## 1 Introduction
During the last decade, many other authors compared the performance of the different single and two-phase models with constant or temperature-dependent properties in the context of nanofluid flows [16; 17; 18; 19; 20; 21]. A detailed state-of-the-art review has been besides recently proposed by [PERSON] and [PERSON] [10]. [PERSON] et al. [17] compared the predictions of single and two-phase models (discrete phase model) with constant or temperature-dependent properties for a laminar forced convection flow of \(Al_{2}\)O\({}_{3}\)/water-based nanofluids. They concluded that models with temperature-dependent properties lead to higher values of the heat transfer coefficient and Nusselt number, while decreasing the wall shear stress. With variable properties and for a volume fraction of \(Al_{2}\)O\({}_{3}\) nanoparticles equal to 4%, similar results have been found using single- and two-phase models with a maximum difference of 11%. On the contrary, [PERSON] et al. [18] showed that the mixture model performs better than the single-phase model and the Eulerian one. [PERSON] et al. [19] compared three different two-phase models and a single-phase model to the experiments of [PERSON] and [PERSON] [11] for \(Al_{2}\)O\({}_{3}\)/water-based nanofluids. The mixture, Volume of Fluid (VOF) and Eulerian models provided very similar results for the thermal field, while the single-phase model strongly underestimated the heat transfer coefficient. No clear consensus arises then from these former studies on the choice of the appropriate single- or two-phase flow models. Some attempts have also been achieved to investigate the influence of constant or variable thermophysical properties on the performances of single-phase flow models. In that, [PERSON] et al. [16] showed that the model with constant properties tends to underestimate the wall shear stress and overestimate the heat transfer coefficient. [PERSON] et al. [20] found that the single-phase model with constant physical properties provides an acceptable agreement with the experimental data and the temperature-dependent model improves the predictions of the discrete two-phase flow model for low volume fractions in nanoparticles, typically \(\varphi=0.03\%\). On the contrary, at higher particle concentrations \(\varphi=3.5\%\), the two-phase flow performs best. Numerical modeling of laminar convective nanofluid flows even in relatively simple geometries remains very challenging, since the choice of the single- or two-phase flow models appears to be very case dependent. Analytical models have also been developed to investigate the entropy generation in similar configurations. For example, one could cite the recent work of [PERSON] et al. [22], who investigated the entropy generation of \(Al_{2}O_{3}\)-water nanofluid turbulent forced convection in a pipe with constant wall temperature by means of a second law analysis. They showed in particular that the type of inlet conditions greatly influences the mechanisms responsible for entropy generation. Such analysis could be then very helpful to optimize nanofluid flows from an exergetic point of view. The present paper focuses on the convective heat transfer in a cylindrical pipe for laminar flows of \(Al_{2}O_{3}\)/water-based nanofluids. This choice is justified by the large number of former works using this nanofluid in a similar flow configuration (laminar or developing flows in a pipe with constant heat flux) [23,24]. Moreover, such nanofluid is of a particular interest due to its non-corrosive properties and its good thermal conductivity enhancement using very low volume fractions in nanoparticles. For examples, [PERSON] and [PERSON] [25] obtained an enhancement of 13% using only a volume fraction equal to 0.4% and [PERSON] et al. [26] measured an increase of 34% for a nanoparticle diameter equal to 33 nm and a volume fraction of 3%. The reader can refer to the reviews by [PERSON] and [PERSON] [10,27] for more details about the thermal enhancement using nanofluids.