Numerical investigation of MHD flow and heat transfer of hybrid nanofluid over a Darcy–Forchheimer shrinking sheet using the Reiner–Philippoff model with significant viscous dissipation and Joule heating effects

Hybrid nanofluids have attracted significant attention due to their enhanced thermal performance in engineering applications. This study numerically investigates the effects of magnetohydrodynamics (MHD), Joule heating, thermal radiation, heat source, and nanoparticle volume fractions on the flow and heat transfer characteristics of a copper-titanium dioxide/sodium alginate (Cu-TiO₂/SA) hybrid nanofluid over a Darcy-Forchheimer porous shrinking sheet using the non-Newtonian Reiner-Philippoff model. The governing partial differential equations were transformed into ordinary differential equations via similarity transformations and solved using the MATLAB bvp4c solver. Dual solutions were obtained for the shrinking case, and stability analysis confirmed that the first solution is physically stable. The results indicate that increasing nanoparticle concentration enhances fluid viscosity and skin friction while slightly reducing the heat transfer rate due to the dominance of conductive heat transport. The Cu-TiO₂/SA hybrid nanofluid enhances heat transfer by 34.2%, outperforming TiO₂ (4.5%) and Cu (29.1%) nanofluids. Thermal radiation and suction effect further improve heat transfer, whereas higher Eckert number and heat source parameters reduce thermal performance. Meanwhile, the effects of a magnetic field have been shown to reduce the fluid velocity. These findings highlight the importance of controlling physical parameters to optimize heat transfer in non-Newtonian hybrid nanofluid systems.