ON STEADY THIRD GRADE FLUIDS EQUATIONS

Description

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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Sujets

• Vector field
• Existential quantification

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 47
• Langue: English

Extrait

ONSTEADYTHIRDGRADEFLUIDSEQUATIONS ADRIANAVALENTINABUSUIOC,DRAGO¸SIFTIMIEANDMARIUSPAICU Abstract. LetΩbeasimplyconnected,bounded,smoothdomainof R 2 or R 3 .Weconsiderthe equationofsteadymotionofathirdgradefluidinΩwithhomogeneousDirichletboundaryconditions. WeprovethatthemonotonicitytechniqueusedbyPaicu[22]inthefullspaceforunsteadymotion 4 4,1 allowstoobtaintheexistenceofaW 0 solutionprovidedthattheforcingbelongstoW − 1, 3 .The sizeoftheforcingisarbitrary. Keywords: Non-Newtonianfluids,thirdgradefluids. 1. Introduction Fluidsofgradethreeareasubclassofthefamilyoffluidsofcomplexitythreeforwhichthe constitutivelawisgivenbytheformula (1)T= − pI+νA+α 1 A 2 +α 2 A 2 +β 1 A 3 +β 2 (AA 2 +A 2 A)+β | A | 2 A whereA 1 ≡ A,A 2 andA 3 arethefirstthreeRivlin-Ericksentensors(orrate-of-straintensors)defined recursivelyby A=A 1 = r u+( r u) t ,A n =A ˙ n − 1 +( r u) t A n − 1 +A n − 1 r u, wherethedotdenotesthematerialderivativeanduisthevelocityfield.Relation(1)ariseswhen thefluidisassumedincompressibleandtheconstitutivelawispolynomialofdegreelessthan3in thefirstthreeRivlin-Ericksentensors. Toproveamathematicaltheoryofexistenceanduniquenessofsolutionsfortheconstitutivelaw(1) withnorestrictionsonthematerialcoefficientsν,α 1 ,α 2 ,β 1 ,β 2 ,β(otherthantheobviouscondition ν ≥ 0)seemstobeoutofreach.Onecanfindconditionsonthesecoefficientseitherthrough theoreticalinvestigationsorfromexperimentaldata. TheoreticalconditionswerefoundbyFosdickandRajagopal[16].Theseauthorsperformeda thermodynamicstudyanddeducedthatthematerialcoefficientsshouldsatisfytherestrictionsν ≥ 0,α 1 ≥ 0,β 1 =β 2 =0,β ≥ 0and | α 1 +α 2 |≤ (24νβ) 1/2 .Unfortunately,experimentaldataseems tobeatoddswiththetheoreticalresults.Indeed,virtuallyallexperimentaldataexhibitsnegative valuesofα 1 .Thereasonwhythisapparentcontradictionoccursisbeyondthescopeofthepresent paper;wereferthereaderto[15,17,19]foradiscussioninvolvingbothpointsofview.Wealso remarkthatνshouldbepositiveandthatbothsignsofthecoefficientβ(ifβ>0thefluidissaid shearthickeningwhileforβ<0thefluidissaidshearthinning)areobservedinexperimentsalthough thesignminusforβseemstobemorefrequent,see[25,Fig.7,22,23]and[4,Table6.2-1]. Inorderforthemodeltobemathematicallyamenablewewillassumethatβ 1 =β 2 =0asin[16] butwewillallowsignsofcoefficientsinagreementwiththeexperimentaldata.Inparticular,α 1 may benegativeinourresults(seethestatementsofTheorems1and2belowforthepreciserestrictions imposedonthecoefficients). Weareconcernedinthisworkwiththesteadymotionofathirdgradefluidinaboundeddomain ΩendowedwithhomogeneousDirichletboundaryconditions.Giventheconditionβ 1 =β 2 =0,the equationofmotionbecomes − ν 4 u+u ∙r u − α 2 div(A 2 ) − α 1 div(u ∙r A+L t A+AL) − βdiv( | A | 2 A)=f −r p,inΩ, (2)divu=0,inΩ, u=0,on∂Ω. 1