Analytical study of couple stress fluid flow past a semipermeable cylinder with an embedded solid core–biporous medium

This study presents an analytical solution for the creeping flow of a couple stress fluid within a porous medium that encloses a semipermeable cylinder containing an embedded solid core. The flow field is modeled with a uniform velocity far from the cylinder, ensuring the proper application of boundary conditions. The fluid motion within the porous medium is described by the Brinkman and Darcy equations, each accounting for distinct permeability characteristics. Specifically, the Brinkman equation with permeability k1 governs the region outside the semipermeable cylinder, while the Darcy equation with permeability k2 applies within the semipermeable region. To improve the accuracy and applicability of the model, the governing equations are formulated using cylindrical polar coordinates, reflecting the structural configuration of the system. The stream function approach is employed to reduce the complexity of the governing equations, resulting in a sixth-order differential equation for the Brinkman region and a second-order differential equation for the Darcy region. The analytical solutions, derived via the separation of variables technique, are expressed in terms of modified Bessel functions. These solutions provide explicit mathematical representations for essential flow characteristics, including normal and tangential velocities, couple stress components, pressures, stresses, and drag forces on the semipermeable cylinder. A graphical analysis examines the variations in radial velocity, pressure, and drag force with respect to parameters such as permeability (ξ1, ξ2), separation parameter (γ), and couple stress parameter (λ). Additionally, streamline patterns are investigated to gain deeper insights into flow behavior. This study contributes to fluid mechanics by advancing the understanding of couple stress fluid flows in porous media and their implications for complex porous structures.