Gauss–Legendre weighted residual method on the stability of Eyring–Powell hybrid nanofluid flow past a shrinking surface with slip conditions

Purpose This study aims to investigate the dual-solution and stability characteristics of Eyring–Powell hybrid nanofluid flow over a shrinking surface, motivated by applications in coating, cooling and biomedical fluid processes where flow stability is essential. The influence of velocity slip, thermal slip, magnetic field and nanoparticle concentration on skin friction and heat transfer is examined. Design/methodology/approach Similarity transformations are used to derive the governing equations, which are solved using the Gauss–Legendre weighted residual method (GLWRM). By incorporating Gauss–Legendre quadrature into the Galerkin residual framework, the method efficiently accommodates the nonlinear Eyring–Powell rheology. Temporal stability analysis is performed to assess the physical validity of dual solutions. Convergence and comparison studies confirm numerical reliability. Findings Dual steady-state solutions exist within a limited suction range and the critical suction value separating solution branches is fwc=2.1682. Temporal stability confirms that the upper branch is stable, whereas the lower branch is unstable. Higher Eyring–Powell parameters enhance wall shear and heat transfer, while velocity slip and magnetic field reduce both. Nanoparticle loading and buoyancy increase skin friction but weaken thermal transport. GLWRM demonstrates fast convergence at N = 15 with residual errors≤10−6. Originality/value A novel application of GLWRM is presented for nonlinear hybrid nanofluid stability problems, offering an efficient alternative to BVP4C. The study provides clear physical criteria for maintaining stable non-Newtonian hybrid nanofluid flow over shrinking surfaces.