In recent years, there has been much interest in the effects of porosity and surface roughness (SR) or geometric irregularities between two moving plates under hydrodynamic lubrication. Porous bearings are used extensively in wide range of equipment, including computers, office equipment, home appliances, electric motors, and vehicles. In light of the importance of the aforementioned applications, we explored how SR and porous materials affect annular discs under the condition of a squeeze film. A five-point Gauss quadrature integral formula has been used to examine the characteristics of annular discs and a small perturbation method has been used to discretise the governing Rabinowitsch fluid flow (RFF) equations. The impact of nonlinear parameters on the behaviour of porosity and SR have been visualised in terms of film pressure (FP), load carrying capacity (LCC), and squeeze response time (SRT) of annular discs. Under the conditions of pseudoplastic and dilatant fluids, the effects of SR and porous materials between annular discs have been estimated in the form of the film pressure, LCC, and SRT and are presented in this manuscript as tables and graphs. According to the findings, the performance of an annular disc is significantly affected by porous material and radial roughness patterns. In addition, when RFF is carried through a rough surface and porous media, the performance is found to improve for dilatant fluids but suffer for pseudoplastic fluids.