Nonuniform Couette–Poiseuille Shear Flow with a Moving Lower Boundary of a Horizontal Layer

Abstract

The exact solution of the Navier–Stokes equations with the equation of incompressibility is obtained. This solution describes a steady isobaric and gradient nonuniform shear flow of a viscous incompressible fluid. The fluid motion at a constant pressure is induced by the motion of the lower boundary of an infinite horizontal fluid layer. A nonuniform Poiseuille-type fluid flow is considered with simultaneously specified constant horizontal pressure gradient and velocities. A shear isothermal flow of a viscous incompressible fluid is described by an overdetermined system of partial differential equations. The Navier–Stokes equations are integrated exactly in the Lin–Sidorov–Aristov class. The velocity and pressure fields are linear forms with respect to two coordinates (horizontal, or longitudinal). The coefficients of the linear form depend on the third (vertical or transverse) coordinate. Due to the structure of the exact solution, the incompressibility equation is satisfied automatically. Thus, the role of the “odd” equation is played by the equation of incompressibility. An analysis of the obtained polynomial exact solution of the Navier–Stokes equations for an incompressible fluid is performed. The nonuniform Couette-type flow is characterized by a fourth-degree polynomial. The modified Poiseuille flow is described by a fifth-degree polynomial. An analysis of the localization of the roots of the polynomial showed the existence of a nonmonotonic profile of the specific kinetic energy with two zero values. This means the existence of countercurrents in the fluid.