## 4 Results and Discussions  
The mixed convective flow of micropolar fluids with alumina and Titania nanoparticles across a horizontal circular cylinder is investigated in a saturated porous medium. The effects of governing parameters with the convective boundary condition are analyzed for assisting and opposing flows. Both local similarity and non-similarity equations with the selected boundary conditions are solved numerically. The results of the local Nusselt number are compared with the existing results in Table 1 and are found in excellent agreement. The effects of buoyancy and material parameters on the dimensionless velocity are presented in Figures 2 and 3 for (a) assisting and (b) opposing flows, respectively. The other parameters are kept constant. In the case of assisting flow, the dimensionless velocity of both Al\({}_{2}\)O\({}_{3}\) and TiO\({}_{2}\) base micropolar fluids increases within the hydrodynamic boundary layer with a mixed convection parameter, see Figure 2a,b. The velocity satisfies both boundary conditions. On the other side, the material parameter \(\Delta\) reduces the dimensionless velocity. This is because the velocity of micropolar fluids is coupled with the macroscopic rotation of the fluid particles. For both micropolar nanofluids, we observe the same behavior. For opposing flows, the dimensionless velocity shows opposite behavior, Figure 3a,b. It decreases with the mixed convection and the material parameters inside the boundary layer. For smaller values of both parameters, the dimensionless velocity increases from the surface to the boundary layer. However, for larger values of both parameters, the velocity decreases close to the surface and then increases and converges. The effects of the same parameters on the microrotation velocity are depicted in Figure 4 for assisting flows and in Figure 5 for opposing flows of micropolar nanofluids. In the hydrodynamic boundary layer, the variation of microrotation velocity with the mixed convection and material parameters is negligible for both nanofluids, Figure 4a,b. For smaller values of the material parameter, the microrotation velocity is almost zero in both cases. In fact, both velocity and temperature fields become independent of the microstructure of the fluid. However, as the values of the material  
Figure 3: Variation of dimensionless velocity with \(\Delta\) and \(\lambda\) for opposing flow of micropolar fluid with (a) Al\({}_{2}\)O\({}_{3}\) and (b) TiO\({}_{2}\) nanoparticles. Figure 2: Variation of dimensionless velocity with \(\Delta\) and \(\lambda\) for assisting flow of micropolar fluid with (a) Al\({}_{2}\)O\({}_{3}\) and (b) TiO\({}_{2}\) nanoparticles. parameter increase, the velocity goes in the negative direction and then approaches the boundary layer. For smaller values of \(\Delta\), the microrotation velocity is also negligible in case of opposing flows for both nanofluids, Figure 5a,b. However, for large values of \(\Delta\), the microrotation velocity shows oscillatory behavior in both cases. The impacts of mixed convection and material parameters on the dimensionless temperature are presented in Figure 6 for assisting flows and in Figure 7 for opposing flows of micropolar nanofluids. In the thermal boundary layer, the dimensionless temperature decreases with both parameters for both nanofluids, Figure 6a,b, respectively. In both cases, the temperature at the surface is highest and decreases to ambient temperature. The boundary layer thickness is found to be the same for both nanofluids. It is important to note that the dimensionless temperature is almost independent of the mixed convection parameter. For opposing flows of micropolar nanofluids, the dimensionless temperature increases with \(\lambda\) and decreases with \(\Delta\) in both cases, Figure 7a,b. However, this difference increases with a decrease in \(\lambda\). The surface temperature is highest and decreases to ambient temperature. The rate of convergence increases with a decrease in \(\lambda\) for both cases.