## 5 Conclusions  
Laminar forced convection flows of water-based nanofluids through a uniformly heated tube were revisited here using direct numerical simulations. The single-phase and mixture models with constant and temperature-dependent properties were compared to the experimental data of [PERSON] and [PERSON] [11] and to the numerical simulations of [PERSON] et al. [19]. The mixture model with temperature-dependent  
Figure 16: Influence of the Reynolds number \(Re\) on the average friction coefficient \(f_{av}\). Results obtained for all values of \(\varphi\), \(d_{np}\) and all types of nanofluids. properties was shown to perform best with a close agreement to the experimental data. The former simulations of [PERSON] et al. [19] using the same model were significantly improved with the use of an appropriate mesh grid. The numerical model was then used confidently and extensively to investigate the influences of the Reynolds number (\(600\leq Re\leq 1600\)), the concentration in nanoparticles (\(\phi\leq 1.6\%\)) and their diameter (\(42\leq d_{np}\leq 200\) nm) on the hydrodynamic and thermal fields. \(Al_{2}O_{3}/\)water based nanofluids have been considered first before evaluating the thermal performances of other nanoparticles such as: \(Cu\), \(C\), \(CuO\), \(TiO_{2}\), and \(SiO_{2}\). For \(Al_{2}O_{3}/\)water based nanofluids, the average heat transfer coefficient increased linearly with the nanoparticle concentration for all Reynolds numbers. At \(Re=1600\), the local heat transfer coefficient increased in average by \(29\%\), \(46\%\) and \(74\%\) for \(\varphi=0.006\), \(0.01\) and \(0.016\), respectively. Increasing the nanoparticle concentration led to a more homogenous temperature field, impeding the hot temperature region observed at the top of the pipe wall for pure water flows. The flow field revealed two recirculation regions for all \((r,\theta)\) planes, only weakly influenced by \(\varphi\). The maximum value of the axial velocity component observed at \((r/R\simeq-0.2,\theta=-90^{\circ})\) was also weakly affected by \(\varphi\). The volume fraction in nanoparticles affected significantly the streamwise vorticity of the two recirculation cells. The flow and temperature fields exhibited a more homogeneous behavior. A particular attention was also paid to the sedimentation of the nanoparticles, which, as expected, increased for large size or high density nanoparticles. Finally, empirical correlations to predict both the Nusselt number and the average friction coefficient have been provided, summarizing all simulations presented here (in the range of \(Pr\geq 1\) for \(Nu\)). Further calculations are now required to extend the present simulations to the turbulent flow regime using large-eddy simulations. Further developments are also planned to improve the numerical model to take into account more complex phenomena like the thermophoresis effect and particle-particle interactions. The authors would like to thank the NSERC chair on industrial energy efficiency funded by Hydro-Quebec, CamnetEnergy and Rio Tinto Alcan established at Universite de Sherbrooke for the period 2014-2019. The financial support of the CREEPUS research center and the HPC resources of the Compute Canada network are also gratefully acknowledged. [PERSON] performed the numerical simulations; [PERSON] and [PERSON] performed the physical analysis and wrote the paper. The authors declare no conflict of interest. The following abbreviations are used in this manuscript: