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 Nelin. Dinam., 2019, Volume 15, Number 3, Pages 271–283 (Mi nd659)

Nonlinear physics and mechanics

Nonlinear Gradient Flow of a Vertical Vortex Fluid in a Thin Layer

V. V. Privalovaab, E. Yu. Prosviryakovab, M. A. Simonova

a Ural Federal University named after the first President of Russia B.N. Yeltsin, ul. Mira 19, Ekaterinburg, 620002 Russia
b Institute of Engineering Science UB RAS, ul. Komsomolskaya 34, Ekaterinburg, 620049 Russia

Abstract: A new exact solution to the Navier – Stokes equations is obtained. This solution describes the inhomogeneous isothermal Poiseuille flow of a viscous incompressible fluid in a horizontal infinite layer. In this exact solution of the Navier – Stokes equations, the velocity and pressure fields are the linear forms of two horizontal (longitudinal) coordinates with coefficients depending on the third (transverse) coordinate. The proposed exact solution is two-dimensional in terms of velocity and coordinates. It is shown that, by rotation transformation, it can be reduced to a solution describing a three-dimensional flow in terms of coordinates and a two-dimensional flow in terms of velocities. The general solution for homogeneous velocity components is polynomials of the second and fifth degrees. Spatial acceleration is a linear function. To solve the boundaryvalue problem, the no-slip condition is specified on the lower solid boundary of the horizontal fluid layer, tangential stresses and constant horizontal (longitudinal) pressure gradients specified on the upper free boundary. It is demonstrated that, for a particular exact solution, up to three points can exist in the fluid layer at which the longitudinal velocity components change direction. It indicates the existence of counterflow zones. The conditions for the existence of the zero points of the velocity components both inside the fluid layer and on its surface under nonzero tangential stresses are written. The results are illustrated by the corresponding figures of the velocity component profiles and streamlines for different numbers of stagnation points. The possibility of the existence of zero points of the specific kinetic energy function is shown. The vorticity vector and tangential stresses arising during the flow of a viscous incompressible fluid layer under given boundary conditions are analyzed. It is shown that the horizontal components of the vorticity vector in the fluid layer can change their sign up to three times. Besides, tangential stresses may change from tensile to compressive, and vice versa. Thus, the above exact solution of the Navier – Stokes equations forms a new mechanism of momentum transfer in a fluid and illustrates the occurrence of vorticity in the horizontal and vertical directions in a nonrotating fluid. The three-component twist vector is induced by an inhomogeneous velocity field at the boundaries of the fluid layer.

Keywords: Poiseuille flow, gradient flow, exact solution, counterflow, stagnation point, vorticity

 Funding Agency Grant Number Russian Science Foundation 19-19-00571 The work was supported by the Russian Scientific Foundation (project 19-19-00571).

DOI: https://doi.org/10.20537/nd190306

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MSC: 76F02, 76F45, 76M45, 76R05, 76U05
Accepted:23.08.2019

Citation: V. V. Privalova, E. Yu. Prosviryakov, M. A. Simonov, “Nonlinear Gradient Flow of a Vertical Vortex Fluid in a Thin Layer”, Nelin. Dinam., 15:3 (2019), 271–283

Citation in format AMSBIB
\Bibitem{PriProSim19} \by V. V. Privalova, E. Yu. Prosviryakov, M. A. Simonov \paper Nonlinear Gradient Flow of a Vertical Vortex Fluid in a Thin Layer \jour Nelin. Dinam. \yr 2019 \vol 15 \issue 3 \pages 271--283 \mathnet{http://mi.mathnet.ru/nd659} \crossref{https://doi.org/10.20537/nd190306} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=4021369}