K. V. Gubareva, E. Yu. Prosviryakov, A. V. Eremin

INHOMOGENEOUS COUETTE POISEUILLE FLOW OF A VISCOUS INCOMPRESSIBLE FLUID IN AN INFINITE HORIZONTAL LAYER WITH PERMEABLE BOUNDARIES

This paper presents an exact analytical solution to the boundary value problem for a steady flow of a viscous incompressible fluid in a plane channel formed by two parallel permeable plates. Particular emphasis is placed on the combined effect of three physical key factors, namely a constant longitudinal pressure gradient, uniform normal fluid flow through permeable boundaries, and an inhomogeneous boundary condition on the upper wall, where streamwise velocity is specified as a linear function of the transverse coordinate. The classical full no-slip condition is implemented on the lower boundary. The system of steady-state Navier–Stokes equations is reduced to two independent 2nd-order ordinary differential equations, which are solved by the method of superposition. The obtained closed-form analytical formulas for the velocity profiles allow a detailed analysis of the effect of the pressure gradient, penetration rate, and the parameters of the inhomogeneous boundary condition on the flow structure. The solution strictly satisfies all the specified boundary conditions and the continuity equation. It extends the classical Poiseuille and Couette problems to the case of permeable boundaries and non-uniform slip, thus providing a valuable theoretical tool for modeling processes in microfluidics, filtration, and lubrication.

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