This article aims to investigate the generation of entropy for the magnetized coupled stress fluid passed through a permeable stretching cylinder that creates the condition of convective heat transfer. Additionally, in the uneven heat source in the flow field, we also analyzed our research. The properties of heat transfer taking into account in the perspective of thermal radiation. The main nonlinear partial differential equations (NPDE) become ordinary nonlinear differential equations by adopting the corresponding dimensionless variables. The recognized repeated shooting technique combined with the fourth-order standard Runge-Kutta integration solution to evaluate the resulting nonlinear ODE numerically. The velocity, temperature profile, streamline, number of local entropy generations, Bejan number, local friction coefficient, and Nusselt number effects of the new flow parameters are explained by graphs and tables. The flow system’s physical properties and the correlation between the parameters were clarified by using statistical methods. The results show that the Bejan number is reduced due to the magnetic source. Generation of entropy promotes the growth of the magnetic field and the Brinkman number, but the coupling stress aspect shows a double effect. The coupled stress parameter reduces the number of Nusselt by a fraction of 4.46%, while in the attendance of a magnetic field, the radiant heat transfer rate increases at a rate of 2.65%. In the attendance of a magnetic field, skin friction reduces the coupling stress factor by a rate of 24.67%. Besides, the current outcomes have been verified by previously published studies and are very acceptable. For the stretched cylinder, which has a curvature parameter of γ = 1.0, and for the flat, stretched surface, γ = 0.0, the thickness of the momentum and the thermal boundary layer are greater.