Numerical Study of Entropy in a Non‐Newtonian Fluid Flow Through a Microchannel With Electromagnetic and Pollutant Effects

ABSTRACT This study presents an in‐depth analysis of the flow and thermodynamic behavior of Casson couple‐stress fluid in a vertical microchannel under the combined influence of Hall current, uniform heat source/sink, variable thermal conductivity, Brownian motion, thermophoresis, and pollutant discharge, subject to convective boundary conditions. The governing equations are transformed using similarity variables and solved numerically via the shooting technique coupled with the Runge–Kutta–Fehlberg (4,5th) scheme. The originality of this study lies in the simultaneous consideration of non‐Newtonian rheology, electromagnetic effects, variable transport properties, and pollutant dynamics, a combination rarely addressed in microfluidic studies. The thermal field demonstrates a direct sensitivity to heat generation, where the temperature rises with increasing heat source strength, while it declines when variable thermal conductivity is allowed to vary, owing to enhanced thermal diffusion. The concentration within the medium is elevated by Brownian motion and external pollutant source effects, both of which promote particle dispersion and accumulation, whereas thermophoretic forces act in the opposite manner by driving particles away from hotter regions, thereby suppressing concentration. Entropy generation responds dually to Brownian motion, thermophoresis, Casson parameter, and variable thermal conductivity, indicating the sensitive interplay of flow, mass, and thermal transport mechanisms. The Bejan number profile illustrates spatial variations in the relative contributions of heat transfer and fluid friction irreversibilities under the impact of heat source, variable thermal conductivity, Brownian motion, and thermophoresis parameters. These results provide a valuable approach to the design and optimization of microchannel systems in biomedical, chemical, and energy applications, in settings where precise heat control, mass, and pollutant transport are critical.