Inhomogeneous Nusselt–Couette–Poiseuille Flow

A new type of steady flow of a liquid thin film flowing down an inclined plane is studied. Two-dimensional inhomogeneous flows of the Nusselt type are considered. Depending on the boundary conditions at the free interface, which is assumed to be a nondeformable interface, inhomogeneous fluid flows generalize the exact solutions of Nusselt, Couette, and Poiseuille. The generalizations and modifications of classical flows considered in the article are described by an overdetermined system that is comprised of the Navier–Stokes equations and the continuity equation. A nontrivial exact solution of the overdetermined system is shown, which characterizes the inhomogeneous motion of a vertical swirling fluid. Velocities and shear stresses described by polynomials are analyzed. The study of hydrodynamic fields shows that they have a complex stratification. The fluid flow moving on an inclined plane may contain four regions with counterflows. Shear stresses across the layer thickness have different signs and can change sign twice.