## 4 Conclusions  
In this work, we used the CFD method to study the flow of slurries described by a Bingham model subjected to a temperature difference between the channel boundary wall and the inlet incoming fluid. Since the flow is accompanied by the heat transfer, its flow characteristics, including the flow structure, are obviously different from those without a temperature difference. As a conclusion, we obtained the following important points:  
* As the fluid cools, the viscosity increases, and the inlet-outlet pressure drop increases. And no matter the shapes of the channels, as the length of the channel or the temperature difference increases, or as the flow rate decreases, the heat transfer increases, which is represented by a narrower maximum temperature zone at the channel outlet.
* In the straight channels, the heat transfer is merely the part because of the thermal conductivity, whereas in the curved channels, the heat transfer also has the component that the fluid velocity carries. As a result, at the beginning of the channel bend, the temperature biases towards the inner wall, since at this cross-section, the central velocity points towards the inner wall. In contrast, in the sections on the second half of the semicircle of the bend, the temperature biases toward the outer wall, which again comforts the direction of the fluid velocity.
* Temperature influences the flow structure in the stream-wise direction in this way: in the more cooled fluid, the maximum velocity zone is wider than that in the less cooled fluid because of the larger viscosity; the flow turns more sharply near the bend; the overlap of the streamlines occurs at the earlier position within the bend.
* The temperature influences the flow structure in the transverse direction in this way: the rotational flow in the more cooled fluid is much stronger, particularly near the edge of the channel, which is represented by the larger vorticity magnitude and the stronger strength of the vortex flow and the denser streamlines. Furthermore, when the fluid is in a higher temperature or has a smaller viscosity, the maximum and the minimum vortex cores are easily concentrated at the same side of the cross section. In contrast, under a lower temperature, the maximum and the minimum vorticity cores are separated to opposite sides. In summary, we attribute these effects to the wider unyielded central plug in the lower temperature and the larger characteristic length of the flow structures with a smaller viscosity.
* The pressure loss coefficient is found to decrease as the flow rate increases, and the energy dissipation rate due to the viscosity follows the same trend, implying that the pressure loss mainly results from the viscous energy dissipation in the laminar regime. This converse relation is against the previous studies because of the yield stress of our fluid and the not fully developed flow in the strongly curved bend, in which the vorticity weighs more on the inner-wall side and is hindered by the outer-wall pointed velocities. In this case, as the flow rate increases, the vorticity magnitude increases by a smaller magnitude compared to that in the fully developed flow, where the vorticity weights more on the outer-wall side.  
For this research article, [PERSON] and [PERSON] were in charge of performing simulations. [PERSON] was in charge of writing the paper. [PERSON] was in charge of advising. [PERSON] was in charge of the testing and verification of the UDF code used in Fluent. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Natural Science Foundation of China (Grant Nos. 12102456, 52078477), and the Fundamental Research Funds for the Central Universities (Grant No. 202 IQN1101), and the National Key Research and Development Program (2022 YFC2904102).  
The study was approved by the China University of Mining and Technology (Xuzhou).  
The authors declare no conflict of interest.