Dual Solutions and Temporal Stability of MHD Al₂O₃–Cu/Water Hybrid Nanofluid Flow over a Bidirectional Shrinking Sheet in a Porous Medium

This study investigates the dual solution existence and the associated temporal stability of magnetohydrodynamic hybrid nanofluid (HNF) flow over a bidirectional shrinking sheet. The working fluid consists of an Al₂O₃–Cu/water HNF subjected to suction, magnetic field effects, and porous medium resistance. The reduced coupled ordinary differential governing equations are solved using the Galerkin weighted residual method. A temporal stability analysis based on eigenvalue formulation is conducted to distinguish physically realizable solutions from unstable ones. The effects of emerging parameters on the skin friction coefficients and the Nusselt number are examined with the concentration given on magnetic effect. The results reveal the dual solution existence in a critical range of suction values, beyond which no solutions exist, indicating boundary-layer separation. Stability analysis confirms that only the upper-branch solution is stable and physically admissible. Enhanced magnetic field strength, delay boundary-layer separation and improve thermal performance. The present results show excellent agreement with previously reported limiting cases, demonstrating the accuracy and robustness of the proposed method.