Williamson MHD nanofluid flow via a porous exponentially stretching sheet with bioconvective fluxes

The recent study concentrates on magnetohydrodynamics (MHD). Williamson fluid drifts over an exponentially extending sheet. The porous medium is also crucial to improving thermal efficiency. The flow pattern model considers the inspirations of Joule heating, heat generation, viscous dissipation, and thermal radiation. This study also comprises the activation energy, bio-convectional, and gyrotactic microorganism phenomena. Furthermore, the Brownian and thermophoresis effects of nanoparticles are taken into consideration. Using proper similarity transformation, PDEs of the impetus, temperature, concentricity, motility microbe density, and boundary constraints upgrade into a non-linearly ordinary differential equations (ODEs) mode. Using MATLAB, transformed non-dimensional ODEs are dealt with using shooting procedures and results of significant physical strictures using a built-in bvp4c solver. Finally, it is elaborated and briefly explored numerically and visually can find interesting physical strictures versus the velocity gradient, temperature gradient, solutal gradient of species, and microbes' gradient. Incorporating microorganisms and nanoparticles into Williamson fluids can create novel materials with tailored properties for diverse applications. However, it's crucial to consider stability, biocompatibility, and environmental impact when designing these advanced fluid systems.