Exact Quadratic Polynomial Solution for Describing Inhomogeneous Couette–Poiseuille Flow in an Infinite Horizontal Layer with Permeable Boundaries

The paper investigates steady flow of a viscous incompressible fluid in a plane channel with permeable parallel walls. In contrast to classical formulations, not only the velocity value but also its first two spatial gradients are specified at the upper boundary. This approach enables modeling flows with local inhomogeneity along the channel. The lower wall is stationary and satisfies the no-slip condition. A constant pressure gradient of arbitrary sign and a uniform normal flow through both boundaries are taken into account. The problem is solved analytically in dimensionless form, where the Reynolds number, the permeability-based Reynolds number, and the dimensionless pressure gradient play the determining role. Asymptotic analysis is carried out for the limiting cases of weak and strong permeability. Based on the structure of the exact solution, an estimate for the boundary layer thickness under injection is derived. The results are verified by numerical simulations for real fluids and demonstrate the transition from a viscosity-dominated to a convection-dominated flow regime.