Inhomogeneous Gradient Poiseuille Flows of a Vertically Swirled Fluid

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.

Keywords:

exact solution , overdetermined system , Poiseuille flow , vertically swirling fluid , countercurrents , stratification , reduced symmetry , Hopf bifurcation