Analysis of Ti-Alloy Nanoparticles in Paraffin Oil for 3D MHD Darcy-Forchheimer Flow Over a Bi-Directional Stretching Surface

The three-dimensional model of Darcy-Forchheimer flow in a convection system consisting of Ti-alloy nanoparticles (TiO₂) suspended in paraffin oil is mathematically constructed using fluid mechanics and partial differential equations (PDEs). A novel aspect of this study is the application of similarity transformation techniques to convert complex PDEs into a system of ordinary differential equations (ODEs), which are then solved using the Runge–Kutta 4th order method with the shooting technique. This unique approach provides deeper insights into the effects of magnetohydrodynamics (MHD), porosity, heat source, stretching surface, and radiation on bi-directional velocity and temperature profiles. The results demonstrate that Ti-alloy nanoparticles significantly enhance the thermal conductivity of the base fluid, leading to a 34.7% increase in temperature profiles compared to conventional fluids. The presence of a magnetic field induces a Lorentz force, reducing the bi-directional velocity by 18.5% while increasing fluid temperature by 22.9%. An increase in the porosity parameter results in a 15.3% reduction in velocity due to higher resistance, whereas the temperature profile shows a corresponding rise of 26.1%. Furthermore, an increase in the Forchheimer parameter reduces velocity by 21.6%, while the radiation parameter enhances heat transfer by 29.4%. These findings highlight the superior heat transfer efficiency of Ti-alloy-based nanofluids, making them highly suitable for applications in thermal energy storage, solar energy systems, and industrial cooling technologies.