,aty=0,(19)u(1)=Uh,v(1)=0,aty=h(x),(20)whereU0andUhdenotethetangentialvelocitiesaty=0andaty=h(x).From(18)–(20)itfollowsthatv(1)(x,y)0.(21)Wehavemoreoverthatp(1)=p(1)(x)from(17).UsingEquation(16)andtheboundaryconditions(19)and(20)foru(1)weconcludethatUhU0y(yh(x))dp(0)u(1)(x,y)=U0+y+.(22)h(x)2dxTheequationsatthenextorderreadas2!2!2dp(1)11?u(1)?2u(1)?u(1)?2u(1)=4(n4)a+(n2)adx22?y?y2?y?y2!23?u(1)?2u(1)?2u(2)+a+25,(23)?y?y2?y2?p(2)?2u(1)=,(24)?y?x?y?v(2)?u(1)=,(25)?y?x:..Fluids2019,4,985of17v(2)=u(2)=0,aty=0,(26)v(2)=Uhh0(x),u(2)=0aty=h(x).(27)Equations(24)–(27)canbeexpressedasaone-dimensionalStokessystemfor(v(2),p(2)).Assumingthatthissystemissolvable,itfollowsfrom(23)and(25)thatu(2)satis?esthesecond-orderODE(iny)!2?2u(2)dp(1)3?u(1)?2u(1)2=(n2)a2,(28)?ydx4?y?yu(2)=0,aty=0,(29)u(2)=0,aty=h(x).(30)Ontheotherhand,theStokessystem(24)–(27)patibilityconditionZh(x)(1)?u0hdy=h(x)U(31)0?xissatis?,thisamountstoZh(x)d(1)udy=0,(32)dx0TheclassicalReynoldsequationforp(0)isthenaconsequenceof(22),namely"#dh3dp(0)1dh=(U0+Uh).(33)dx12dx2dxTheequationsatO(e0)in(1),atO(e1)in(2)andatO(e2)in(9)and(10)readas?2u(3)2=...,(34)?y?p(3)?2v(3)+2=...,(35)?y?y?v(3)?u(2)=,(36)?y?xu(3)=v(3)=0,aty=0,(37)(3)102h(3)u=hU,v=0,aty=h(x),(38)2whereby...(35)–(38)formaone-dimensionalStokessystemfor(v(3),p(3))andthevelocitycomponentu(3)satis?esthesecond-orderODE?2u(3)2=...,(39)?yu(3)=0,aty=0,(40)(3)102hu=hU,aty=h(x).(41)2:..Fluids2019,4,986of17arisingfromEquation(34)andfromtheboundaryconditions(37)and(38).Recallthattermsindicatedby...-dimensionalStokessystemfor(v(3),p(3))issolvableifandonlyifthefollowingcompatibilityconditionissatis?edZh(x)?u(2)dy=0.(42)0?xUsingtheLeibnizintegralrule,esZh(x)d(2)udy=0.(43)dx0Integratingbyparts,oneseesthatZh(x)1Zh(x)?2u(2)u(2)dy=y(yh(x))dy.(44)020?y2Substituting(28)intothepreviousexpressionleadstoaReynoldstypeequationforthe?rst-orderpressurecorrection"#20!13(1)(0)(0)2dhdpa(n2)******@h024dpA=5h4(UU)+hdx12dx320dxdx0!132(0)(0)******@h024dpA5+h220(UU)+3h.(45)=O(e2)Usingagaintheansatz(13)and(14)andmakingthesameassumptionsasinthepreviouscase,exceptforthematerialparametera0whichiswrittenasa0=ae2,weobtainatO(e3)in(1)that?p(0)=0,.,p(0)isindependentofy.?yAtO(e2)in(1),atO(e1)in(2)andatO(e0)in(9)and(10),weobtain,respectively0!1n42!3(0)(1)222(1)(1)22(1)******@a?uA4?ua?u?u5=1+2+(n1)2,(46)dx2?y?y2?y?y?p(1)=0,(47)?y?v(1)=0,(48)?yu(1)=U0,v(1)=0,aty=0,(49)u(1)=Uh,v(1)=0,aty=h(x),(50)whereU0andUhdenotethetangentialvelocitiesaty=0andaty=h(x).From(48)–(50)itfollowsthatv(1)(x,y)0,andfrom(47),wealsoseethatp(1)=p(1)(x).:..Fluids2019,4,987of17Atthenextorderine,theequationsare0!1n60!1(1)(1)2(1)(2)(1)22(1)2dpa?u?u?******@a?uA@?uA=(n4)2+1a(n1)+2dx4?y?y?y2?y?y0!1n24"!#(1)22(2)(1)(2)2(1)(1)2(2)***@a?uA?ua?u?u?u?u?u++12+(n1)22+2,(51)2?y?y2?y?y?y?y?y0!1n24(2)(1)2(1)(1)2(1)?******@a?uA?u?u?u=+1a(n2)2?y2?y?x?y?y0!211?2u(1)a?u(1)+***@1(n3)AA,(52)?x?y2?y?v(2)?u(1)=,(53)?y?xv(2)=u(2)=0,aty=0,(54)v(2)=Uhh0(x),u(2)=0aty=h(x).(55)TheStokessystem(52)–(55)patibilityconditionZh(x)(1)?u0hdy=h(x)U(56)0?xissatis?ed