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 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]: Year: Volume: Issue: Page: Find

 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2017, Volume 21, Number 4, Pages 736–751 (Mi vsgtu1568)

Mathematical Modeling, Numerical Methods and Software Complexes

A large-scale layered stationary convection of a incompressible viscous fluid under the action of shear stresses at the upper boundary. Temperature and presure field investigation

N. V. Burmashevaab, E. Yu. Prosviryakova

a Institute of Engineering Science, Urals Branch, Russian Academy of Sciences, Ekaterinburg, 620049, Russian Federation
b Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg, 620002, Russian Federation

Abstract: In this paper a new exact solution of an overdetermined system of Oberbeck–Boussinesq equations that describes a stationary shear flow of a viscous incompressible fluid in an infinite layer is under study. The given exact solution is a generalization of the Ostroumov–Birich class for a layered unidirectional flow. In the proposed solution, the horizontal velocities depend only on the transverse coordinate z. The temperature field and the pressure field are three-dimensional. In contradistinction to the Ostroumov–Birich solution, in the solution presented in the paper the horizontal temperature gradients are linear functions of the $z$ coordinate. This structure of the exact solution allows us to find a nontrivial solution of the Oberbeck–Boussinesq equations by means of the identity zero of the incompressibility equation. This exact solution is suitable for investigating large-scale flows of a viscous incompressible fluid by quasi-two-dimensional equations. Convective fluid motion is caused by the setting of tangential stresses on the free boundary of the layer. Inhomogeneous thermal sources are given on both boundaries. The pressure in the fluid at the upper boundary coincides with the atmospheric pressure. The paper focuses on the study of temperature and pressure fields, which are described by polynomials of three variables. The features of the distribution of the temperature and pressure profiles, which are polynomials of the seventh and eighth degree, respectively, are discussed in detail. To analyze the properties of temperature and pressure, algebraic methods are used to study the number of roots on a segment. It is shown that the background temperature and the background pressure are nonmonotonic functions. The temperature field is stratified into zones that form the thermocline and the thermal boundary layer near the boundaries of the fluid layer. Investigation of the properties of the pressure field showed that it is stratified into one, two or three zones relative to the reference value (atmospheric pressure).

Keywords: Oberbeck–Boussinesq system, shear flow, convection, exact solution, polynomial solution, root localization, thermocline, field stratification

 Funding Agency Grant Number Fund for Assistance to Small Innovative Enterprises in the Scientific and Technical Sphere 12281ÃÓ/2017 This work was supported by the Foundation for Assistance to Small Innovative Enterprises in Science and Technology (the UMNIK program), agreement no. 12281GU/2017.

DOI: https://doi.org/10.14498/vsgtu1568

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Bibliographic databases:

UDC: 517.958:532.51
MSC: 76F02, 76F45, 76M45, 76R05, 76U05
Revised: December 15, 2017
Accepted: December 18, 2017
First online: December 29, 2017

Citation: N. V. Burmasheva, E. Yu. Prosviryakov, “A large-scale layered stationary convection of a incompressible viscous fluid under the action of shear stresses at the upper boundary. Temperature and presure field investigation”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 21:4 (2017), 736–751

Citation in format AMSBIB
\Bibitem{BurPro17} \by N.~V.~Burmasheva, E.~Yu.~Prosviryakov \paper A large-scale layered stationary convection of a incompressible viscous fluid under the action of shear stresses at the upper boundary. Temperature and presure field investigation \jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.] \yr 2017 \vol 21 \issue 4 \pages 736--751 \mathnet{http://mi.mathnet.ru/vsgtu1568} \crossref{https://doi.org/10.14498/vsgtu1568} \zmath{https://zbmath.org/?q=an:06964885} \elib{http://elibrary.ru/item.asp?id=32712835} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. E. Yu. Prosviryakov, “Dynamic equilibria of a nonisothermal fluid”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 22:4 (2018), 735–749
2. N. V. Burmasheva, E. Yu. Prosviryakov, “Investigation of temperature and pressure fields for the Marangoni shear convection of a vertically swirling viscous incompressible fluid”, 12th International Conference on Mechanics, Resource and Diagnostics of Materials and Structures, MRDMS 2018 (Ekaterinburg, Russian Federation, 21–25 May 2018), AIP Conference Proceedings, 2053, 2018, 040012
3. N. V. Burmasheva, E. Yu. Prosviryakov, “Investigation of a velocity field for the Marangoni shear convection of a vertically swirling viscous incompressible fluid”, 12th International Conference on Mechanics, Resource and Diagnostics of Materials and Structures, MRDMS 2018 (Ekaterinburg, Russian Federation, 21–25 May 2018), AIP Conference Proceedings, 2053, 2018, 040011
4. V. V. Privalova, E. Yu. Prosviryakov, “Couette-Hiemenz exact solutions for the steady creeping convective flow of a viscous incompressible fluid with allowance made for heat recovery”, Vestn. Samar. Gos. Tekhnicheskogo Univ.-Ser. Fiz.-Mat. Nauka, 22:3 (2018), 532–548
5. N. V. Burmasheva, E. Yu. Prosviryakov, “Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 23:2 (2019) (to appear)
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