## 3 Mathematical Analysis
The impacts of the Eckert parameter on the temperature behavior \(\beta(\eta)\) for both types of nanoliquids that are elucidated in Figure 5. The temperature \(\beta(\eta)\) arises for opposing condition and then asymptotically vanishes far from the surface. For alike movement of the nanofluids and the wedge, the behavior of the temperature is almost inconsequential. It is a prominent fact that the thermal conductivity of the host liquids (water and ethylene glycol) is not up to the mark. However, mixing the cobalt and ferrite nanoparticles can enhance the thermal conductance of these liquids. The resultant composition then has better thermal enhancement characteristics. For this, the volume fraction factor \(\phi\) plays the role of key ingredient These effects are elaborated in Figure 6 for both nature of the nanofluids. It can be seen that, for static wedge flow, the temperature alterations are more obvious in comparison with the opposing flow. The temperature of the nanofluids shows asymptotic pattern far from the wedge. Figure 4: Alterations in the temperature distribution for (**a**) CoFe\({}_{2}\)O\({}_{4}\) – H\({}_{2}\)O; (**b**) CoFe\({}_{2}\)O\({}_{4}\) – C\({}_{2}\)H\({}_{6}\)O\({}_{2}\); (c) MnZnFe\({}_{2}\)O\({}_{4}\) – H\({}_{2}\)O; (**d**) MnZnFe\({}_{2}\)O\({}_{4}\) – C\({}_{2}\)H\({}_{6}\)O\({}_{2}\) nanofluids by varying \(Rd\). The significance of the nondimensional physical quantities in the shear stresses and the heat transfer rate cannot be ignored. Since water and ethylene glycol taken as a regular liquids. For these liquids, the Prandtl number has a fixed value and is described in [52], respectively. Figures 7-11 elaborated the impacts of aforementioned parameters on the local heat transfer. The variations in the local heat transfer rate by altering the radiation and volume fraction parameter described in Figure 7. The transfer rate is rapid for strong thermal radiation parameter Rd. For Ethylene glycol based nanofluids, drops in the local heat transfer rate are noted. For opposing flow, less amount of heat transfers at the wedge surface in comparison with the static and moving wedge, respectively. For smaller \(\phi\), the heat transfer rate is quite slow. Figure 8 highlights the impacts of fraction factor \(\phi\) on the heat transfer at the wedge. The higher values of \(\phi\) lead to increasing the behavior of the heat transfer. The up turns in the nusselt number are inspected for alike nanofluids flow. For the static wedge condition, the heat transfer slowly varies in comparison with opposing and assisting flow situations. The heat transfer quite rapidly arises for EG based nanofluid and effects of Rd and Hartree pressure elaborated in Figure 9.  
The impacts of Hartree pressure versus Eckert number on the heat transfer rate are organized in Figure 10. The heat transfer reduces for Hartree pressure versus Eckert number Ec. By increasing the Eckert parameter, the heat transfers at the wedge surface arises drops. For EG based nanofluids, heat transfer promptly declines. Figure 11 elaborated on the influences of Hartree pressure versus volume fraction factor \(\phi\). The heat transfer enhances by altering the Hartree pressure and the volume fraction \(\phi\) of the nanoparticles. For the opposing flow situation, heat transfers rapidly at the wedge surface. Further, it is examined that, for smaller \(\lambda\) and \(\phi\), the heat transfer slowly intensifies. Figure 8: _Cont._  
Figure 8: Alterations in the local nusselt number for (**a**) \(\mathrm{CoFe_{2}O_{4}-H_{2}O}\); (**b**) \(\mathrm{CoFe_{2}O_{4}-C_{2}H_{6}O_{2}}\); (**c**) \(\mathrm{MnZnFe_{2}O_{4}-H_{2}O}\); (**d**) \(\mathrm{MnZnFe_{2}O_{4}-C_{2}H_{6}O_{2}}\) nanofluids by varying \(\phi\). Figure 11: Cont.