## 3 Results and Discussion  
The effects of the Reynolds number (Re = 250, 500, 1000, 1500, or 1750), Al\({}_{2}\)O\({}_{3}\) nanoparticle volume fraction (\(\phi\) = 0, 0.01, 0.03, 0.05, or 0.1) or Prandtl number (Pr = 0.765, 0.759, 0.750, 0.742, or 0.729), Knudsen number (Kn = 0, 0.01, 0.03, 0.05, 0.07, or 0.1), and aspect ratio (x/D\({}_{\rm h}\) > 200) on the entrance length (L/D\({}_{\rm h}\)), velocity ratio (U/U\({}_{\rm m}\)), skin friction coefficient (C\({}_{\rm f}\)), and Nusselt number (Nu) are presented in this section at the entrance region. The entrance region was decided based on the value of (x/D\({}_{\rm h}\)) that corresponded to the fully developed value (U/U\({}_{\rm m}\)), as defined in Equation (16). Results from within the entrance region were considered in the current study only. The slip flow was modeled based on Equations (12) and (13), and it was dependent on the value of the mean free path value (\(\lambda\)) that is defined in Equation (11). The temperature value in Equation (11) was dependent on the constant value of wall heat flux (q\({}^{\prime\prime}\)) that was provided at the wall boundary condition. Therefore, three different values of q\({}^{\prime\prime}\) = 5, 10, and 50 W/m\({}^{2}\) were studied, and the results are shown in Figures 3-7. The value of U/U\({}_{\rm m}\) as a function of the inverse of the Graetz number (Gz\({}^{-1}\) = x/(D\({}_{\rm h}\) Re Pr)) increased with an increase in the axial distance from a value of 1.0 at Gz\({}^{-1}\) = 0.0 to a value of 1.53 at Gz\({}^{-1}\) = 0.07 for q\({}^{\prime\prime}\) = 5 W/m\({}^{2}\), as shown in Figure 3a. At q\({}^{\prime\prime}\) = 5 or 10 W/m\({}^{2}\), the value of U/U\({}_{\rm m}\) was similar, with minimal differences. This meant that the increase in the fluid temperature in Equation (11) was small enough not to create a major effect on the resulting value of \(\lambda\), which was reflected in the value of slip velocity defined in Equation (12). However, this was not the case for q\({}^{\prime\prime}\) = 50 w/m\({}^{2}\), where the increase in the nanofluid temperature resulted in a noticeable difference in U/U\({}_{\rm m}\) in comparison with the other two values. The highest difference occurred at Kn = 0.1 and was less than 1% when compared with the results at q\({}^{\prime\prime}\) = 5 W/m\({}^{2}\). Figure 4: Velocity profile at two distances from inlet: (**a**) 10D; (**b**) fully developed. Figure 5: Values of (Re C\({}_{\rm f}\)) at different wall flux (q\({}^{\prime\prime}\)) value functions of (**a**) \(\times\)(D\({}_{\rm h}\) Re Pr); (**b**) Kn. Figure 6: Nanofluid temperatures at different wall flux (q\({}^{\prime\prime}\)) value functions of aspect ratio (x/D\({}_{\rm h}\)): (**a**) nanofluid bulk temperature (T\({}_{\rm m}\)); (**b**) pipe wall temperature (T\({}_{\rm w}\)). Figure 3b shows that the slip wall boundary represented by \(\mathrm{Kn}\) affected the fully developed value of \(\mathrm{U/U_{m}}\) by lowering its value from 2.0 at \(\mathrm{Kn}\) = 0.0 (no-slip wall) to the value of 1.55 at \(\mathrm{Kn}\) = 0.1 (slip wall). The other values were found to be located in between these values in a nonlinear pattern due to the conservation of mass. Consequently, a constant average velocity was maintained because of the constant thermophysical property assumption. The combination of the continuity and the momentum equations caused the non-linearity in the \(\mathrm{U/U_{m}}\) vs. \(\mathrm{Kn}\) curve. As the slip velocity value increased due to the increase in the \(\mathrm{Kn}\) value, the fluid velocity value at the wall increased and became higher than zero. Therefore, the maximum velocity value at the pipe centerline would decrease, resulting in a lowered velocity ratio value, as shown in Figure 4.  
At an entrance length of 10 pipe diameters (10D), the differences in the velocity profile were minimal for all investigated wall heat fluxes due to the relatively low increase in the nanofluid temperature, as shown in Figure 4a.