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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]:
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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)  

This article is cited in 1 scientific paper (total in 1 paper)

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 12281ГУ/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:

Document Type: Article
UDC: 517.958:532.51
MSC: 76F02, 76F45, 76M45, 76R05, 76U05
Received: October 17, 2017
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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    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  mathnet  crossref  elib
  • Вестник Самарского государственного технического университета. Серия: Физико-математические науки
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