Analytical Solution of the Navier–Stokes Equations for Describing Inhomogeneous Couette Flow with a Quadratic Velocity Profile in a Layer with Permeable Boundaries

Introduction. Flow control in microfluidic systems, membrane technologies, and porous bearings requires an understanding of the synergy between boundary permeability, their spatial inhomogeneity, and the viscosity of the working fluid. Each of these factors is actively studied separately. However, a comprehensive analytical description of their combined effect on the flow is needed. No such publications exist. The presented article fills this gap. Research objectives are as follows: to obtain an analytical solution for the velocity field in Couette flow with permeable boundaries and a nonlinear boundary condition; to study the formation of hydrodynamics under the influence of permeability (α), dynamic viscosity (μ), linear ( A ) and quadratic ( B ) inhomogeneity of the boundary condition. Materials and Methods . The analytical solution is based on the stationary Navier–Stokes equations for an incompressible Newtonian fluid, with a quadratic expansion of velocity along the transverse coordinate. The axial, linear, and quadratic modes of the velocity profile were investigated using numerical modeling in MATLAB. For a stationary, laminar, isothermal flow of a Newtonian viscous incompressible fluid, the distance between permeable plates was h = 1 m. The lower plate was stationary, while the upper plate moved with a velocity of W = 0.3 m/s. The liquid filtration rate was V w = 0.001 m/s, and μ = 0.01 Pa·s for A = ±0.03 s –1 and B = ±0.005 m –1 ·s –1 . Water, motor oil, and crude oil were studied at temperatures of 20 °C, 40 °C, or 60 °C. For this case, h = 0.02 m, W = 0.05 m/s, A = 0.1 s –1 , B = 0.02 m –1 ·s –1 , V w = 0.0005 m/s. Depending on the fluid and temperature, μ ranged from 0.05 to 9.15·10 –3 Pa·s. Results. Asymmetry of the flow, deviation from the channel axis, and variability of the vorticity amplitude ω y were visualized. Zero filtration velocity was observed at the lower plate in the plane z = 0 and increased with this parameter, reaching a maximum at z = h (distance between the plates). For water, the streamlines exhibited minimal deviation from the horizontal, while for oil at 20 °C, they curved near the upper wall. Two-dimensional vorticity fields for water, oil, and petroleum at various temperatures were compared. Weak ω y and reduced viscosity resulted in negative values ω y for water and petroleum. For oil, the situation was reversed: positive values corresponded to elevated ω y . Discussion. The calculation results allow us to conclude: − changing the sign of A inverts the directions of the maxima for velocity and vorticity; − the sign of B determines the curvature of the isolines; − the thickness of the layer with the maximum velocity gradient changes by two orders of magnitude when transitioning from water to oil. The identified patterns are explained by the physical meaning of the parameters: A defines the macroscopic flow asymmetry, B governs the distribution of the transverse flow, and viscosity, through α, controls the depth of boundary perturbations. Conclusion. For the first time, an exact analytical solution to the stationary Navier–Stokes equations was obtained for generalized Couette flow of a Newtonian fluid between permeable plates with a quadratic velocity profile at the boundary. A parametric analysis has shown that coefficient A determines the asymmetry of the velocity and vorticity fields, while B determines their nonlinearity. Viscosity controls the thickness of the shear layer: for high-viscosity media, the velocity drop is localized near the walls, while for low-viscosity media, the profile is linear. The results provide a foundation for applications in microfluidics, membrane technologies, and tribology. Future prospects are associated with accounting for non-Newtonian fluid properties, unsteady regimes, and flow stability.