The three-dimensional stationary flow problem at small Reynolds numbers

L. I. Sazonov

Izvestiya. Mathematics , Tome 75 (2011) no. 6, pp. 1185-1214.

Résumé

We study the problem of finding convergent series representations of solutions of exterior problems for the stationary Navier–Stokes system at small Reynolds numbers $\operatorname{Re}$. Our approach enables us to treat some cases of boundary conditions that increase as $\operatorname{Re}\to 0$.

Détail

Keywords: stationary Navier–Stokes system, exterior domain, hydrodynamical potentials, resistance force.

Bibliographie

[1] J. Leray, “Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique”, J. Math. Pures Appl. (9), 12 (1933), 1–82 | Zbl

[2] O. A. Ladyzhenskaya, “Investigation of the Navier–Stokes equation for a stationary flow of an incompressible fluid”, Transl. Ser. 2. Amer. Math. Soc., 25 (1963), 173–197 | MR | Zbl | Zbl

[3] R. Finn, “On the steady-state solutions of the Navier–Stokes equations. III”, Acta Math., 105:3–4 (1961), 197–244 | DOI | MR | Zbl

[4] I. Proudman, J. R. A. Pearson, “Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder”, J. Fluid Mech., 2:3 (1957), 237–262 | DOI | MR | Zbl

[5] M. van Dyke, Perturbation methods in fluid mechanics, Academic Press, New York–London, 1964 | MR | Zbl | Zbl

[6] T. M. Fischer, G. C. Hsiao, W. L. Wendland, “Singular perturbations for the exterior three-dimensional slow viscous flow problem”, J. Math. Anal. Appl., 110:2 (1985), 583–603 | DOI | MR | Zbl

[7] V. I. Yudovich, “O problemakh i perspektivakh sovremennoi matematicheskoi gidrodinamiki”, Uspekhi mekhaniki, 1:1 (2002), 61–102

[8] L. I. Sazonov, “The justification of an asymptotic expansion for the solution of the two-dimensional flow problem at small Reynolds numbers”, Izv. Math., 67:5 (2003), 975–1006 | DOI | MR | Zbl

[9] D. A. Maltsev, L. I. Sazonov, Asimptoticheskoe razlozhenie resheniya trekhmernoi zadachi obtekaniya pri malykh chislakh Reinoldsa, dep. v VINITI No 1356-V04

[10] D. A. Maltsev, L. I. Sazonov, “Postroenie asimptoticheskogo razlozheniya resheniya trekhmernoi zadachi obtekaniya pri malykh chislakh Reinoldsa”, Izv. vuzov. Severo-Kavkazskii region. Matematika i mekhanika sploshnoi sredy. Spetsvypusk, 2004, 164–169 | Zbl

[11] D. A. Maltsev, L. I. Sazonov, Obosnovanie asimptotiki resheniya trekhmernoi statsionarnoi zadachi obtekaniya pri malykh chislakh Reinoldsa, dep. v VINITI No 1560-V2005

[12] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren Math. Wiss., 224, Springer-Verlag, Berlin, 1983 | MR | MR | Zbl | Zbl

[13] S. M. Nikolśkiǐ, Approximation of functions of several variables and imbedding theorems, Springer-Verlag, New York–Heidelberg, 1975 | MR | MR | Zbl | Zbl

[14] P. I. Lizorkin, “O multiplikatorakh integralov Fure v prostranstvakh $L_{p,\theta}$”, Teoremy vlozheniya i ikh prilozheniya, Trudy simpoziuma po teoremam vlozheniya (Baku, 1966), Nauka, M., 1970, 137–141 | MR

[15] L. I. Sazonov, “On the asymptotics of the solution to the three-dimensional problem of flow far from streamlined bodies”, Izv. Math., 59:5 (1995), 1051–1075 | DOI | MR | Zbl

[16] O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Gordon and Breach, New York–London, 1963 | MR | MR | Zbl | Zbl

[17] N. M. Gunther, La théorie du potentiel et ses applications aux problèmes fondamentaux de la physique mathématique, Gauthier-Villars, Paris, 1934 | MR | Zbl

[18] R. Temam, Navier–Stokes equations, Stud. Math. Appl., 2, North-Holland, Amsterdam–New York, 1979 | MR | MR | Zbl | Zbl

[19] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, de Gruyter, Paris, 1969 | MR | MR | Zbl | Zbl

[20] L. Hörmander, The analysis of linear partial differential operators, v. 1, Grundlehren Math. Wiss., 256–257, Springer-Verlag, Berlin, 1983 | MR | MR | Zbl | Zbl