Geodesic
On steady motion of viscoelastic fluid of Oldroyd type
Sbornik. Mathematics, Tome 205 (2014) no. 6, pp. 763-776 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider a mathematical model describing the steady motion of a viscoelastic medium of Oldroyd type under the Navier slip condition at the boundary. In the rheological relation, we use the objective regularized Jaumann derivative. We prove the solubility of the corresponding boundary-value problem in the weak setting. We obtain an estimate for the norm of a solution in terms of the data of the problem. We show that the solution set is sequentially weakly closed. Furthermore, we give an analytic solution of the boundary-value problem describing the flow of a viscoelastic fluid in a flat channel under a slip condition at the walls. Bibliography: 13 titles.
Keywords: non-Newtonian fluids, viscoelastic media, Oldroyd model, flow in a channel.
Mots-clés : Navier slip condition 
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E. S. Baranovskii. On steady motion of viscoelastic fluid of Oldroyd type. Sbornik. Mathematics, Tome 205 (2014) no. 6, pp. 763-776. http://geodesic.mathdoc.fr/item/SM_2014_205_6_a0/

[1] K. R. Rajagopal, “On some unresolved issues in non-linear fluid dynamics”, Russian Math. Surveys, 58:2 (2003), 319–330 | DOI | DOI | MR | Zbl

[2] C. L. M. H. Navier, “Memoire sur le lois du mouvement des fluides”, Mem. Acad. Roy. Sci. Paris, 6 (1823), 389–416

[3] M. Reiner,, “Rheology”, Handbuch der Physik, herausgegeben von S. Flügge, v. 6, Elastizität und Plastizität, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1958, 434–550 | MR | Zbl

[4] G. Astarita, G. Marucci, Principles of non-Newtonian fluid mechanics, McGraw-Hill, New York, 1974, 296 pp.

[5] D. A. Vorotnikov, “On the existence of weak stationary solutions of a boundary value problem in the Jeffreys model of the motion of a viscoelastic medium”, Russian Math. (Iz. VUZ), 48:9 (2004), 10–14 | MR

[6] E. S. Baranovskii, “An inhomogeneous boundary value problem for the stationary motion equations of Jeffreys viscoelastic medium”, J. Appl. Ind. Math., 7:1 (2013), 22–28 | DOI

[7] V. G. Zvyagin, D. A. Vorotnikov, “Approximating-topological methods in some problems of hydrodynamics”, J. Fixed Point Theory Appl., 3:1 (2008), 23–49 | DOI | MR | Zbl

[8] C. Guillopé, J.-C. Saut, “Existence results for the flow of viscoelastic fluids with a differential constitutive law”, Nonlinear Anal., 15:9 (1990), 849–869 | DOI | MR | Zbl

[9] L. D. Landau, E. M. Lifshitz, Course of theoretical physics, v. 6, Fluid mechanics, 2nd ed., Pergamon Press, Oxford, 1987, xiv+539 pp. | MR | MR | Zbl

[10] J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, v. 1, Travaux et Recherches Mathematiques, 17, Dunod, Paris, 1968, xx+372 pp. | MR | Zbl | Zbl

[11] V. G. Litvinov, Dvizhenie nelineino-vyazkoi zhidkosti, Nauka, M., 1982, 375 pp. | MR | Zbl

[12] R. Temam, Navier–Stokes equations. Theory and numerical analysis, Stud. Math. Appl., 2, rev. ed., North-Holland Publishing Co., Amsterdam–New York, 1979, x+519 pp. | MR | MR | Zbl | Zbl

[13] I. V. Skrypnik, Methods for analysis of nonlinear elliptic boundary value problems, Transl. Math. Monogr., 139, Amer. Math. Soc., Providence, RI, 1994, xii+348 pp. | MR | MR | Zbl | Zbl