In current paper, we considered the 3-D occurrence of magneto Williamson fluid subjected to linear penetrating expanding sheet. The Williamson liquid is taken into account under the essence of bio convection bordering with Joule heating, heat production. Moreover, Brownian movement combined with thermophoresis diffusion was also analyzed in current study. Central PDEs are modified into system of ODEs on the basis of appropriate similarity transformation utilization. For solution purpose, an optimal approach namely HAM was opted. The critical review based on all involved multifarious constraints across the non- dimensional velocity contour ( along x − axes and also y − axes ) , energy and concentration distribution. Moreover, across the motile organism distribution was also conducted graphically and depicts total covenant with the former published research work. The drag force effect, Nusselt number, Sherwood number, and motile organism density in tabular form is also proposed in this study. Moreover, Motile organism profile graphically demonstrates decreased demeanor for considered bio convection Lewis number, Peclet number, and also temperature difference constraint whereas results of motile organism density augmented in tabular version for the considered bio convection Lewis number, Peclet number, and also temperature difference constraint.

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Optimal homotopic strategy for thermal and mass transport in Williamson model flow PDEs under heat generation/absorption and Joule heating