Recent advances in modified Arrhenius activation energy and bioconvection in Williamson nanofluid flow over a bidirectional surface

Non-Newtonian flows have applications in food combination, plasma flow, inherent and organic fluids, antibiotics, and lubrication through oils and greases. This study explores the bidirectional flow of Williamson nanofluid in a porous medium, incorporating thermophoresis, Brownian motion, bioconvection effects, and Arrhenius activation energy over a nonlinear stretching surface. The governing equations are transformed into a dimensionless form using similarity transformations and numerically solved via MATLAB's bvp4c shooting scheme. Results indicate that increasing the Williamson parameter $\lambda$ and porosity parameter $\epsilon$ reduces velocity, with a 10% rise in $\lambda$ leading to an 8% velocity reduction. Temperature increases with the thermophoresis parameter $\left(Nt\right)$ where a 15% increase in $Nt$ results in a 7% temperature rise. The Nusselt number improves with a higher Prandtl number $\mathrm{Pr}$ increasing by 10% when $\mathrm{Pr}$ rises from 5 to 7, while the Sherwood number declines with stronger Brownian motion. These findings provide key insights into heat and mass transfer mechanisms, contributing to advancements in industrial cooling, biomedical applications, and nanofluid-based thermal systems.