<p>This paper presents a numerical study of the squeeze flow of non-Newtonian couple-stress fluids between parallel discs, a phenomenon essential in engineering applications such as lubrication systems, polymer processing, engine bearings, and joint prostheses. Unlike Newtonian fluids, these fluids require microcontinuum models to account for polar effects, such as couple stresses, providing a more accurate description of their dynamics. A fourth-order finite difference method was used to solve the governing equations, with a variable transformation applied to reduce the biharmonic equation to two coupled Poisson equations. An inverse method, based on a fixed-point technique, was employed to iteratively adjust the radial pressure gradient to satisfy the boundary conditions. The results show that couple-stress parameters significantly influence velocity distribution, pressure profiles, load-carrying capacity, shear stress, and frictional forces, highlighting the potential of these fluids to enhance performance in terms of load support and friction properties compared to classical Newtonian fluids.</p> Graphical abstract <p></p>