Stability and multiple solutions of MHD-quadratic radiative tri-hybrid nanofluid flow in a Darcy–Forchheimer porous medium over a rotating shrinking disk

Purpose The purpose of this study is to explore the dual solutions of an unstable three-dimensional flow and quadratic radiative heat transfer involving a tri-hybrid nanofluid over a rotating shrinking disk, considering the influence of a Darcy–Forchheimer porous medium, an external magnetic field and heat source/sink. The tri-hybrid nanofluid is developed by dispersing molybdenum disulfide (MoS2), silicon dioxide (SiO2) and graphene oxide (GO) into a water (H2O) base fluid. Design/methodology/approach Similarity transformation is employed to reduce the governing partial differential equations to a system of nonlinear ordinary differential equations, which are solved numerically using MATLAB’s bvp4c solver. The analysis reveals multiple solutions (upper and lower branch solutions) for the similarity equations. The effects of various parameters on radial and azimuthal velocity, temperature, radial and azimuthal skin friction and the Nusselt number are analyzed through comprehensive tables and figures. Stability analysis confirms that the lower branch solutions are unstable, where as the upper branch solutions are stable and practically realizable. An increase of 1% in the overall nanoparticle volume fraction leads to maximum increases of 2.94% in radial skin friction, 3.10% in azimuthal skin friction and 1.99% in the Nusselt number. The study also highlights improvements in heat transfer rates driven by quadratic thermal radiation, unsteady radial shrinkage and Biot number effects. In addition, velocity slip and magnetic field parameters significantly influence radial and azimuthal velocity profiles. Entropy production increases with quadratic radiation, magnetic field and porous medium characteristics. Notably, the opposing trends observed in the lower-branch solutions are generally more pronounced than in the upper-branch solutions. Practical implications The findings underscore the potential of tri-hybrid nanofluids in industrial and electrical applications, mainly as heat transfer fluids in rotating machinery, gas turbine rotors and air purification systems. Originality/value The novelty of this work lies in the inclusion of quadratic thermal radiation and a heat source/sink, which significantly enhance the thermal conductivity and heat transfer properties of the base fluid, offering potential improvements for heat exchange systems.