## 3 Results and Discussions
### Inverse Thermal Flux from the Wall to the Fluid  
In some real circumstances, the slurry's temperature is lower than the environment due to the hot weather, and as a result, the thermal flux flows from the pipe wall to the inner slurry. We simulated such a case, in which the temperature on the pipe wall is constantly 300 K whereas the slurry coming from the inlet is 283 K. The inlet velocity is 0.5 m/s. Figure 15 shows the temperature distributions at the five slices in the bend and the outlet. Under such a thermal boundary condition, the heat flux is from the wall to the fluid. Therefore, opposite to the cases in the cooler pipe, the lower temperature is biased to the inner wall at the beginning of the bend, while it is biased to the outer wall at the end of the bend. The velocity magnitudes and the vectors at the five cross sections within the bend are shown in Figure 16. Compared with the results in the case of the inverse temperature flux at the same flow rate (Figure 12, upper row), because of the lower temperature and thus the higher viscosity of the fluid in this case, the central velocity becomes smaller. The vorticity distributions (Figure 17) are not quite different from the case with the inverse thermal flux (Figure 13, upper row). There is only a minor deviation that the streamlines are denser near the channel edge in in this case with 300/283 K, especially at the 45\({}^{\circ}\) section. The same phenomenon is also observed, comparing the cases of 273/300 K and 300/300 K at the \begin{table}
\begin{tabular}{c c c} \hline \(\varepsilon\)/Q\(\Delta P_{loss}\) & 0.5 m/s & 1 m/s \\ \hline
300/300 (K) & 1.1080 & 1.0160 \\
283/300 (K) & 1.1187 & 1.0235 \\
273/300 (k) & 1.1254 & 1.0281 \\ \hline \end{tabular}
\end{table} 
Table 2: Energy dissipation rate in different cases. Figure 14: Correlated friction factor \(f_{c}/f_{p}\) as a function of Dean number \(De\) at different temperature conditions (**left**). Correlated friction factor \(f_{c}/f_{p}\) as a function of Nusselt number \(Nu\) at different temperature conditions (**right**). same flow rates (Figure 8). This is mainly due to the almost unyielded plug flow at the center of the cross section. The fluid's temperature and the viscosities are lower in the case of 300/283 K compared with the case of 283/300 K. Thus, the plug is wider and the fluid within the plug is more rigid, resulting in a greater difficulty of creating a rotational flow inside, and therefore the most rotational flow is outside the plug and near the edge. Comparing both cases of 283/300 K and 300/283 K with the case of 300/300 K (Figure 7, upper row), it is found that the deflection of the minimum or the maximum vorticity core is switched to different sides from the case with zero thermal flux to the cases with non-zero thermal flux, regardless of the direction of the heat transfer. Under a temperature difference, at the 0\({}^{\circ}\) section, the minimum vorticity is always biased to the outer-wall side, whereas it is closer to the inner-wall side with no thermal flux. The same trend is observed at the 180\({}^{\circ}\) section. In conclusion, at both 0\({}^{\circ}\) and 180\({}^{\circ}\) sections, the maximum and minimum vorticity zones are always concentrated on the same side under zero thermal flux, or at the smaller fluid viscosity. On the other hand, with the temperature difference boundary conditions or the larger fluid viscosity, the maximum and minimum vorticity zones are less concentrated and thus distributed on the opposite sides. To understand this, when the fluid's viscosity is small, its flow structures, including the vorticity maximum or the  
Figure 16: The velocity magnitude and the velocity vectors at the five sections of the U-shaped channel. The inlet velocity of 0.5 m/s. The temperature boundary condition is: 300 K on the wall and 283 K of the incoming fluid at the inlet. Figure 17: The vorticity magnitude and the projected streamlines at the five sections of the U-shaped channel. The inlet velocity of 0.5 m/s.