## 3 Results and Discussions
### Pressure Loss and Energy Dissipation Due to the Bend  
In industries, it is important to accurately predict the pressure drop in the pipe bend for optimizing the design of the complex geometries of the pipes. For the fully developed flow in the bend with a large curvature radius, the friction coefficient has a ready-use expression. However, in the bend with a small curvature radius, such as the one in the present article, the flow is not fully developed. Moreover, our fluid is plastic and there is an unyielded plug at the center of the channel. All these factors may induce abnormal phenomenon, and the existing formula of the friction may not be applicable. Therefore, this section will analyze the pressure loss, friction coefficient, and the energy dissipation rate induced by the bend, to compare with the previous studies and to provide comprehensive understandings. The extra pressure loss induced by the bend is the difference between the total pressure drop and that of a straight pipe with a same length, which is defined by [41]:  
\[\Delta P_{loss}=\Delta P_{t}-\frac{dP}{dx}(L_{s}+L_{b}). \tag{6}\]  
\(\Delta P_{t}\) is the total pressure drop of the U-shaped channel, \(\frac{dP}{dx}\) is the pressure gradient of the straight channel with the same length, \(L_{s}\) is the length of the straight part of the U-shaped channel, \(L_{b}\) and is the length of the bend part. Here, we employ a dimensionless pressure loss coefficient \(\xi=\Delta P_{loss}/0.5\rho V^{2}\) to represent the influences of the flow rate and the temperature. The values of \(\xi\) are summerized in Table. 1. When calculating \(\frac{dP}{dx}\), we used the following relation between \(\frac{dP}{dx}\) and \(V\) for the Bingham fluid [45]:  
\[V(r)= \begin{cases}\frac{1}{2}\Big{(}\frac{\Delta P}{2 bL}\Big{)}\big{(}R-R_{p} \big{)}^{2}&r<r_{p}\\ \frac{1}{2}\Big{(}\frac{\Delta P}{2 bL}\Big{)}\big{(}R-R_{p}\big{)}^{2}\Big{[}1 -\Big{(}\frac{r-R_{p}}{R-R_{p}}\Big{)}^{2}\Big{]}&\text{otherwise}\end{cases} . \tag{7}\]  
\(R\) and \(R_{p}=\frac{2 Lr_{p}}{\Delta P}\) are the channel radius and the radius of the unyielded plug, respectively. \(V=\frac{1}{R}\int_{0}^{R}V(r)\) is the average flow velocity. As Table 1 shows, \(\xi\) surprisingly decreases with the flow rate, and increases as the fluid cools more because of the increase in the viscosity. According to the previous studies of laminar flow of non-Newtonian fluids in a bend [32], it is reasonable to suppose that the pressure drop in the bend is almost totally associated to the energy dissipation rate \(\varepsilon\) due to the viscosity. Herein, we took \(\varepsilon\) along  
Figure 13: The vorticity magnitude and the projected streamlines at the five sections of the U-shaped channel. The upper row shows the data at the inlet velocity of 0.5 m/s at the temperature conditions of 283 K on the wall and 300 K of the incoming fluid at the inlet. The lower row shows the results at the inlet velocity of 1 m/s at the temperature conditions of 283 K on the wall and 300 K of the incoming fluid at the inlet. the horizontal diameter of the cross section at the middle of the channel as the average to calculate the total energy dissipation rate in the whole bend part:  
\[\varepsilon=\frac{Vol_{b}}{D}\int_{0}^{D}\eta|\overset{\rightarrow}{\omega}|^{2} dl, \tag{8}\]  
where \(\overset{\rightarrow}{\omega}\) is the vorticity vector and \(Vol_{b}\) is the volume of the whole bend part of the channel. Moreover, we made \(\varepsilon\) dimensionless through dividing it by \(Q\Delta P_{loss}\), in which \(Q\) is the volumetric flow rate. The values of \(\varepsilon/Q\Delta P_{loss}\) in different cases are listed in Table 2, and it shows the same tendency as \(\zeta\), which confirms our supposition that the viscosity mainly dissipates the energy in the bend. Actually, in the early studies, it was widely accepted that the energy loss due to the bend, which was sometimes represented by the friction coefficient \(f_{c}\), increases as the Dean number \(De\) that is proportional to the fluid velocity.