Abstract:
This article discusses the solvability of an overdetermined system of heat convection equations in the Boussinesq approximation. The Oberbeck–Boussinesq system of equations, supplemented by an incompressibility equation, is overdetermined. The number of equations exceeds the number of unknown functions, since non-uniform layered flows of a viscous incompressible fluid are studied (one of the components of the velocity vector is identically zero). The solvability of the non-linear system of Oberbeck–Boussinesq equations is investigated. The solvability of the overdetermined system of non-linear Oberbeck–Boussinesq equations in partial derivatives is studied by constructing several particular exact solutions. A new class of exact solutions for describing three-dimensional non-linear layered flows of a vertical swirling viscous incompressible fluid is presented. The vertical component of vorticity in a non-rotating fluid is generated by a non-uniform velocity field at the lower boundary of an infinite horizontal fluid layer. Convection in a viscous incompressible fluid is induced by linear heat sources. The main attention is paid to the study of the properties of the flow velocity field. The dependence of the structure of this field on the magnitude of vertical twist is investigated. It is shown that, with nonzero vertical twist, one of the components of the velocity vector allows stratification into five zones through the thickness of the layer under study (four stagnant points). The analysis of the velocity field has shown that the kinetic energy of the fluid can twice take the zero value through the layer thickness.
This work was supported by the Foundation for Assistance to Small Innovative Enterprises in Science and Technology (the UMNIK program, agreement 12281GU/2017).
Citation:
N. V. Burmasheva, E. Yu. Prosviryakov, “Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 23:2 (2019), 341–360
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\by N.~V.~Burmasheva, E.~Yu.~Prosviryakov
\paper Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation
\jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]
\yr 2019
\vol 23
\issue 2
\pages 341--360
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\crossref{https://doi.org/10.14498/vsgtu1670}
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Linking options:
https://www.mathnet.ru/eng/vsgtu1670
https://www.mathnet.ru/eng/vsgtu/v223/i2/p341
This publication is cited in the following 11 articles:
Evgenii Yu. Prosviryakov, Larisa S. Goruleva, Mikhail Yu. Alies, “A class of exact solutions of the Oberbeck-Boussinesq equations with the Rayleigh dissipative function”, CPM, 26:2 (2024), 164
S. A. Berestova, E. Yu. Prosviryakov, “An Inhomogeneous Steady-State Convection
of a Vertical Vortex Fluid”, Rus. J. Nonlin. Dyn., 19:2 (2023), 167–186
N. V. Burmasheva, E. Yu. Prosviryakov, “Tochnoe reshenie tipa Kuetta – Puazeilya dlya ustanovivshikhsya kontsentratsionnykh techenii”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 164, no. 4, Izd-vo Kazanskogo un-ta, Kazan, 2022, 285–301
N. V. Burmasheva, E. Yu. Prosviryakov, “Exact solutions to the Navier–Stokes equations describing stratified fluid flows”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 25:3 (2021), 491–507
Vyach. V. Bashurov, E. Yu. Prosviryakov, “Steady thermo-diffusive shear Couette flow of incompressible fluid. Velocity field analysis”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 25:4 (2021), 763–775
N. V. Burmasheva, E. A. Larina, E. Yu. Prosviryakov, “Techenie tipa Kuetta s uchetom idealnogo skolzheniya na kontakte s tverdoi poverkhnostyu”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2021, no. 74, 79–94
N. V. Burmasheva, V. V. Privalova, E. Yu. Prosviryakov, “Layered Marangoni convection with the Navier slip condition”, Sadhana-Acad. Proc. Eng. Sci., 46:1 (2021), 55
N. V. Burmasheva, E. Yu. Prosviryakov, “Exact solutions to the Oberbeck-Boussinesq equations for shear flows of a viscous binary fluid with allowance made for the Soret effect”, Bull. Irkutsk State Univ.-Ser. Math., 37 (2021), 17–30
N. V. Burmasheva, E. Yu. Prosviryakov, “Convective layered flows of a vertically whirling viscous incompressible fluid. Temperature field investigation”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 24:3 (2020), 528–541
N. V. Burmasheva, E. Yu. Prosviryakov, “Klass tochnykh reshenii dlya dvumernykh uravnenii geofizicheskoi gidrodinamiki s dvumya parametrami Koriolisa”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 32 (2020), 33–48
N. V. Burmasheva, E. Yu. Prosviryakov, MECHANICS, RESOURCE AND DIAGNOSTICS OF MATERIALS AND STRUCTURES (MRDMS-2019): Proceedings of the 13th International Conference on Mechanics, Resource and Diagnostics of Materials and Structures, 2176, MECHANICS, RESOURCE AND DIAGNOSTICS OF MATERIALS AND STRUCTURES (MRDMS-2019): Proceedings of the 13th International Conference on Mechanics, Resource and Diagnostics of Materials and Structures, 2019, 030021
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