Implementation of homotopy analysis method for entropy-optimized two-phase nano fluid flow in a bio convective non-Newtonian model with thermal radiation

The analytical and numerical simulation for the two-phase magneto-hydrodynamics flow in entropy-optimized nano fluid past a stretching sheet with extended Darcy-Forchheimer porous medium and microorganisms is explored. The key focus is to analyze the combined influences of viscous dissipation, chemical reaction, mass suction velocity, and convective boundary conditions. These effects play a pivotal role in the manufacturing process that involves coatings as well as the processing of polymer in addition to enhancing the cooling efficacy in chemical reactors and biomedical equipment. The irreversibility aspects of thermal transfer are also explored by the Bejan number. The ongoing investigation is conducted by both the numerical (using the Mathematical ND solve command) and analytical approach (employing the homotopy analysis method (HAM)). Therefore, the key results include the augmented entropy generation with the escalating values of radiation, Brinkman number, magnetic field, and non-Newtonian parameter. On the other hand, the Bejan number shows a similar elevated pattern for the same dimensionless factors. Furthermore, the escalating rate of mass transmission is observed with Schmidt number, Prandtl number, and thermophoresis, whereas the boosting rate of motile transmission is noted with Peclet number, Schmidt number, and bio-convection Lewis number. Also, the magnitude of the rate of thermal transport near the surface exhibits an ascending pattern with the posited Brownian motion, thermophoresis, and radiation