## 4 Results and Discussions
Figure 4: Variation of dimensionless rotational velocity with \(\Delta\) and \(\lambda\) for assisting flow of micropolar fluid with (**a**) Al\({}_{2}\)O\({}_{3}\) and (**b**) TiO\({}_{2}\) nanoparticles. Figure 5: Variation of dimensionless rotational velocity with \(\Delta\) and \(\lambda\) for opposing flow of micropolar fluid with (**a**) Al\({}_{2}\)O\({}_{3}\) and (**b**) TiO\({}_{2}\) nanoparticles. The variation of skin friction along the surface with \(\Delta\) and \(\lambda\) is depicted in Figure 8a for assisting flow and in Figure 8b for opposing the flow of micropolar-Al\({}_{2}\)O\({}_{3}\) nanofluids. As expected, the friction is zero at the front stagnation point and increases along the surface up to maximum and then decreases up to the rear stagnation point for both flows. In assisting flows, the maximum friction increases with both mixed convection and material parameters. However, in opposing flows, the maximum friction decreases with increasing mixed convection parameter. The material parameter does not affect the opposing flow. The same effects of mixed convection and material parameters on the skin friction can be observed in Figure 9a for assisting flow and in Figure 9b for opposing flow of micropolar-TiO\({}_{2}\) nanofluid. Due to higher density of TiO\({}_{2}\) nanoparticles, the skin friction for TiO\({}_{2}\) nanofluid is little bit higher than Al\({}_{2}\)O\({}_{3}\) nanofluid. The effects of solid volume fraction of TiO\({}_{2}\) nanoparticles and Biot number on the skin friction are presented in Figure 10a for assisting flow and in Figure 10b for opposing flow of micropolar nanofluids. The skin friction at the front stagnation point is zero and increases up to the maximum along the cylinder surface and then decreases up to the rear stagnation point in case of assisting flow. However, in opposing flow, skin friction becomes zero before the rear stagnation point. As expected, the skin friction increases with the solid volume fraction of TiO\({}_{2}\) nanoparticles due to the higher density of nanoparticles (Table 2). The small values of Biot number generate small temperature gradients within the cylinder, and the temperature gradients increase with an increase in  
Figure 6: Variation of dimensionless temperature with \(\Delta\) and \(\lambda\) for assisting flow of micropolar fluid with (**a**) Al\({}_{2}\)O\({}_{3}\) and (**b**) TiO\({}_{2}\) nanoparticles. Figure 7: Variation of dimensionless temperature with \(\Delta\) and \(\lambda\) for opposing flow of micropolar fluid with (**a**) Al\({}_{2}\)O\({}_{3}\) and (**b**) TiO\({}_{2}\) nanoparticles. The variation of Nusselt number along the cylinder surface with mixed convection and material parameters is illustrated in Figure 11a for assisting flow and in Figure 11b for opposing the flow of micropolar-Al\({}_{2}\)O\({}_{3}\) nanofluids. In both cases, the local Nusselt number is found to be maximum at the front stagnation point and decreases along the surface towards the rear stagnation point. In case of assisting flow, the local Nusselt numbers are found to be higher than the opposing flow. In both cases, the local Nusselt number decreases with increasing material parameters. However, the mixed convection parameters have opposite effect on both flows, see Figure 11a,b. The variation of local Nusselt number along the surface with solid volume fraction of nanoparticles and Biot number is demonstrated in Figure 12a for assisting flow and in Figure 12b for opposing flow of micropolar-TiO\({}_{2}\) nanofluids. Again, the maximum local Nusselt number is found to be at the front stagnation point decreases along the surface. In both cases, the local Nusselt number increases with an increase in the solid volume fraction of TiO\({}_{2}\) nanoparticles and Biot number.