<p>The present work concentrates on the effect of anisotropic permeability on a couple stress fluid between the annular space of two vertical coaxial rotating cylinders filled with an anisotropic porous medium. The novelty of this research lies in the examination of rotating cylinders containing an anisotropic porous medium and the application of dual boundary conditions for the couple stress fluid. Both vertical cylinders rotate with different angular velocities. The directional permeability of an anisotropic porous medium characterized by a permeability ratio and an anisotropic angle. Using the Brinkman model of couple stress fluid, the velocity profile, vorticity profile, and shear stress distribution are determined analytically under two different boundary conditions: (i) the fluid vorticity rate at both cylinders are analogous to corresponding cylinders’ angular velocities, and (ii) the couple stress of the fluid is negligible at both cylinders. The results indicate that increasing the permeability ratio and anisotropic angle leads to a decrease in fluid velocity, with the maximum velocity achieved under zero couple stress conditions. An increase in the couple stress parameter and Darcy number promotes flow, and rotational parameters positively affect velocity. The graphical representations of vorticity and shear distribution across varying parameters are also presented. The specific cases regarding the geometry and the impact of parameters on their flow are also discussed. The model has practical applications for lubricants used in industrial settings for porous bearings and rotating filter separators.</p>