Stability and dual solutions for Darcy-Forchheimer flow of Casson fluid over a non-linear curved surface with homogeneous and heterogeneous reactions

Shrinking boundary-layer flows often admit dual solutions. Identifying the physically realizable branch is essential for establishing safe operating limits. In the present study, the steady two-dimensional Darcy-Forchheimer flow of a Casson fluid over a non-linearly shrinking permeable curved surface has been investigated by considering the viscous dissipation and cubic autocatalytic homogeneous-heterogeneous reactions. The governing equations are reduced to a coupled non-linear system by similarity transformations, which is solved numerically by using the MATLAB BVP5C collocation solver. The lower solution branch is traced through continuation in the shrinking parameter, while temporal stability is examined through the smallest eigenvalue of the associated linearized perturbation problem. Benchmark and limiting-case comparisons show excellent agreement with previously published analytical and numerical results. Dual solutions are found to exist above a critical shrinking parameter, which shifts from to as the shrinking index increases, thereby widening the admissible operating range. This range is further extended by suction and Darcy permeability, whereas Forchheimer inertia reduces it. The results of the parameter perturbation analysis indicate that the skin friction is most sensitive to the shrinking index, with a variation of for the upper branch and for the lower branch. The overall Nusselt number is maximally varied by and less than with variations in curvature, Casson, and Eckert numbers, while the Sherwood number shows the maximum sensitivity to the heterogeneous response parameter of . The positive eigenvalue for the upper branch confirms that the upper-branch solution is the only physically stable. These results provide quantitative bounds on stability and identify key control parameters relevant to polymer extrusion, curved-roll coating processes, and reactive transport in porous media.