Numerous engineering and industrial applications rely on base fluids that are too common to meet the needs of industrial operations. Research in the area of nanotechnology is accelerating dramatically to overcome this. When copper nanoparticles are added to base fluid of water, the nanofluid's thermal conductivity increases. The novelty of the present research is heat transfer comparison between linear and nonlinear stretching sheets over 3D Darcy–Forchheimer Cu–H₂O nanofluid flow caused by a power-law stretch. Additionally, nonlinear solar radiation, magnetic field, linear reaction rate, heat source/sink, Brownian motion, activation energy, thermophoresis, and suction/blowing are considered with convective boundary constraints. A set of complex coupled partial differential equations represent the physical system. Pertaining to similarity transformations higher-order dimensionless ordinary differential equations is obtained. The numerical solution is estimated with the help of the shooting technique which comprises the Runge–Kutta Felhberg (RKF45) and Newton–Raphson method using MAPLE software. Flow aspects were analysed at different relevant parameters. The outcome supports that in the case of suction, the Nusselt number delivers increasing values as the copper nanoparticles increase relative to injection. The results of the current research fulfil the needs of industries such as glass-fibre and rubber sheets production, melt-spinning, wire drawing, etc.