Numerous non-Newtonian fluid models are formulated as an alternative to Newton's Law of viscosity, which describes absolute viscous sources as a function of shear rate rather than constantly. In this paper, the Reiner-Philippoff fluid (RPF) model is considered out of many existing models since it can capture certain non-Newtonian characteristics in certain cases. An adopted underlying mathematical model employs a partial differential system, which is reduced to the system of ordinary differential system with the help of suitable similarity variables. The resultant system is computed with the Runga-Kutta-Fehlberg (RK-F) method to boost-up the confidence of its accuracy. It is found that the porosity and Darcy-Forchheimer parameters improve the flow rate of RPF. On the other hand, heat transfer is positively affected by thermal slip in contrast to the outcomes of radiation parameter, which, however, are inauspicious. Furthermore, the mass diffusion rate is noticed to be significantly influenced by the Schmidt number along with the chemical reaction parameter. In conclusion, all the results are expected to benefit numerous transport phenomena of industrial, biomedical, and thermal sciences.