Dual solutions for the Darcy–Forchheimer porous medium with convective heat transfer effect on rotating hybrid nanofluid

In this research, the three-dimensional (3D) rotating $${\mathrm{Cu}} - {\mathrm{A}}{{\mathrm{l}}_2}{{\mathrm{O}}_3}$$/hybrid nanofluid (HNF) developed in Darcy–Forchheimer porous medium (DFPM) has been examined numerically across the stretching/shrinking surface in impact of the convective heat transfer and heat source/sink effect. Moreover, the impact of Forchheimer, suction, and MHD parameters on magnitudes of reduced skin friction and heat transfer were examined using the Tiwari-Das model. The governing equations of the considered problem are transmuted into the appropriate structure of ordinary differential equations (ODEs) by employing the linear similarity variables. The resulting model of ODEs is numerically solved utilizing the “three-stage Labatto III-A technique” provided in the “MATLAB software's” bvp4c solver. In addition, to make evaluations, the current numerical results are compared with those from previous studies. Dual branches are created by the ODE governance system. With the aid of stability analysis, a single stable branch is identified. Reduced $${\phi _2}$$ in both branches has been shown to increase the reduced heat transfer rate. In addition, the outcomes demonstrated that the number of branches is dependent on the parameter ranges for suction, porosity, and magnetic for a given value of the parameter for rotation. Moreover, temperature profile enhanced as increased the amount of heat source strength. Dual branches are obtained for specified assortments of the related parameters. Moreover, when the solid volume of copper was improved, the strength of reduced heat transfer intensified toward the Forchheimer parameter and declined in MHD and suction effects for both branches. Consequently, the findings demonstrate that branch duality occurs for $${F_S} \le {F_{Sci}}$$, $$M \ge {M_{ci}}$$, and $$S \ge {S_{ci}}$$, but no fluid flow is conceivable when $${F_S} > {F_{Sci}}$$, $$M < {M_{ci}}$$, and $$S < {S_{ci}}$$. Future research on dual solutions in rotating HNFs may concentrate on improved fluid models, turbulent flows, biological applications, and experimental validation. These developments will help to build more efficient energy systems, biological heat transfer applications, and industrial thermal management solutions.