Geodesic
On exact solutions of quasi-hydrodynamic system that don't satisfy the Navier-Stokes and Euler systems
Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 2 (2021), pp. 5-15 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The quasi-hydrodynamic system was proposed by Sheretov Yu.V. in 1993. The known exact solutions of this system in the overwhelming majority of cases satisfy either the Navier-Stokes equations or the Euler equations. This paper describes a new class of exact solutions of quasi-hydrodynamic system that satisfy neither the Navier-Stokes equations, nor the Euler equations. The corresponding exact solutions of the Navier-Stokes system are obtained from the constructed solutions by passing to the limit at $c_s\to +\infty$, where $c_s$ is the sonic velocity in the fluid.
Keywords: Navier-Stokes system, Euler system, principle of superposition.
Mots-clés : quasi-hydrodynamic system, exact solutions 
@article{VTPMK_2021_2_a0,
     author = {V. V. Grigoryeva and Yu. V. Sheretov},
     title = {On exact solutions of quasi-hydrodynamic system that don't satisfy the {Navier-Stokes} and {Euler} systems},
     journal = {Vestnik Tverskogo gosudarstvennogo universiteta. Seri\^a Prikladna\^a matematika},
     pages = {5--15},
     year = {2021},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTPMK_2021_2_a0/}
}
TY  - JOUR
AU  - V. V. Grigoryeva
AU  - Yu. V. Sheretov
TI  - On exact solutions of quasi-hydrodynamic system that don't satisfy the Navier-Stokes and Euler systems
JO  - Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika
PY  - 2021
SP  - 5
EP  - 15
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/VTPMK_2021_2_a0/
LA  - ru
ID  - VTPMK_2021_2_a0
ER  - 
%0 Journal Article
%A V. V. Grigoryeva
%A Yu. V. Sheretov
%T On exact solutions of quasi-hydrodynamic system that don't satisfy the Navier-Stokes and Euler systems
%J Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika
%D 2021
%P 5-15
%N 2
%U http://geodesic.mathdoc.fr/item/VTPMK_2021_2_a0/
%G ru
%F VTPMK_2021_2_a0
V. V. Grigoryeva; Yu. V. Sheretov. On exact solutions of quasi-hydrodynamic system that don't satisfy the Navier-Stokes and Euler systems. Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 2 (2021), pp. 5-15. http://geodesic.mathdoc.fr/item/VTPMK_2021_2_a0/

[1] Lojtsyanskij L. G., Fluid and Gas Mechanics, Nauka Publ., Moscow, 1987, 840 pp. (in Russian)

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

[3] 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

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

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

[6] Wang C. Y., “Exact solutions of the unsteady Navier–Stokes equations”, Applied Mechanics Reviews, 42:11, Part 2 (1989), S269–S282 | DOI | MR | Zbl

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

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

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

[10] Sheretov Yu. V., “On the solutions of Cauchy problem for quasi-hydrodynamic system”, Herald of Tver State University. Series: Applied Mathematics, 2020, no. 1, 84–96 (in Russian) | DOI

[11] Sheretov Yu. V., “On classes of exact solutions of quasi-hydrodynamic system”, Herald of Tver State University. Series: Applied Mathematics, 2020, no. 2, 5–17 (in Russian) | DOI

[12] Sheretov Yu. V., “On the construction of exact solutions of two-dimensional quasi-hydrodynamic system”, Herald of Tver State University. Series: Applied Mathematics, 2021, no. 1, 5–20 (in Russian) | DOI

[13] Vladimirov V. S., Equations of Mathematical Physics, Nauka Publ., Moscow, 1971, 512 pp. (in Russian) | MR

[14] Stenina T. V., Elizarova T. G., Kraposhin M. V., “Regularized equations for disk pump simulation problems in OpenFOAM implementation”, Keldysh Institute of Applied Mathematics Preprints, 2020, 066, 30 pp. (in Russian) | DOI

[15] Stenina T. V., Elizarova T. G., Kraposhin M. V., “Regularized equations for disk pump simulation problems in OpenFOAM implementation”, Keldysh Institute of Applied Mathematics Preprints, 2020, 066, 30 pp. (in Russian) | DOI

[16] Balashov V. A., Zlotnik A. A., “An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations”, Journal of Computational Dynamics, 7:2 (2020), 291–312 | DOI | MR | Zbl

[17] Balashov V. A., Savenkov E. B., “Regularized phase-field model for description of dynamics of “solid-fluid” system”, Keldysh Institute of Applied Mathematics Preprints, 2020, 29 pp. (in Russian) | DOI

[18] Balashov V. A., Zlotnik A. A., “On a new spatial discretization for a regularized 3D compressible isothermal Navier–Stokes–Cahn–Hilliard system of equations with boundary conditions”, Journal of Scientific Computing, 86:33 (2021), 1–30 | DOI | MR