The Innovative Technique for Obtaining the Solutions of Jeffery Hamel Nano-fluids Flow Problem

The study of heat transfer in nanofluid flows is increasingly important in many engineering, medical, and industrial applications. These fluids offer enhanced thermal cooling properties compared to conventional fluids. The research problem lies in the challenges of solving the Jeffrey-Hammel flow model for nanofluids, which includes coupled nonlinear differential equations that describe the thermal and hydrodynamic behavior of this type of flow, taking into account the influence of multiple factors such as the type, size, and concentration of nanoparticles. This research aims to propose a new hybrid analytical technique that combines the Laplace transform and the q-homotopy analysis technique with the convolution theory and Padé approximation (LCP-q-HAM), to provide accurate and improved solutions for a complex mathematical model. These solutions are compared with numerical results to verify their effectiveness. The effectiveness of the proposed method is verified by comparing the analytical results with numerical solutions using the bvp4c algorithm in the Maple environment. The results were presented in the form of tables and graphs to display the velocity and temperature distributions under the influence of physical variables such as Reynolds number (Re), Prandtl number (Pr), Eckert number (Ec), and open-angle (φ). However, water (H2 O) was used as the base fluid with three different types of nanoparticles: copper (Cu), titania (TiO₂), and alumina (Al₂O₃). The results showed that the proposed LCP-q-HAM technique provides more accurate solutions and faster convergence than conventional methods. The type of nanoparticles also appeared to have a significant impact on enhancing heat transfer, with copper nanoparticles achieving the highest surface friction coefficient and the best thermal performance compared to other studied particles, highlighting the importance of selecting the appropriate nanomaterial for heat transfer applications.