On common exact solutions of Navier-Stokes and quasi-hydrodynamic systems for nonstationary flows

Yu. V. Sheretov

Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 3 (2017), pp. 13-25

Résumé

It is shown that the Trkal method for constructing uniformly-helical solutions of nonstationary Navier-Stokes equations is applicable for quasi-hydrodynamic system. A wider class of flows, obeying the generalized Gromeki-Beltrami condition, is considered. Examples of exact solutions, common to the Navier-Stokes and quasi-hydrodynamic systems, but not satisfying the Euler equations, are given.

Détail

Keywords: Navier-Stokes and Euler systems, Trkal method, generalized Gromeka-Beltrami condition, helical flows.
Mots-clés : quasi-hydrodynamic equations, exact solutions

Bibliographie

[1] Landau L. D., Lifshits E. M., Hydrodynamics, Nauka Publ., Moscow, 1986, 736 pp. (in Russian) | MR

[2] Loytsyansky L. G., Fluid and Gas Mechanics, Nauka Publ., Moscow, 1987, 840 pp. (in Russian) | MR

[3] Shmyglevskii Yu. D., Analytical Investigations of Gas and Fluid Dynamics, Editorial URSS Publ., Moscow, 1999, 232 pp. (in Russian) | MR

[4] Riley N., Drazin P. G., The Navier-Stokes equations: A classification of flows and exact solutions, Cambridge University Press, Cambridge, 2006, 196 pp. | MR | Zbl

[5] Pukhnachev V. V., “Symmetries in the Navier-Stokes equations”, Advances in Mechanics, 2006, no. 1, 6-76 (in Russian)

[6] Trkal V., “A note on the hydrodynamics of viscous fluids”, Czechoslovak Journal of Physics, 44:2 (1994), 97-106 | DOI

[7] Bogoyavlenskij O. I., “Infinite families of exact periodic solutions to the Navier-Stokes equations”, Moscow Mathematical Journal, 3:2 (2003), 263-272 | MR | Zbl

[8] Vereshchagin V. P., Subbotin Yu. N., Chernykh N. I., “Some solutions of continuum equations for an incompressible viscous medium”, Proceedings of the Steklov Institute of Mathematics, 287, suppl.1 (2014), 208-223 | DOI | MR

[9] Kovalev V. P., Prosviryakov E. Yu., Sizykh G. B., “Obtaining examples of exact solutions of the Navier-Stokes equations for helical flows by the method of summation of velocities”, Proceedings of Moscow Institute of Physics and Technology, 9:1 (2017), 71-88 (in Russian)

[10] Sheretov Yu. V., “About the uniqueness of the solutions for one dissipative equation system of hydrodynamic type”, Mathematical Modeling, 6:10 (1994), 35-45 (in Russian) | MR | Zbl

[11] Sheretov Yu. V., Continuum Dynamics under Spatiotemporal Averaging, “Regular and Chaotic Dynamics” Publ., Moscow, Izhevsk, 2009, 400 pp. (in Russian)

[12] Sheretov Yu. V., Regularized Hydrodynamic Equations, Tver State University, Tver, 2016, 222 pp. (in Russian)

[13] Sheretov Yu. V., “On the exact solutions of stationary quasi-hydrodynamic equations in cylindrical coordinates”, Vestnik TvGU. Seriya: Prikladnaya Matematika [Herald of Tver State University. Series: Applied Mathematics], 2017, no. 1, 85-94 (in Russian) | DOI

[14] Sheretov Yu. V., “On the common exact solutions of stationary Navier-Stokes and quasi-hydrodynamic systems, not satisfying to Euler equations”, Vestnik TvGU. Seriya: Prikladnaya Matematika [Herald of Tver State University. Series: Applied Mathematics], 2017, no. 2, 5-15 (in Russian) | DOI

[15] Zherikov A. V., Application of Quasi-Hydrodynamic Equations: Mathematical Modeling of Viscous Incompressible Fluid, Lambert Academic Publishing, Saarbrücken, 2010, 124 pp. (in Russian)

[16] Elizarova T. G., Milyukova O. Yu., “Numerical simulation of viscous incompressible flow in a cubic cavity”, Computational Mathematics and Mathematical Physics, 43:3 (2003), 453-466 (in Russian) | MR | Zbl

[17] Arnold V. I., Khesin B. A., Topological Methods in Hydrodynamics, Springer-Verlag, New York, 1998, 376 pp. | MR | Zbl

[18] Elizarova T. G., Shirokov I. A., Regularized Equations and Examples of their Use in the Modeling of Gas-Dynamic Flows, MAKS Press Publ., Moscow, 2017, 136 pp. (in Russian)