Statistical and mathematical modeling of viscoelastic 3D Casson nanofluid flow with activation energy and motile microbes induced by exponential sheet bounding with Darcy-Forchheimer porous medium

This study scrutinizes the numerical exploration of viscoelastic Casson nanofluid flow induced by a Darcy-Forchheimer exponential sheet. For novelty of problem the impact of motile microbes and activation energy are taken into account. The Casson model is employed to capture the non-Newtonian behavior of the nanofluid, while the Darcy-Forchheimer formulation accounts for the permeable medium effects. Micropolar fluids is used for viscoelastic material. Moreover, the occurrence of swimming microorganism introduces biological complexity to the system. The activation energy parameter is crucial in understanding the thermal behavior of the nanofluid. Governing PDEs of micropolar fluid are converted into ODEs via similarity variables. Through numerical simulations via MATLAB, the interplay between these factors is examined, shedding light on their combined influence on flow characteristics such as velocity \(\text{f}{\prime}\) $\text{f}\prime$ , temperature distribution \(\theta\) $\theta$ , concentration profiles \(\phi\) $\varphi$ , and density distribution \(\chi\) $\chi$ . It is noted that temperature profile θ is increased for growing value of magnetics parameter \(M\) $M$ and reverse trend is noted in temperature profile for improving value of Prandtl number \(Pr\) $\mathit{Pr}$ and thermal radiation \(Rd\) $\mathit{Rd}$ . The velocity distribution \(\text{f}{\prime}(\eta )\) $\text{f}\prime (\eta )$ , \(\text{g}{\prime}(\eta )\) $\text{g}\prime (\eta )$ increases with the increase in micropolar parameter \(K\) $K$ . Density of motile microbes is decreased for growing value of Peclet number \(Pe\) $\mathit{Pe}$ .