On construction of exact solutions of stationary quasi-hydrodynamic system using Lin's substitution

Yu. V. Sheretov

Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 1 (2023), pp. 36-48

Résumé

New exact solutions of stationary quasi-hydrodynamic (QHD) system that do not satisfy either the Navier–Stokes equations or the Euler equations are described. With the help of Lin's substitution, the indicated QHD system was reduced to the system of ordinary differential equations, which was fully integrated. For $c_s\to +\infty$, where $c_s$ is the sound velocity in the liquid, the constructed solutions tend to the corresponding exact solutions of the Navier-Stokes system.

Détail

Mots-clés : Quasi-Hydrodynamic system, exact solutions, Lin's substitution.
Keywords: Navier-Stokes system

Bibliographie

[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)

[3] Riley N., Drazin P. G., The Navier–Stokes equations: A classification of flows and exact solutions, Cambridge University Press, Cambridge, 2006, 196 pp.

[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

[7] Aristov S. N., Knyazev D. V., Polyanin A. D., “Exact solutions of the Navier-Stokes equations with linear dependence of the velocity components on two spatial variables”, Theoretical foundations of chemical technology, 43:5 (2009), 547–566 (in Russian)

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

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

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

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

[12] Sheretov Yu. V., “On common exact solutions of Navier-Stokes and quasi–hydrodynamic systems for nonstationary flows”, Herald of Tver State University. Series: Applied Mathematics, 2017, no. 3, 13–25 (in Russian) | DOI

[13] 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

[14] 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

[15] 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

[16] Grigoreva V. V., Sheretov Yu. V., “On exact solutions of quasi-hydrodynamic system that don't satisfy the Navier-Stokes and Euler systems”, Herald of Tver State University. Series: Applied Mathematics, 2 (2021), 5–15 (in Russian) | DOI

[17] Grigoreva V. V., Sheretov Yu. V., “a new class of exact solutions of Quasi-Hydrodynamic system, generated by eigenfunctions of two-dimensional Laplace operator”, Herald of Tver State University. Series: Applied Mathematics, 2022, no. 1, 5–17 (in Russian) | DOI

[18] Sheretov Yu. V., “Superposition principle for solutions of quasi-hydrodynamic system”, Herald of Tver State University. Series: Applied Mathematics, 2022, no. 2, 60–73 (in Russian) | DOI

[19] Harun M. H., Didane D. H., Sayed Abdelaal.M. A., Zaman I., Manshoor B., “Exact solutions to the Navier–Stokes equations for nonlinear viscous flows by undetermined coefficient approach”, Journal of Complex Flow, 3:2 (2021), 38–41

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

[21] 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

[22] Balashov V. A., “Dissipative spatial discretization of a phase field model of multiphase multicomponent isothermal fluid flow”, Computers and Mathematics with Applications, 90:112 (2021), 124 | DOI

[23] Zlotnik A. A., Fedchenko A. S., “On the properties of a quasi-gas dynamic system of equations of a homogeneous gas mixture with a common regularizing velocity”, Differential equations, 58:3 (2022), 346–360 (in Russian)

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

[25] Kraposhin M. V., Ryazanov D. A.Elizarova.T. G., “Numerical algorithm based on regularized equations for incompressible flow modeling and its implementation in OpenFOAM”, Computer Physics Communications, 271:1 (2022), 108216 | DOI