The Inhomogeneous Couette Flow of a Micropolar Fluid

In this paper we consider the steady inhomogeneous shear flow of a viscous incompressible fluid taking into account the possibility of solid-body rotation of a representative volume. Mathematically, the contribution of couple stresses manifests itself in an increase in the order of the system of governing differential equations. We discuss problems of the existence of an exact solution within the framework of the class of functions linear in some of the coordinates. It is shown that the problem of overdetermination of the system of equations, which is traditional for models describing shear flows, does not arise for the chosen class of solutions. An exact solution is constructed for the velocity field of the flow. Also, an exact solution of the boundary-value problem describing adhesion and superadhesion on the boundaries of the flow region is analyzed in dimensionless form. It is shown that these exact solutions are capable of describing stagnation regions observed in real fluids and the effect of increase in velocities.