<p>The purpose of this study is to investigate the thermal behaviour of a Jeffrey non-Newtonian fluid as it flows along a vertical plate immersed in a non-Darcy porous medium. The impacts of the Forchheimer inertial parameter are included into the model to describe the non-linear drag forces, the non-uniform heat source as well as the influence of suction at the vertical plate surface. In wide stream of applications including oil recovery, geothermal energy systems, and chemical reactors, in all these areas, the regulation of heat transmission is essential. Hence, this work offers a useful insight for engineering applications by providing a full knowledge of the behaviour of non-Newtonian fluids in such complex systems as well as the capacity to forecast how these fluids would behave. The governing partial differential equations for momentum and heat transport are transformed into scale free partial differential equation forms through appropriate similarity transformations. The Keller box method, an efficient finite difference methodology for solving boundary layer equations with high precision, is then used to solve the scale free equations to get the desired results. It has been noticed that the local friction rate slopes as the values of the relaxation to retardation time ratio (λ) raise, but it reduces as the Deborah number increase. Increase in the wall transpiration declines both the fluid temperature and concentration. When the tangential coordinate is increased, the temperature and concentration both rises, but the flow rate decreases. The findings are very important for the substantial improvement of polymer processing industries that include non-Newtonian fluids in porous media.</p>