Nonlinear Gradient Flow of a Vertical Vortex Fluid in a Thin Layer
2019, Vol. 15, no. 3, pp. 271-283
Author(s): Privalova V., Prosviryakov E. Y., Simonov M. A.
This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License
Author(s): Privalova V., Prosviryakov E. Y., Simonov M. A.
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.
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