Numerical exploration of MHD bioconvective Williamson-Maxwell nanoliquid flow due to an exponentially elongated porous sheet

This research delves into the intriguing realm of non-Newtonian fluids in conjunction with microorganisms, presenting a mathematical model tailored to analyze heat and mass transfer within Williamson-Maxwell nanofluids hosting gyrotactic microbes. The study investigates how these fluids behave under the influence of multiple factors such as magnetic fields, thermal radiation, chemical reactions, and dissipation effects. Employing a set of similarity invariants, the governing equations are transformed into ordinary differential equations, which are then solved using a fourth-order R-K scheme. The findings, presented graphically, offer insights into various flow parameters and are complemented by pertinent physical explanations. The influence of magnetic flux ([Formula: see text]), Buoyancy ratio ([Formula: see text]), Peclet number ([Formula: see text]), and Schmidt number ([Formula: see text]) on various physical parameters are shown graphically. Notably, the research reveals that while increasing the external magnetic field impedes fluid motion, it enhances thermal and density layers. Additionally, a higher bioconvective Schmidt number is shown to reduce microbial density. These observations hold significant implications for applications involving nanofluids and microorganisms across biomedical, pharmaceutical, biofuels, and other sectors. Overall, this study contributes valuable knowledge to the understanding and potential utilization of complex fluid systems in diverse industrial contexts.