## 4 Results and Discussion
The influence of Brownian force on the thermal conductivity enhancement is strongly debated in the scientific community, some authors assume that it plays a key role [43; 44], while others ignore its effect [42]. Note that the same phenomena are observed for the two other values of the Reynolds number. Figure 8: Axial variations of the heat transfer coefficient ratio \(h_{nf}/h_{bf}\) for \(d_{np}=42\) nm, \(Re=1600\) and three volume fractions in \(Al_{2}\)O\({}_{3}\) nanoparticles: \(\varphi=0.006\), 0.01 and 0.016. Figure 10 displays the corresponding streamlines colored by the axial velocity component \(w\) at four cross-sections along the pipe. Due to the increased temperature at the wall, a secondary flow is observed. It consists of a pair of symmetrical counter-rotating vortices with respect to the tube axis. These vortices are induced by buoyancy forces: an upward flow restricted in a thin layer along the wall rises up a warm fluid and a downward flow along the tube axis drops a cool fluid [45; 46]. For pure water (\(\varphi=0\)), Figure 10a shows clearly that the buoyancy force already appears at \(z=0.2\) m and induces the secondary flow. The contours of the axial velocity component are axisymmetric at this axial position for all nanoparticle concentrations and the recirculation cells are symmetric. This indicates that the velocity is not yet affected by the buoyancy force, which is due to the fact that, at this location, the circumferential temperature gradients are very small. Figure 9: Maps of temperature \(T\) at four axial positions: \(z=0.2\) m (**a**,**e**,**i**,**m**); \(z=0.4\) m (**b**,**f**,**j**,**n**); \(z=0.6\) m (**c**,**g**,**k**,**o**); and \(z=0.8\) m (**d**,**h**,**l**,**p**). Results obtained for \(d_{np}=42\) nm, \(Re=1600\) and four volume fractions of \(Al_{2}O_{3}\) nanoparticles: \(\varphi=0\) (**a**–**d**); \(0.006\) (**e**–**h**); \(0.01\) (**i**–**l**); and \(0.016\) (**m**–**p**). When the fluid moves further downstream, the recirculations are slightly shifted and moves across the median plane, above the plane at \(z=0.4\) m and just below the plane at \(z=0.8\) m, for all values of \(\varphi\). The maximum value of \(w\) is also slightly shifted downward below the horizontal tube axis when moving to the pipe outlet. This shift results from the important increase in the intensity of the secondary flow. This loss of axisymmetry is due to both the boundary layer development and the increasing influence of the buoyancy force, which becomes more pronounced along the tube, increasing then the strength of the secondary flow. At \(z=0.8\) m, the axial velocity contours exhibit an ellipsoid-shaped form for \(\varphi=0\). At the same time, the circular streamlines in the lower half of the tube indicate a weaker secondary flow, while the curved ones in the upper half indicate that a hot fluid  
Figure 10: Streamline patterns colored by the axial velocity component \(w\) at four axial positions: \(z=0.2\) m (**a**,**e**,**i**,**m**); \(z=0.4\) m (**b**,**f**,**j**,**n**); \(z=0.6\) m (**c**,**g**,**k**,**o**); and \(z=0.8\) m (**d**,**h**,**l**,**p**). Results obtained for \(d_{np}=42\) nm, \(Re=1600\) and four concentrations of \(Al_{2}O_{3}\) nanoparticles: \(\varphi=0\) (**a**–**d**); \(0.006\) (**e**–**h**); \(0.01\) (**i**–**l**); and \(0.016\) (**m**–**p**). is confined and accompanied by a more intense secondary flow. At \(z=0.8\) m, the form of the axial velocity contours change from an ellipsoid-shaped pattern for \(\varphi=0\) to a rather more circular form for \(\varphi=0.016\) as illustrated in Figure 10d,p, respectively. For \(\varphi=0.016\), the shape of the velocity contours remains practically unchanged indicating a fully developed region from z = 0.4 m. This result agrees well with the previous observations confirming that the nanoparticles suppress the buoyancy force induced secondary flow, stabilizes the flow with a strong homogeneity of the fluid temperature within the tube.