Rotational Flow of Brinkman Couple-Stress Fluids in Eccentric Spherical Annuli with Slip

This study investigates the low-Reynolds-number rotation of two eccentric spheres within an incompressible fluid that accounts for both micro-rotational effects and the presence of a porous medium. The region between the spheres contains a rigid, stationary skeleton, and we model the resulting flow using the Brinkman-extended Darcy approach. We derive the governing field equations, which account for the resistance to rotation through specific fluid viscosity parameters, while also incorporating surface slip effects on the interior sphere. The system of equations is solved using a semi-analytical boundary collocation method, where the fluid motion is expressed through a series of mathematical expansions satisfied at discrete points along the spherical boundaries. Our numerical results demonstrate that the hydrodynamic torque exerted on the spheres is highly sensitive to the porous environment. Specifically, increasing the permeability of the medium from 0.001 to 0.5 results in a substantial torque increase of approximately 210%. Additionally, the fluid’s resistance to micro-rotation acts as a torque-enhancing factor, with a variation in viscosity parameters from 0.02 to 0.45 inducing a 5.25% increase in torque under conditions of high eccentricity. These findings were validated against established benchmarks for standard fluids in non-porous media, showing excellent agreement.