## 5 Conclusions  
This paper presents a fast and accurate prediction model to calculate the flow distribution in compact parallel channels in order to achieve uniform distribution of high-viscosity slurry fluid flow in compact parallel channels, which can be used in multi-nozzle array direct printing technology to make the flow in each nozzle consistent.  
Through numerical simulation analysis and experiments, the study revealed the flow distribution law of Herschel-Bulkley fluids (high viscosity slurry fluids) in each manifold of the compact parallel channels, indicating the relationship between the non-uniformity flow coefficient \(\Phi\) and parameters of the compact parallel channels. The main conclusions are as follows:  
1. When the Herschel-Bulkley fluid flows slowly in the compact parallel channels, within the parameter range of this study, changes in \(V\) and \(\Delta P\) have negligible effects on the fluid flow distribution in the compact parallel channels, and \(V\) and \(\Delta P\) have similar effects on the results of flow distribution, thus we can choose one inlet velocity (\(V\)) and inlet pressure (\(P\)) to study.
2. The maximum flow occurs in channels near the inlet and at the minimum flow occurs at the end of the cavity region. The flow decreases sequentially and the results are different from those of Newtonian fluids. In order to create uniform flow at each channel, the inlet diameter (\(D\)) and channel length (\(L\)) should be increased and the channel diameter (\(d\)) and channel spacing (\(S\)) should be reduced as much as possible within the specified dimensional parameters of the structure design. The strength of the effect on the uniform flow is: inlet diameter (\(D\)) > channel diameter (\(d\)) > channel length (\(L\)) \(\approx\) channel spacing (\(S\)).
3. The non-uniformity flow coefficient (\(\Phi\)) was defined to analyze the flow distribution in parallel channels, and the area ratio parameter (\(A\)) was defined to uniformly represent changes in the three parameters (\(N\), \(D\), \(d\)). According to Figure 10, \(\Phi\) has a specific functional relationship with \(A\), \(S\), and \(L\), which we can obtain by calculation. The functional relationship helps to design the compact parallel channel structure in order to achieve the specified flow difference.  
The flow distribution theory of fluids in compact parallel channels is complicated and explaining all complex flows theoretically is difficult. The research methods and processes of this paper, which can be used to study the flow of other fluid models in compact parallel channels, can help improve the understanding of flow behaviors of fluids. Additionally, the prediction model provides easy-to-use design guidance in engineering applications. This study can also be used in multi-nozzle 3D printing, additive manufacturing, and so on. The flow distribution of more channels and more materials is a further research direction.  
Conceptualization, Z.W. and Y.C.; Methodology, Z.W.; Investigation, Z.W.; Data Curation, Z.W.; Visualization, Z.W.; Writing--Original Draft, Z.W.; Project administration, S.W. and Y.C.; Writing--Review & Editing, Y.C., S.W., J.Z., Z.Q., Y.L., H.Y. and J.S.; Supervision, C.S. and H.Y.; Resources, J.Z.; Funding acquisition, C.S. and H.Y. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the Opening Project of the National Enterprise Technology Center of Guangxi Bosco Environmental Protection Technology Co., Ltd., Nanning 530007, China (grant no. GXU-BFY-2020-012), the Guangxi Bagui Scholars Project (No. 2019A02), and the Projects of Innovation and Development Multiplication Plan of Guangxi University (grant no. 202201343 and 202201369).  
Not applicable.  
The authors declare no conflict of interest.