Computational study of two and three dimensions non-Newtonian MHD fluid flow through non-Darcian porous regime with gyrotactic motile microbes

Magnetohydrodynamic (MHD) bioconvective transport of non-Newtonian fluids through porous media has applications in thermal management, bioengineering, materials processing, and environmental fluid systems, where complex rheology, magnetic forces, heat generation, and microorganism motion can all significantly impact transport characteristics. This study examines the steady two-dimensional MHD bioconvective flow of a Casson-Williamson fluid over an infinite vertical permeable surface embedded in a non-Darcy porous medium. It considers Darcy-Forchheimer resistance, Joule heating, thermal radiation, heat generation, Brownian motion, thermophoresis, chemical reaction, and gyrotactic motile microorganisms. The governing nonlinear partial differential equations are converted into a coupled system of ordinary differential equations via appropriate similarity transformations and numerically solved with the MATLAB bvp4c solver. The novelty of the current study examines Darcy-Forchheimer porous resistance, Joule heating, and gyrotactic bioconvection inside the Casson-Williamson fluid framework, resulting in a thorough description of linked momentum, heat, mass, and microbe movement. The results demonstrate that the magnetic, porous, Forchheimer, and Casson factors diminish fluid velocity, while thermal and solutal buoyancy characteristics increase it. Increasing the Prandtl number lowers the temperature, but thermal radiation, viscous dissipation, and heat production raise it. The concentration profile falls with increasing Schmidt number, Brownian motion, and chemical reaction parameters, but thermophoresis increases concentration. Furthermore, the density of motile microorganisms falls as the bioconvective Lewis number, Peclet number, and motile parameter increase. The results provide insights on how to control coupled heat, mass, and microorganism transport in electrically conducting non-Newtonian fluids, and they could help with the design and optimisation of porous-medium thermal management, bioengineering, materials-processing, and environmental transport systems.