## 3 Numerical Simulation
### The Influence of Inlet and Outlet Conditions and Structural Parameters  
Due to the research background being the flow distribution in a compact structure, the units of the structure parameters and inlet velocity were given in millimeters. The numerical simulation study of a single variable was carried out for the six parameters mentioned above, \(V\), \(\Delta P\), \(D\), \(d\), \(S\), and \(L\), to analyze and determine the effect of changes in each parameter on \(\Phi\) and \(\Gamma\). The initial parameters were set as described in Table 1: \(V=1\) mm/s, \(\Delta P=0.1\) MPa, \(D=3\) mm, \(d=1\) mm, \(S=2\) mm, and \(L=4\) mm. The parametric values and conditions were based on experiments. At the beginning of the numerical simulation, the high-viscosity metal slurry fluids completely filled the entire structure. The dimensions of the structure were given in millimeters, and the velocity of flow was slow and measured in millimeters. In the numerical simulation study of the parameters listed in Table 1, Figure 4 presents the results of test cases 1 and 2, while test cases 3-6 results are shown in Figure 5.  
As can be seen from Figure 4, keeping other parameters constant and only changing the inlet velocity (\(V\)) resulted in the non-uniformity flow coefficient (\(\Phi\)) slowly decreasing as \(V\) increased, indicating that the distribution of the flow in each channel tended to be uniform as \(V\) increased. Although \(\Phi\) changed with \(V\), the maximum difference was only \(\Delta\Phi=0.006\). Similarly, the change in inlet and outlet differential pressure (\(\Delta P\)) had a similar phenomenon as the inlet velocity (\(V\)). The change in inlet velocity (\(V\)) and differential pressure (\(\Delta P\)) had negligible effects on the flow distribution in the compact parallel channels. This was consistent with the conclusion in ref. [16], that is, flow velocity was not the main factor affecting the flow distribution at low flow velocity conditions. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Test Case** & \(V\) **[mm/s]** & \(P_{2}\) **[MPa]** & \(D\) **[mm]** & \(d\) **[mm]** & \(S\) **[mm]** & \(L\) **[mm]** \\ \hline
1 & [1:6:1] & 0.1 & 3 & 1 & 2 & 4 \\
2 & 0 & \(\Delta P\) [0.1:0.6:0.1] & 3 & 1 & 2 & 4 \\
3 & 1 & 0.1 & [2:6:1] & 1 & 2 & 4 \\
4 & 1 & 0.1 & 3 & [0.6:1.4\(\pm\)0.2] & 2 & 4 \\
5 & 1 & 0.1 & 3 & 1 & [2:4:0.5] & 4 \\
6 & 1 & 0.1 & 3 & 1 & 2 & [4:8:1] \\ \hline \end{tabular}
\end{table} 
Table 1: Parameter values. Figure 3: Grid independence verification. Additionally, the difference between the two curves was extremely small, indicating that \(V\) and \(\Delta P\) had similar influences on the results. Therefore, to reduce the calculation and combine with the actual conditions of this study, the subsequent numerical simulation used an inlet velocity \(V=1\) mm/s and outlet pressure \(P_{2}=0.1\) MPa. The influences of the changes in inlet diameter (\(D\)), channel diameter (\(d\)), channel spacing (\(S\)), and channel length (\(L\)) on the flow ratio (\(\Gamma\)) are respectively given in Figure 5. Keeping other parameters constant and changing only \(D\) or \(L\), the value of \(\Gamma\) for the same  
Figure 5: Influence of structural parameters on flow ratio (\(\Gamma\)). (**a**): The effect of changing \(D\) on \(\Gamma\); (**b**): The effect of changing \(d\) on \(\Gamma\); (**c**): The effect of changing \(S\) on \(\Gamma\); (**d**): The effect of changing \(L\) on \(\Gamma\). channel gradually decreased as the value of the parameters increased. This indicated that the flow distribution at each channel tended to be uniform, and the relationships between \(D\) and \(L\) and the variation of \(\Gamma\) showed a negative correlation. The results of transforming \(d\) and \(S\) were the opposite of the above; their relationships with the variation of \(\Gamma\) were positively correlated.