Heat Transfer Analysis of Alumina+Copper/Sodium Alginate Radiative Hybrid Nanofluid in Darcy‐Forchheimer Flow Over a Nonlinear Stretching/Shrinking Porous Surface

This numerical study investigates dual solution branches of sodium alginate‐based hybrid nanofluid flow over exponentially stretching and shrinking surfaces. Main objective is to analyze how solid volume fraction affects stretching and shrinking behavior, and to evaluate variations in skin friction coefficient and heat transfer rate under suction and permeability effects. Additionally, influences of permeability, magnetic field strength, and viscous dissipation on velocity and temperature profiles of hybrid nanofluid are thoroughly examined. By applying an exponential similarity transformation, governing partial differential equations are reduced to ordinary differential equations. These transformed equations are solved numerically using the three‐stage Lobatto III‐A formula via MATLAB's bvp4c solver. Results confirm the presence of dual (non‐unique) solutions within specific parameter ranges, with non‐uniqueness arising as controlling parameters approach critical suction or shrinking limits. An increase in permeability parameter decreases heat transfer and skin friction for the upper branch, while a similar decreasing trend is noted for the lower branch. Temperature gradients reduce with higher permeability, whereas higher Eckert numbers amplify thermal effects. Increasing copper nanoparticle volume fraction suppresses heat transfer for the upper branch but enhances it for the lower branch. Overall, findings offer valuable insights for optimizing fluid flow and thermal management in engineering applications.