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Some theoretical results concerning non-Newtonian fluids of the Oldroyd kind. (English) Zbl 0914.76006

The paper deals with incompressible viscoelastic fluids satisfying the Oldroyd constitutive law which include Newtonian fluids as a special case. The authors prove local and global existence, uniqueness and stability results. Nevertheless, the global in time results are proven only for small data.

MSC:

76A10 Viscoelastic fluids
35Q35 PDEs in connection with fluid mechanics
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References:

[1] G. Astarita - G. Marrucci , ” Principles of Non-Newtonian Fluid Mechanics ”, McGraw Hill , New York , 1974 . · Zbl 0316.73001
[2] J. Baranger - D. Sandri , Finite element approximation of viscoelastic fluid flow: Existence of approximate solutions and error bounds , Numer. Math. 63 ( 1992 ), 13 - 27 . Article | MR 1182509 | Zbl 0761.76032 · Zbl 0761.76032
[3] M.J. Crochet - A.R. Davies - K. Walters , ” Numerical Simulation of Non-Newtonian Flow ”, Elsevier , Amsterdam , 1985 . MR 801545 | Zbl 0583.76002 · Zbl 0583.76002
[4] R. Diperna - P.-L. Lions , Ordinary differential equations, transport theory and Sobolev spaces , Invent. Math. 98 ( 1989 ), 511 - 547 . MR 1022305 | Zbl 0696.34049 · Zbl 0696.34049
[5] E. Fernández-Cara - F. Guillén - R.R. Ortega , Existence et unicité de solution forte locale en temps pour des fluides non newtoniens de type Oldroyd (version LS - Lr) , C. R. Acad. Sci. Paris. Sér. I Math. 319 ( 1994 ), 411 - 416 . MR 1289322 | Zbl 0808.76005 · Zbl 0808.76005
[6] A. Friedman , ” Partial Differential Equations ”, Holt - Rinehart - Winston , New York , 1976 . MR 454266 | Zbl 0224.35002 · Zbl 0224.35002
[7] H. Giesekus , A unified approach to a variety of constitutive models for polymer fluids based on the concept of configuration dependent molecular mobility , Rheol. Acta 21 ( 1982 ), 366 - 375 . Zbl 0513.76009 · Zbl 0513.76009
[8] Y. Giga - H. Sohr , Abstract LP estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains , J. Funct. Anal. 102 ( 1991 ), 72 - 94 . MR 1138838 | Zbl 0739.35067 · Zbl 0739.35067
[9] C. Guillopé - J.-C. Saut , Existence results for the flow of viscoelastic fluids with a differential constitutive law , Nonlinear Anal. Vol. 15 , No. 9 , ( 1990 ), 849 - 869 . MR 1077577 | Zbl 0729.76006 · Zbl 0729.76006
[10] C. Guillopé - J.-C. Saut , Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type , Math. Mod. Numer. Anal. Vol. 24 , No. 3 , ( 1990 ), 369 - 401 . Numdam | MR 1055305 | Zbl 0701.76011 · Zbl 0701.76011
[11] O.A. Ladyzhenskaya , ” The Mathematical Theory of Viscous Incompressible Flow ”, Gordon and Breach , New York , 1969 . MR 254401 | Zbl 0184.52603 · Zbl 0184.52603
[12] R.G. Larson , A critical comparison of constitutive equations for polymer melts , J. Non-Newtonian Fluid Mech. 23 ( 1987 ), 249 - 269 .
[13] J. Leray , Sur le mouvement d’une liquide visqueux emplissant l’espace , Acta Math. 63 ( 1934 ), 193 - 248 . JFM 60.0726.05 · JFM 60.0726.05
[14] J.L. Lions , ” Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires ”, Dunod , Gauthier-Villars , Paris , 1969 . MR 259693 | Zbl 0189.40603 · Zbl 0189.40603
[15] J.G. Oldroyd , On the formulation of rheological equations of state , Proc. Roy. Soc. London Ser. A 200 ( 1950 ), 523 - 541 . MR 35192 | Zbl 1157.76305 · Zbl 1157.76305
[16] J.G. Oldroyd , Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids , Proc. Roy. Soc. London Ser. A 245 ( 1958 ), 278 - 297 . MR 94085 | Zbl 0080.38805 · Zbl 0080.38805
[17] R.R. Ortega , Thesis , University of Seville ( Spain ), 1995 .
[18] N. Phan Thien - R.I. Tanner , A new constitutive equation derived from network theory , J. Non-Newtonian Fluid Mech. 2 ( 1977 ), 353 - 365 . Zbl 0361.76011 · Zbl 0361.76011
[19] M Renardy , Existence of slow flows of viscoelastic fluids with differential constitutive equations , Z. Angew. Math. Mech. 65 ( 1985 ), 449 - 451 . MR 814684 | Zbl 0577.76014 · Zbl 0577.76014
[20] M. Renardy - W.J. Hrusa - J.A. Nohel , ” Mathematical Problems in Viscoelasticity ”, Longman , London , 1987 . MR 919738 | Zbl 0719.73013 · Zbl 0719.73013
[21] D. Sandri , Approximation par éléments finis d’écoulements de fluides viscoélastiques: Existence de solutions approchées et majoration d’erreur II. Contraintes continues , C. R. Acad. Paris Sér. I Math. 313 ( 1991 ), 111 - 114 . MR 1119920 | Zbl 0737.76048 · Zbl 0737.76048
[22] R. Témam , ” Navier-Stokes Equations, Theory and Numerical Analysis ”, North-Holland , Amsterdam , 1977 . MR 609732 | Zbl 0383.35057 · Zbl 0383.35057
[23] A. Valli , Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method , Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10 ( 1983 ), 607 - 647 . Numdam | MR 753158 | Zbl 0542.35062 · Zbl 0542.35062
[24] A. Valli , Navier-Stokes equations for compressible fluids: global estimates and periodic solutions , Proc. Sympos Pure Math. 45 ( 1986 ), 467 - 478 . MR 843633 | Zbl 0601.35094 · Zbl 0601.35094
[25] K. WALTERS (ed.), ” Rheometry: Industrial Applications ”, J. Wiley and Sons , 1980 .
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