Exact Solution of the Equations for Inhomogeneous Couette–Poiseuille Shear Flow with Rayleigh Friction

Abstract

we obtained a new exact solution of the modified Navier–Stokes equations with linear Rayleigh friction. The solution describes a three-dimensional steady flow of a viscous incompressible fluid in a plane channel and generalizes the classical Couette and Poiseuille flows. The longitudinal velocity component varies linearly with one transverse coordinate, while the corresponding coefficients vary exponentially with the second transverse coordinate. Using water as an example, we performed a numerical analysis for different values of the Rayleigh friction coefficient. We investigated the effect of the friction parameter on the thickness of the near-wall layer and on the ratio between viscous dissipation and energy dissipation caused by Rayleigh friction. The results show that at low values of the friction coefficient, the flow remains close to the classical solutions. At high values of the friction coefficient, thin boundary layers form, and Rayleigh friction becomes the dominant dissipation mechanism. The proposed solution extends the family of exact solutions in fluid dynamics and can be used to model flows in porous media, filtration processes, and geophysical fluid dynamics problems.