## 3 Numerical Simulation
### Prediction of the Influence of \(A\), \(S\), \(L\) on \(\Phi\) Value
#### 3.2.2 Study of the Functional Relationship between \(S\), \(L\), and \(\Phi\)  
Based on the research above, several conclusions may be drawn:  
1. \(\Phi\) and \(A\) have a certain linear relationship while \(S\) and \(L\) are determined.  
2. When changing \(S\) and \(L\), the primary term coefficient (\(a1\)) and constant term (\(b1\)) in the linear relationship vary in a regular manner.  
Numerical simulation methods were used to study the numerical change in \(a1\) and \(b1\) with \(2<S<5\) and \(2<L<10\), as demonstrated in Figure 7. The linear relationship was given by:  
\[\Phi=a1\cdot A+b1 \tag{5}\]  
Figure 7a,b present the trends of \(a1\) and \(b1\), respectively, as L changed when \(S\) remained constant. It can be seen that the results of curve fitting was an exponential function, that is:  
\[y=c\cdot\exp(-x/t)+d \tag{6}\]  
Figure 7c,d show the trends of \(a1\) and \(b1\), respectively, as \(S\) changed when \(L\) remained constant. The relationships were linear, given by:  
\[a1=a_{a}\cdot S+b_{a} \tag{7}\]  
\[b1=a_{b}\cdot S+b_{b} \tag{8}\]  
The variations of \(a1\) and \(b1\) with \(S\) and \(L\) are displayed in Figure 8. The two functional relationships described above were unified, and a significant trend in numerical variation is shown. We know that smaller \(S\) and larger \(L\) will make smaller \(a1\) and larger \(b1\), thus causing uniform flow distribution in the parallel channels.  
Figure 7: Study of the relationship between \(S\), \(L\), and \(a1\), \(b1\). (**a**): \(a1\) changes with respect to \(S\) and \(L\), \(L\) is the abscissa; (**b**): \(b1\) changes with respect to \(S\) and \(L\), \(L\) is the abscissa; (**c**): \(a1\) changes with respect to \(S\) and \(L\), \(S\) is the abscissa; (**d**): \(b1\) changes with respect to \(S\) and \(L\), \(S\) is the abscissa.  
The linear relationship was analyzed and the results are summarized in Figure 9. The relationship between the primary term coefficient (\(a_{a}\)) and L was exponential (Figure 9a), given by:  
\[a_{a}=0.2\cdot\exp(-L/4)+0.06 \tag{9}\]  
The relationship between the primary term coefficient (\(a_{b}\)) and \(L\) was constant (Figure 9c) because the values all fluctuated around \(-0.0051\). When \(-0.0051\) was used instead, the maximum error after calculation of \(\Phi\) was \(0.00105\). This is an acceptable value that has little effect on the calculation of \(\Phi\), so in this research we think:  
\[a_{b}=-0.0051 \tag{10}\]  
Figure 8: The variations of \(a1\) and \(b1\) with \(S\) and \(L\). **(a)**: \(a1\) changes with respect to \(S\) and \(L\); **(b)**: \(b1\) changes with respect to \(S\) and \(L\).  
Figure 9: Analysis of the linear relationship. **(a)**: The relationship of \(a_{a}\) about \(L\); **(b)**: The relationship of \(b_{a}\) about \(L\); **(c)**: The relationship of \(a_{b}\) about \(L\); **(d)**: The relationship of \(b_{b}\) about \(L\).  
The relationships between the constant terms \(b_{a}\) and \(b_{b}\) and \(L\) were exponential (Figure 9b,d), given by:  
\[b_{a}=0.32\cdot\exp(-L/3)-0.01 \tag{11}\]  
\[b_{b}=0.05\cdot\exp(-L/3.2)+0.00123 \tag{12}\]  
Now, substituting each parameter into Equation (5), an approximate solution of \(\Phi\) can be obtained:  
\[\begin{array}{l}\Phi=\{[0.2\cdot\exp(-L/4)+0.06]\cdot S+0.32\cdot\exp(-L/3)- 0.01\}\cdot A\\ -0.0051\cdot S+0.05\cdot\exp(-L/3.2)+0.00123\end{array} \tag{13}\]  
In summary, the Herschel-Bulkley fluid flow distribution in the compact parallel channels had obvious regularity. In practical engineering applications, the complex fluids theory formula is abandoned and the flow distribution law of different fluids in different structures can be studied directly by numerical simulation calculation.