Geodesic
A new model in the theory of “creeping” flows
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 3 (2014), pp. 19-27 Cet article a éte moissonné depuis la source Math-Net.Ru

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Common way of Navier–Stokes equations linearization at small Reynolds number is Stokes model, which proposes completely neglected of convective members in these equations. However, when considering the plane problem of flow past an arbitrary contour, Stokes equations do not have solutions. This fact is known as the “paradox of Stokes”. For overcome the “paradox of Stokes” the Oseen approximation is used, where a part of speed components in quadratic terms is replaced with speed constants of outside flow, while the rest equation members are rejected as being “small” ones. The paper suggests linearization of Navier–Stokes equations against velocity field of perfect liquid flowing the body around. The system of linearized Navier–Stokes equations under such approach is a linear one with variable coefficients. The offered method is used for description of plane problems of body uniform flow with viscous incompressible liquid. It turns out for plane problems that in case flow stream function and potential are selected as independent variables, the problem is formulated in the form of equations for vorticity, speed and pressure components, and is successfully solved numerically. To develop numerical method transition to new coordinates is used — to potential and stream function of contour flow of perfect liquid. The problem of circular cylinder uniform flow with viscous incompressible liquid is analyzed as a model problem of flat body flow. Results of resistance coefficient prediction for the circular cylinder flow problem are compared with experimental data and results of Oseen theory. Bibliogr. 12. Il. 4.
Keywords: Navier–Stokes equations, small Reynolds numbers, linearization of Navier–Stokes equations, new model of “creeping” flows. 
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S. G Kadyrov; E. N. Afanasov; V. N. Zverkov. A new model in the theory of “creeping” flows. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 3 (2014), pp. 19-27. http://geodesic.mathdoc.fr/item/VSPUI_2014_3_a1/

[1] Buleev N. L., “Numerical method for solving two- and three-dimensional diffusion equations”, Mat. Sb. (N. S.), 51(93):2 (1960), 227–238 | Zbl

[2] Van Dyke M., Perturbation methods in fluid mechanics, Per. s angl. V. A. Smirnova, ed. A. A. Nikol'skii, Mir, M., 1967, 310 pp.

[3] Kochin N. E., Kibel I. A., Roze N. V., Theoretical hydromechanics, Textbook for high schools, v. 2, 4 edition revised and enlarged, Fizmatgiz, M., 1963, 728 pp.

[4] Lamb H., Hydrodynamics, per. s angl. A. V. Germogenova, V. A. Kudrjavceva, ed. N. A. Slezkin, Gos. izd-vo tehn.-teor. lit., M.–L., 1947, 928 pp.

[5] Landau L. D., Lifshitz E. M., Course of Theoretical Physics, in 10 v., v. VI, Hydrodynamics, 5 edition, stereotip., Fizmatgiz, M., 2001, 736 pp.

[6] Novozhilov V. V., “The use of potential solutions in the theory of viscous fluid”, Reports of the USSR, 290:6 (1968), 1320–1323

[7] Slezkin N. A., Dynamics of viscous incompressible fluid, Textbook for public universities, Gostekhizdat, M., 1955, 520 pp.

[8] Shkadov V. Y., Zapryanov Z. D., Viscous flow, Izd-vo Mosk. un-ta, M., 1984, 200 pp.

[9] Shlikhting H., Boundary layer theory, Per. s nem. G. A. Vol'perta, ed. L. G. Lojcjanskii, Nauka, M., 1974, 712 pp.

[10] Dennis S. C. R., Shimshoni M., The steady flow of a viscous fluid past a circular cylinder, Aero. Res. Counc. Lond. Current Paper No 797, 1965, 44 pp.

[11] Taneda S., “Experimental investigation of the wakes behind cylinders and plates at low Reynolds numbers”, J. Phys. Soc. Japan, 11:3 (1956), 302–307 | DOI

[12] Tritton D. J., “Experiments on the flow past a circular cylinder at low Reynolds numbers”, J. of Fluid Mechanics, 6:4 (1959), 547–567 | DOI | Zbl