Inhomogeneous Gradient Poiseuille Flows of a Vertically Swirled Fluid

Document Type : Research Paper


1 Sector of Nonlinear Vortex Hydrodynamics, Institute of Engineering Science of Ural Branch of the Russian Academy of Sciences, 34 Komsomolskaya st., Ekaterinburg, 620049, Russia

2 Academic Department of Information Technologies and Control Systems, Ural Federal University, 19 Mira st., Ekaterinburg, 620049, Russia

3 Plekhanov Russian University of Economics, Scopus number 60030998, 36 Stremyanny lane, Moscow, 117997, Russia

4 Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine


An exact solution is proposed for describing the steady-state and unsteady gradient Poiseuille shear flow of a viscous incompressible fluid in a horizontal infinite layer. This exact solution is described by a polynomial of degree N with respect to the variable y where the coefficients of the polynomial depend on the coordinate z and time t, a boundary value problem for a steady flow has been considered and the velocity field with a quadratic dependence on the horizontal longitudinal (horizontal) coordinate y is considered. The coefficients of the quadratic form depend on the transverse (vertical) coordinate z. Pressure is a linear form of the horizontal coordinates x and y. The exact solution of the constitutive system of equations for the boundary value problem is considered here to be polynomial. The boundary value problem is solved for a non-uniform distribution of velocities on the upper non-deformable boundary of an infinite horizontal liquid layer. The no-slip condition is set on the lower non-deformable boundary. The exact solution obtained is a polynomial of the tenth degree in the coordinates x, y and z. Stratification conditions are obtained for the velocity field, for the stress tensor components, and for the vorticity vector. The constructed exact solution describes the counterflows of a vertically swirling fluid outside the field of the Coriolis force. Shear stresses are tensile and compressive relative to the vertical (transverse) coordinates and relative to the horizontal (longitudinal) coordinates. The article presents formulas illustrating the existence of zones of differently directed vortices.


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