# Numerical Simulation of Williamson Nanofluid Flow over an Inclined Surface: Keller Box Analysis
## 3 Results and Discussion  
This section presents numerical simulations of non-Newtonian liquid called Williamson nano fluid over a slanted surface. The heat generation or absorption effects are incorporated in the energy equation, while the chemical reaction is added in mass transport equation. The converted differential equations by the utilization of supported similarity transformation are solved numerically using the Keller box method. The effects of pertinent flow factors, i.e., chemical reaction effect \(R\) Brownian motion factor \(Nb\), magnetic factor \(M\), local Grashof number \(Gr\), \(Gc\) local modified Grashof number, inclination, i.e., \(\Omega\), Pr, i.e., Prandtl number, thermophoresis assumed by \(Nt\), heat generation or absorption factor \(\lambda_{1}\), Williamson factor \(\gamma_{1}\), \(R\) (chemical reaction), and Lewis number \(Le\), are presented via several diagrams and tables to show their impacts on velocity, temperature, and concentration distributions. In addition, physical descriptions of the involved effects in this study are presented. Figure 2 exhibits that dimensionless velocity is moderated by resistive force, which is produced by the magnetic effect. It is verified that the numerical computations of \(f^{\prime}(\eta)\) for \(Gr\) and \(Gc\), which are displayed in Figures 3 and 4, show a good approval with the reported earlier study by [PERSON] et al. [38]. It is true because the buoyancy effect increases the speed of fluid flow. Figure 5 reveals the inclination impact on the velocity. It is clear that \(f^{\prime}(\eta)\) decreases on growing angle, and the recovered trend is in reasonable agreement with finding by [PERSON] et al. [39]. Physically, when \(\Omega=90^{\circ}\), the buoyancy force effect  
delines; as a result, velocity also decreases. Figure 6 demonstrates that the Williamson factor decreases the velocity field because the boundary layer thickness decreases with growth in non-Newtonian Williamson parameter. Figure 3: Velocity profile versus \(Gr\). Figure 2: Velocity profile versus \(M\). Figure 4: Velocity profile versus \(Gc\). Figure 5: Velocity profile versus \(\Omega\). The magnetic field increases the temperature profile in Figure 7. Figure 8 portrays temperature increases with respect to \(\lambda_{1}\), i.e., heat generation or absorption factor generates heat in the flow field and temperature enhances with in the thermal boundary layer. Figure 9 shows an enhancement in \(Pr\) basis the viscosity enrichment and descent in thickness of thermal boundary layer, finally temperature decreases. The irregular diffusion of particles warms up the boundary layer, because of which the temperature of the fluid increases as shown in Figure 10. Figure 11 exhibits temperature increases against thermophoretic effects. Increasing values of thermophoresis imply a higher shear gradient and temperature difference. Therefore, changes in the thermophoresis parameter help us to improve the temperature difference across the boundary layer. Figure 12 demonstrates that the concentration outline increases as we strengthen the magnetic field. A similar impact on the temperature profile against the magnetic parameter was observed by [PERSON] et al. [40]. Further, Figure 13 reveals that \(\phi(\eta)\) displays a converse relation with \(R\) (chemical reaction). It is notable that, with change in chemical reaction, [PERSON] et al. 's [41] results have been recovered. Figure 14 displays the boundary layer viscosity changes by enhancing Lewis number. A larger Lewis number is related to a lower \(D_{B}\) and vice versa. The boundary layer thickness decreases when \(Nb\) increases, which causes a decline in concentration distribution (see Figure 15). Figure 16 presents the thermophoresis influence on \(\phi(\eta)\), showing direct correspondence with \(Nt\), because nanoparticle concentration is a strong function of \(Nt\), due to which \(\phi(\eta)\) was significantly affected.