ON STEADY THIRD GRADE FLUIDS EQUATIONS

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ON STEADY THIRD GRADE FLUIDS EQUATIONS ADRIANA VALENTINA BUSUIOC, DRAGOS¸ IFTIMIE AND MARIUS PAICU Abstract. Let ? be a simply connected, bounded, smooth domain of R2 or R3. We consider the equation of steady motion of a third grade fluid in ? with homogeneous Dirichlet boundary conditions. We prove that the monotonicity technique used by Paicu [22] in the full space for unsteady motion allows to obtain the existence of a W 1,40 solution provided that the forcing belongs to W ?1, 43 . The size of the forcing is arbitrary. Key words: Non-Newtonian fluids, third grade fluids. 1. Introduction Fluids of grade three are a subclass of the family of fluids of complexity three for which the constitutive law is given by the formula (1) T = ?pI + ?A+ ?1A2 + ?2A 2 + ?1A3 + ?2(AA2 + A2A) + ?|A| 2A where A1 ? A, A2 and A3 are the first three Rivlin-Ericksen tensors (or rate-of-strain tensors) defined recursively by A = A1 = ?u+ (?u) t, An = A˙n?1 + (?u) tAn?1 + An?1?u, where the dot denotes the material derivative and u is the velocity field. Relation (1) arises when the fluid is assumed incompressible and the constitutive law is polynomial of degree less than 3 in the first three Rivlin-Ericksen tensors.

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Vector field Existential quantification

ONSTEADYTHIRDGRADEFLUIDSEQUATIONSADRIANAVALENTINABUSUIOC,DRAGO¸SIFTIMIEANDMARIUSPAICUAbstract.LetΩbeasimplyconnected,bounded,smoothdomainofR2orR3.WeconsidertheequationofsteadymotionofathirdgradeﬂuidinΩwithhomogeneousDirichletboundaryconditions.WeprovethatthemonotonicitytechniqueusedbyPaicu[22]inthefullspaceforunsteadymotion44,1allowstoobtaintheexistenceofaW0solutionprovidedthattheforcingbelongstoW−1,3.Thesizeoftheforcingisarbitrary.Keywords:Non-Newtonianﬂuids,thirdgradeﬂuids.1.IntroductionFluidsofgradethreeareasubclassofthefamilyofﬂuidsofcomplexitythreeforwhichtheconstitutivelawisgivenbytheformula(1)T=−pI+νA+α1A2+α2A2+β1A3+β2(AA2+A2A)+β|A|2AwhereA1≡A,A2andA3aretheﬁrstthreeRivlin-Ericksentensors(orrate-of-straintensors)deﬁnedrecursivelybyA=A1=ru+(ru)t,An=A˙n−1+(ru)tAn−1+An−1ru,wherethedotdenotesthematerialderivativeanduisthevelocityﬁeld.Relation(1)ariseswhentheﬂuidisassumedincompressibleandtheconstitutivelawispolynomialofdegreelessthan3intheﬁrstthreeRivlin-Ericksentensors.Toproveamathematicaltheoryofexistenceanduniquenessofsolutionsfortheconstitutivelaw(1)withnorestrictionsonthematerialcoeﬃcientsν,α1,α2,β1,β2,β(otherthantheobviousconditionν≥0)seemstobeoutofreach.Onecanﬁndconditionsonthesecoeﬃcientseitherthroughtheoreticalinvestigationsorfromexperimentaldata.TheoreticalconditionswerefoundbyFosdickandRajagopal[16].Theseauthorsperformedathermodynamicstudyanddeducedthatthematerialcoeﬃcientsshouldsatisfytherestrictionsν≥0,α1≥0,β1=β2=0,β≥0and|α1+α2|≤(24νβ)1/2.Unfortunately,experimentaldataseemstobeatoddswiththetheoreticalresults.Indeed,virtuallyallexperimentaldataexhibitsnegativevaluesofα1.Thereasonwhythisapparentcontradictionoccursisbeyondthescopeofthepresentpaper;wereferthereaderto[15,17,19]foradiscussioninvolvingbothpointsofview.Wealsoremarkthatνshouldbepositiveandthatbothsignsofthecoeﬃcientβ(ifβ>0theﬂuidissaidshearthickeningwhileforβ<0theﬂuidissaidshearthinning)areobservedinexperimentsalthoughthesignminusforβseemstobemorefrequent,see[25,Fig.7,22,23]and[4,Table6.2-1].Inorderforthemodeltobemathematicallyamenablewewillassumethatβ1=β2=0asin[16]butwewillallowsignsofcoeﬃcientsinagreementwiththeexperimentaldata.Inparticular,α1maybenegativeinourresults(seethestatementsofTheorems1and2belowforthepreciserestrictionsimposedonthecoeﬃcients).WeareconcernedinthisworkwiththesteadymotionofathirdgradeﬂuidinaboundeddomainΩendowedwithhomogeneousDirichletboundaryconditions.Giventheconditionβ1=β2=0,theequationofmotionbecomes−ν4u+u∙ru−α2div(A2)−α1div(u∙rA+LtA+AL)−βdiv(|A|2A)=f−rp,inΩ,(2)divu=0,inΩ,u=0,on∂Ω.1

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