HAM and Numerical solution for .

Nanofluids achieve high thermal transport efficiency by uniformly dispersing small particles in base liquids, significantly enhancing the heat transfer coefficients and making them vital in various thermal engineering applications. The research examines non-uniform thermal conductivity and activation energy critical for accurately describing fluid behaviour. The study incorporates bioconvection to prevent nanoparticle settling and ensure fluid stability through motile microorganisms. The governing partial differential equations are converted into ordinary differential equations that are solved using the Homotopy Analysis Method (HAM), to provide a strong mathematical framework for the analysis. This study finds that the velocity of the fluid decreases with magnetic constraint intensification and time retardation. however, heat transfer increases at higher radiation, and heat absorption/emission parameters but decreases with a higher Prandtl number, while an increased Schmidt number leads to decreased concentration profiles. This paper investigates a nano-Williamson fluid (NWF) flow over an exponentially stretched surface in a permeable medium, considering essential variables such as mixed convection, electromagnetic forces, non-linear thermal radiation, heat production, Joule heating and ohmic dissipation that are essential for understanding its complicated behavior.