## 3 Results and Discussion
However, the nanofluid temperature began to increase as the fluid traveled along the pipe axis toward the fully developed point due to the heat transferred from the pipe wall. The increase in the temperature is proportional to the value of the wall heat flux. As the \(\mathrm{q^{\prime\prime}}\) value increased, the \(\lambda\) value increased and resulted in a higher slip velocity. Consequently, the nanofluid axial velocity would decrease to fulfill the constant mass flow rate restrictions in accordance with the conservation of mass law, as shown in Figure 4b. At constant properties, the skin friction coefficient (\(\mathrm{C_{f}}\)) is defined in Equation (19) and is dependent on the variation in wall shear stress, which is correlated to the viscosity, velocity gradient at the wall, and \(\mathrm{Kn}\) of the slip wall. As shown in Figure 5, the highest value of (\(\mathrm{ReC_{f}}\)) corresponded to the flow with no-slip conditions (\(\mathrm{Kn}\) = 0.0), which resulted in the highest velocity gradient at the wall. As the value of \(\mathrm{Kn}\) increased, the slip velocity value increased, resulting in an increased wall velocity and a reduced velocity gradient at the wall. Consequently, this lowered the shear stress value and the skin friction coefficient (\(\mathrm{C_{f}}\)) as shown in Figure 5a. The highest value was \(\mathrm{C_{f}}\) = 16 at \(\mathrm{Kn}\) = 0.0, which was decreased to \(\mathrm{C_{f}}\) = 8.9 at \(\mathrm{Kn}\) = 0.1, as shown in Figure 5b. However, the effect of increasing the value of \(\mathrm{q^{\prime\prime}}\) from 5 \(\mathrm{W/m^{2}}\) to 50 \(\mathrm{W/m^{2}}\) was significant and resulted in reducing the value of \(\mathrm{C_{f}}\) by 4%. This is due to the increase in the \(\lambda\) value, which resulted in an increased slip effect at the wall and a reduced value of shear stress, which was more than expected. The 4% reduction at \(\mathrm{Kn}\) = 0.1 remains an acceptable variation, and the results from the current simulations remain valid at the entrance region of the circular pipe. \[C_{f}=\frac{\tau_{up}}{\frac{1}{\frac{\gamma}{\rho}\;U^{2}}} \tag{19}\]  
Figure 7: Values of (\(\mathrm{Nu}\)) at different wall flux (\(\mathrm{q^{\prime\prime}}\)) value functions of (\(\mathrm{a}\)) \(\mathrm{x/(D_{h}\;RePr)}\); (\(\mathrm{b}\)) \(\mathrm{Kn}\). The estimation of Nu based on the current CFD simulation was performed by collecting the (T\({}_{\rm w}\)) value from the CFD simulation and calculated nanofluid bulk temperature (T\({}_{\rm m}\)) values based on Equation (15). The value of T\({}_{\rm m}\) depends on the wall heat flux and nanofluid properties that could be represented by Re and Pr, at which it changed linearly with the axial distance (\(\times\)/D\({}_{\rm h}\)) traveled along the pipe by the nanofluid, as shown in Figure 6a. The maximum increases in T\({}_{\rm m}\) at the simulated entrance region, in comparison with the inlet temperature, were 2.8, 5.4, and 23.3 K for q\({}^{\prime\prime}\) = 5, 10, and 50 W/m\({}^{2}\) and occurred at x/D\({}_{\rm h}\) = 92.6, 89.8, and 77.6, respectively. These increased T\({}_{\rm m}\) values resulted in an increased \(\lambda\) value, according to Equation (11), and caused an increase in Kn values based on Equation (10). Therefore, the increase in \(\lambda\) or Kn affected the estimation, by CFD, of the required values of T\({}_{\rm w}\) needed to maintain a constant value of q\({}^{\prime\prime}\) = 5, 10, or 50 W/m\({}^{2}\). The maximum nonlinear increases in T\({}_{\rm w}\) values, in comparison with the inlet temperature, were 4.4, 8.8, and 41.8 K for q\({}^{\prime\prime}\) = 5, 10, and 50 W/m\({}^{2}\) and occurred at x/D\({}_{\rm h}\) = 92.6, 89.8, and 77.6, respectively, as shown by Figure 6b. Consequently, the value of Nu defined in Equation (14) became dependable on the wall heat flux value. The maximum value of Nu\({}_{\rm x}\) at the entrance of the pipe was 4.8 for all simulated q\({}^{\prime\prime}\) values, as shown in Figure 7a.