## 2 Modeling and Computational Details
### Slip Flow Treatment
The difference in reported errors between AS0 and AS3 was 1.0% at Kn = 0.0 and reduced to 0.3% at Kn = 0.1 in favor of mesh AS3 and AS4. Therefore, it was clear that relatively fine meshes AS3 and AS4 provided more accurate results in comparison with the coarse mesh AS0. The slight differences between different meshes showed insensitivity toward mesh element sizes. The dependence of U/U\({}_{\mathrm{m}}\) and ReC\({}_{\mathrm{f}}\), mainly, on Kn has enhanced the tendency toward mesh insensitivity. Furthermore, the assumption of constant properties has reduced the dependency on mesh element sizes. The mesh insensitivity was confirmed by the case of Nu, when all mesh sizes produced same results at all Kn values for Re = 1750, as shown in Figure 2c. This meant that the current mesh element sizes were relatively fine enough for the current investigation. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **AS0** & **AS1** & **AS2** & **AS3** & **AS4** \\ \hline No. in X & 250 & 500 & 1000 & 2000 & 3000 \\ No. in Y & 12 & 25 & 50 & 100 & 150 \\ No of Elements & 3000 & 12,500 & 50,000 & 200,000 & 450,000 \\ Cell to wall distance (m) & \(1.042\times 10^{-3}\) & \(5.000\times 10^{-4}\) & \(2.500\times 10^{-4}\) & \(1.250\times 10^{-4}\) & \(8.333\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mesh details utilized in the current simulation. Comparing the entrance length (L/D\({}_{h}\)) values to those of the no-slip conditions of Shah and London [40] produced different value functions of the utilized mesh element sizes. Mesh AS0 resulted in an error of 5.3%, while mesh AS4 produced an error value equal to 0.01% and mesh AS3 resulted in an error of 0.2%. The main advantage of AS3 and AS4 meshes was the short distance between the wall and the adjacent cell center. Therefore, mesh AS3 was adopted to conduct the rest of simulations of the current study due to the relatively shorter simulation time and the acceptable accuracy. The validation of the current study requires that mesh AS3 would result in an acceptable value of error when compared with reliable published results. Figure 2 showed how the error value increased with the increase in Kn value until it reached the maximum value at Kn = 0.1. When compared to the published analytical results of [PERSON] et al. [35], for values of U/U\({}_{m}\), AS3 produced a very close results, with a maximum error of 0.4% at Kn = 0.1. Similarly, when compared to the published analytical results of [PERSON] et al. [36], for values of ReC\({}_{t}\), AS3 produced close results, with a maximum error of 1.2% at Kn = 0.1. Furthermore, when compared to the published analytical results of [PERSON] et al. [36], for values of Nu, AS3 produced results, with a maximum error of 13.6% at Kn = 0.1, which is considered a relatively high error. The relatively high error value could be due to the dependence of Nu on Pr, \(\gamma\), and Kn, as defined in Equation (18). This dependence is defined by the last term at the denominator and increases as the value of Kn increases. For constant properties, the slight difference  
Figure 2: Mesh sensitivity and validation analysis for the main benchmarked parameters, as follows: (**a**) Velocity ratio (U/U\({}_{m}\)), compared with [PERSON] et al. [35]; (**b**) skin friction coefficient (ReC\({}_{t}\)), compared with [PERSON] et al. [36]; (**c**) Nusselt number (Nu), compared with [PERSON] et al. [36]; (**d**) entrance length (L/D\({}_{h}\)), compared with [PERSON] and [PERSON] [40]. between the constant value of Kn and estimated value by the CFD simulation defined in Equations (10) and (11) resulted in a high error value. It is well-known that as the velocity slip increases, due to the increase in Re and/or Kn, the convective heat transfer increases, too. The values of \(\lambda\) and, consequently, Kn depend on the nanofluid bulk temperature (T\({}_{\rm m}\)) value, proportionally. The current study was based on a constant Tm value for all cases, which contributed to this high value.