Chemically reactive flow of Williamson nanofluid with nonlinear thermal radiation over an exponentially stretching/shrinking surface

This study explore the numerical investigation of Darcy-Forchheimer model with Williamson nanofluid via an exponentially penetrable shrinking surface under the influence of both first and second-degree coefficient for velocity, chemical species, thermal radiation, and variable thermal conductivity. Classical transformations are used to form nonlinear PDEs into a new form of nonlinear ODEs and the resultant system is then resolved through the utilization of the bvp4c function built into MATLAB software. The concurrent impacts of the physical factors on flow, thermal distribution, mass curve, and engineering quantities were examined graphically, tabularly and histrogram analysis. The most notable findings of the current article are the existence of dual solutions for Williamson nanofluids. The outcomes show that Williamson nanofluid factors have positive impacts on the shear stress profile for both solution branches. The variations in Eckert number, radiation and variable thermal factors do not affect the critical point, which remains constant at ${\chi }_{ci}\left(=-0.6081\right)\text{.}$ Moreover, a delay in the boundary layer separation can be observed for the first-order velocity slip factor, which reduces the shear stress. In contrast, boundary layer separation accelerates by the second-order velocity slip factor and improves skin friction. These discoveries over a shrinking surface consisting of multiple solution flow dynamics have shown to be useful insight in real-world scenarios.