Three‐Stage Lobatto Analysis on Hybrid Nanofluids in Darcy–Forchheimer Flow: Enhancing Heat Transfer With Radiation and Nanoparticle Effects

ABSTRACT This study presents a comprehensive numerical investigation of steady Darcy–Forchheimer hybrid nanofluid (HNF) flow over a heated porous surface, with the combined influences of thermal radiation. The presence of microorganisms plays a vital role in suppressing nanoparticle sedimentation, thereby ensuring the suspension's long‐term stability and uniform dispersion. The HNF, formulated by dispersing copper oxide (CuO) and silicon dioxide (SiO 2 ) nanoparticles in water (H 2 O), is selected due to its superior thermal conductivity and potential to enhance energy transport in heat‐intensive applications significantly. The governing system of nonlinear PDEs, derived from conservation laws, is transformed into a set of dimensionless ODEs through similarity transformations and solved using the three‐stage Lobatto IIIA scheme implemented in MATLAB, known for its accuracy and stability in handling complex boundary value problems. A detailed parametric study examines the impacts of Brownian motion, thermophoresis, magnetic field, thermal radiation, Lewis number, and Peclet number on velocity, temperature, concentration, and microorganism density profiles. The results reveal stronger magnetic fields suppress velocity, while thermal radiation and nanoparticle hybridization markedly boost heat transfer rates. Furthermore, higher Lewis and Peclet numbers reduce the motile microorganism density, directly influencing bioconvective transport. These outcomes highlight the dual role of HNFs and microorganisms in achieving enhanced thermal performance and stable dispersion. The novelty of this work lies in integrating Darcy–Forchheimer porous effects, HNF thermal enhancement, and microorganism‐induced stability into a single framework, offering new perspectives for advanced thermal management systems. The findings hold strong potential for applications in compact heat exchangers, renewable energy devices, biomedical systems, and next‐generation cooling technologies.