## 4 Results and Discussion
The same behavior is observed for all volume fractions of nanoparticles indicating that the latter has no remarkable influence on the hydrodynamic field. Nevertheless, the intensity of the secondary flow decreases when increasing \(\varphi\) for all axial positions especially at \(z=0.8\) m as shown in Figure 11. This is consistent with previous observations [47; 48; 49]. The wall layer vorticity also decreases with increased values of \(\varphi\). The weak influence of the nanoparticle concentration on the velocity and temperature fields results from the ability of nanoparticles to homogenize the fluid temperature and therefore impeding buoyancy forces. Only few studies considered the influence of the nanoparticles on the development of the secondary flow and the homogenization of the temperature field. When \(\varphi\) is increased, the molecular diffusion increases accompanied with an increase in the thermal conductivity and a reduction in the specific heat capacity, as proposed by [48]. [PERSON] et al. [49] invoke the role of the Brownian diffusion. Influence of the Nanoparticle Diameter for Al\({}_{2}\)O\({}_{3}\) and Cu/Water-Based Nanofluids  
In real thermal engineering applications, increasing the nanoparticle diameter leads to higher agglomeration effects resulting in the sedimentation of the agglomerates. Such particle-particle interactions are not taken into account in the present model. Only the Brownian motion and the ratio between gravity and buoyancy forces are modeled here. The nanoparticle diameter \(d_{np}\) has then no remarkable effect on the heat transfer coefficients as it will be shown in the following. More interestingly however, the influence of \(d_{np}\) on the axial distributions of the nanoparticle concentration \(\varphi\) is quite remarkable. Figure 12 displays the axial profiles of \(\varphi\) at three radial locations \(r/R=0.98\) (top wall), \(r/R=0\) (axis) and \(r/R=-0.98\) (bottom wall) for two types of nanofluid (copper and alumina water-based), three nanoparticle diameters \(d_{np}=42\), \(100\) and \(200\) nm and for \(Re=1600\) and \(\varphi=1.6\%\). Firstly, the concentration is rather constant along the center line of the tube (\(r/R=0\)) for the two nanofluids and all the nanoparticle diameters, remaining between \(1.6\%\) and \(1.596\%\). For \(d_{np}=42\) nm, along the top wall of the tube, the nanoparticle concentration decreases by \(1.5\%\) and \(3\%\) for \(Al_{2}\)O\({}_{3}\) and Cu nanoparticles, respectively (Figure 12a). As the nanoparticle diameter grows, the concentration decreases along the top wall due to gravity effects. For example, \(\varphi\) is reduced by \(22.5\%\) and \(43.75\%\) for \(Al_{2}\)O\({}_{3}\) and Cu nanoparticles, respectively, for \(d_{np}=200\) nm. By conservation of the average value of \(\varphi\) at a given cross-section, the concentration of nanoparticles along the bottom wall increases. Because of their higher density, this effect is more noticeable for copper/water-based nanofluids as illustrated in Figure 12c,f,i. The latter shows that \(\varphi\) increases by \(0.75\%\), \(17.5\%\) and \(87.5\%\) for \(d_{np}=42\)  
100 and 200 nm, respectively. The concentration of \(Al_{2}O_{3}\) appears quite constant, with an increase of only 2.5% for \(d_{np}=200\) nm. This behavior can be easily explained since the density of copper is twice the alumina one. The influence of the mean diameter of the \(Al_{2}O_{3}\) nanoparticles on the heat transfer is not shown here, but it can be noticed that increasing \(d_{np}\) does not affect the average heat transfer coefficient. For \(\varphi=0.016\) and \(Re=1600\), \(h_{av}=1796.7\) W/m\({}^{2}\) K and 1795.5 W/m\({}^{2}\) K for \(d_{np}=10\) and 200 nm, respectively. However, the average heat transfer coefficient may be strongly modified by the size of the nanoparticles for copper-water nanofluids, which sediment more with only 12% of the particles still in suspension in the base fluid as shown previously for \(d_{np}=200\) nm (Figure 12g,i).