In this paper, the thermal performance of Casson fluid is discussed with variable density while the vertical jet is taken out. The correlation of tri-hybrid nanofluid was utilized whereas base fluid was addressed as ethylene glycol, and suspension of CuO,GO and aluminum oxide were considered because of their superior heat transfer capabilities, such as electronic cooling and heat exchangers. For increased energy efficiency, these nanofluids were also utilized in industrial cooling systems, solar collectors, and automobile radiators. Darcy's law was used with heat sink and viscous dissipation. The development of the mathematical model was visualized in terms of PDEs. The finite element method was used for numerical procedures. The novel aspect included using the Box-Behnken design for optimization and the finite element method to analyze tri-hybrid nanofluid flow over a vertical jet with Darcy-Forchheimer effects, heat source, and viscous dissipation. It was claimed that highly novelty work is discussed. The Box Behnken design was employed for calculating Nusselt number and divergent velocity. We concluded that the motion of nanofluid is enhanced when Forchiermer number and B are enhanced. The temperature profile is boosted when heat sink and Eckert number, D, and the power law index number are enhanced. Ternary hybrid nano-fluid has remarkable achievement in heat transfer rate and divergent velocity than hybrid nanofluid and nanofluid
