## 3 Results and Discussion
The value decreased at different rates based on the applied q\({}^{\prime\prime}\) until it reached the fully developed values of 2.64, 2.64, and 2.53 for q\({}^{\prime\prime}\) = 5, 10, and 50 W/m\({}^{2}\), respectively, at Kn = 0.1. This variation in Nu\({}_{\rm x}\) values was due to the variation in temperature jump values at the wall, as defined in Equation (13), as a result of the variation in the \(\lambda\), Kn, and T\({}_{\rm w}\) values. However, the variation in the Nu\({}_{\rm x}\) of Nu was less than 3%, and it was considered valid to adopt the current approach of defining Kn to simulate the slip flow or temperature jump at the wall boundary condition. The slip flow at the wall was dominated by Kn, which changed the value of slip velocity, as defined in Equation (12) and shown in Figure 8. At the no-slip condition (Kn = 0), the U/U\({}_{\rm m}\) value reached the well-known maximum value of two for a laminar flow inside a circular pipe at the fully developed flow point. Similar trends were observed for the slip flow at which the U/U\({}_{\rm m}\) ratio increased from a value of one to a maximum value at the fully developed flow point. As the Kn value increased, the maximum value of U/U\({}_{\rm m}\) was found to decrease, as shown in Figure 8a. The drop in the U/U\({}_{\rm m}\) ratio reached a minimum value of 1.55 at Kn = 0.1 for all values of Re and Pr. Therefore, the effects of Re or Pr on the U/U\({}_{\rm m}\) ratio at a constant Kn value were negligible, as shown in Figure 8b,c. Figure 9 shows that the slip velocity value increased due to the increase in the Kn value, which caused the nanofluid velocity value at the wall to increase and become higher than 0 m/s. Therefore, the maximum velocity value at the pipe centerline of two would decrease to maintain constant mass flow rates in accordance with the conservation of mass law which resulted in lowering the U/U\({}_{\rm m}\) ratio. At Kn = 0.01 and 10 diameters from the inlet, for instance, the value of the velocity at the wall was equal to 0.01 m/s and increased to be equal to 0.39 m/s at Kn = 0.1, as shown in Figure 9a. The opposite behavior was observed at the centerline velocity, at which the velocity reduced from 0.92 m/s at Kn = 0.01 to 0.77 m/s at Kn = 0.1. The general form of the velocity curves was not parabolic and confirmed that the fully developed flow point is way beyond 10 diameters from the pipe inlet. Figure 9b shows parabolic nanofluid velocity curves resulting from the state of fully developed flow. When compared with the 10 diameter results, the values of the velocity at the wall were reduced to 0.075 m/s and 0.315 m/s at Kn = 0.01 and Kn = 0.1, respectively. However, the centerline velocity values were increased to 1.263 m/s and 1.019 m/s at Kn = 0.01 and Kn = 0.1, respectively. As a result, the U/U\({}_{\rm m}\) ratio increased as the nanofluid traveled downstream from the inlet, and the maximum value that could be reached decreased with the increase in the Kn value. It is well-known that the hydraulic entrance length (L/D\({}_{\text{h}}\)) depends on Re proportional manners for pipe flow at no-slip conditions (Kn = 0). The value of (L/D\({}_{\text{h}}\)), based on current CFD results, was compared with the well-known equation of [PERSON] and London [40] and shown in Figure 10a. The results showed the compliance of CFD results with the equation with a high degree of accuracy for all investigated volume fractions of Al\({}_{2}\)O\({}_{3}\) nanoparticles. Figure 8: Values of (U/U\({}_{\text{m}}\)) at different values of (**a**) Kn; (**b**) Re; and (**c**) Pr. Figure 9: Axial velocity profile distribution function of pipe radial distance (r) at two distances from inlet: (**a**) 10D; (**b**) fully developed.