## 3 Mathematical Analysis  
The model under consideration is nonlinear in nature. For such a type of models, exact solutions are very rare and very difficult to calculate. In such a situation, numerical computation is reliable. Thus, the Runge-Kutta scheme Runge and Kutta (1963) is adopted for particular flow model. In order to apply the aforesaid scheme, firstly the flow model is reduced into a coupled first order Initial Value Problem (IVP). For the said purpose, the following key transformations are used:  
\[y_{1}=F,\ y_{2}=F^{\prime},\ y_{3}=F^{\prime\prime},\ y_{4}=\beta,\ y_{5}= \beta^{\prime}. \tag{16}\]  
To initiate the scheme, firstly transform the model in the following manner:  
\[F^{\prime\prime}=-  
aforementioned physical parameters in the flow characteristics. Three different cases of the nanofluids flow is discussed:  
1. When the wedge and colloidal suspension move in the opposite direction. The parameter \(\gamma<0\) elucidates such flow. 2. When the wedge and colloidal suspension move in the same sense. This type of flow occurs for \(\gamma>0\). 3. The static wedge case is considered for \(\gamma=0\). Further, the numerical values obtained for shear stress and compared with literature which confirms the reliability of the study. The influences of pressure gradient parameter called Hartree pressure plays a vibrant role on the velocity distribution of the nanofluids. Figure 2 shows these effects for assisting, opposing, and static wedge condition. It is examined that, by increasing the pressure, the nanofluids particles move promptly in the region \(0\leq\eta\leq 3\). Near the surface of the wedge, the particles momentum drops due to the friction factor between the surface and the nanofluids particles. The friction between the rest of the layer's declines, which allows the particles move freely. Therefore, the momentum of the particles increases. For the static wedge condition, the velocity slowly arises in comparison with assisting and opposing flow conditions. The opposing motion of the wedge reduces the nanofluids velocity and the movement of the wedge in the direction of the flow increases the velocity field \(F^{\prime}(\eta)\). The changes in the velocity are almost similar for both kinds of the nanofluids. Varying the flow parameters significantly alters the temperature distribution. These parameters are the radiation parameter, Eckert number fraction factor of the nanoparticles, and \(\lambda\). These effects that are portrayed in Figures 3-6 for the nanofluids under consideration. It is examined that the temperature enhances for the flow of water composed by the cobalt and Mn-Zn and drops for ethylene glycol based hosting fluid. These temperature effects are due to varying Hartree pressure and are displayed in Figure 3. For the opposing wedge condition, the nanofluids velocity drops, which is due to the particles slowly becoming colloid and alternately the temperature drops. On the other hand, for the assisting case, the collision between the particles increases due to the alike motion of the nanofluids and the wedge, which promptly enhances the temperature. These effects of the temperature \(\beta(\eta)\), as highlighted in Figure 3 for feasible Hartree pressure gradient parameter \(\lambda\). It is prominent fact that the radiation parameter significantly enhances the temperature and hence it cannot be ignored. Figure 4 portrays the alterations in the temperature distribution \(\beta(\eta)\) for increasing the radiation parameter. It is obvious that, for stronger thermal radiation, the temperature field significantly enhances. When the nanofluids and the wedge move in alike direction, then temperature the distribution becomes rapid in comparison with the static and opposing flow conditions, respectively. The dominating behavior of \(\beta(\eta)\) is noted for ethylene glycol based nanofluids and for water based nanoliquid; these variations are quite slow.