## 4 Results
### Tube Length
#### 4.2.1 The Influence of Tube Length in Cavity-Single Tube Flow
During the inflation process, both the CIDZ and EIDZ areas increase with \(l_{1}/d\). Additionally, due to flow continuity, the region within the tube is equivalent to EIDZ, leading to a significant increase in the inertial force as \(l_{1}/d\) rises, causing \(\overline{m}_{Tough}\) to decrease with increasing \(l_{1}/d\) and resulting in an increase in inflation time. Building on the analysis above, the mechanism by which an increase in \(l_{1}/d\) influences the inertial forces within the cavity is further examined concerning its impact on the axial loads. Figure 14 illustrates the variations in the relative axial force \(\overline{F_{A}}\) at the A end face, the relative axial force \(\overline{F_{B}}\) at the B end face, and their relative axial force difference \(\Delta\overline{F}\) during the process of increasing \(l_{1}/d\) from 0 to 1.5. In the first oscillation cycle, the peak values of the relative axial forces at both end faces and the relative axial force difference are presented in Table 9.  
When \(l_{1}/d\leq 0.25\) occurs, as \(l_{1}/d\) increases, the duration of the axial force acting within the cavity decreases, while the oscillation peak values of the relative axial loads at both end faces and the relative axial load difference exhibit a slight decrease. When \(0.25<l_{1}/d\) occurs, as \(l_{1}/d\) increases, the duration of the axial force acting within the cavity increases, while the oscillation peak values of the relative axial forces at both end faces increase; however, the relative axial force difference decreases. Overall, although the difference in axial loads at both end faces decreases as \(l_{1}/d\) increases, the underlying reasons differ. At various stages of \(l_{1}/d\), it exerts distinct effects on the axial loads within the cavity. When \(l_{1}/d\leq 0.25\) occurs, the tube structure acts as an aperture in the flow path; as \(l_{1}/d\) increases, the CIDZ range diminishes, resulting in a reduction in the inertial forces within the cavity and a weakening of the axial load oscillations. When \(0.25<l_{1}/d\) occurs, as \(l_{1}/d\) increases, the equivalent CIDZ range within \begin{table}
\begin{tabular}{c c c c c c c} \hline \(l_{1}/d\) & 0 & 0.25 & 0.5 & 0.75 & 1 & 1.5 \\ \hline \(\left(\overline{F_{A}}\right)_{\text{min}}\) & 0.64859 & 0.64909 & 0.63235 & 0.57633 & 0.55635 & 0.53778 \\ \hline \(\left(\overline{F_{A}}\right)_{\text{max}}\) & 0.68552 & 0.69384 & 0.70323 & 0.72499 & 0.73845 & 0.75796 \\ \hline \(\left(\overline{F_{B}}\right)_{\text{min}}\) & 0.64495 & 0.64930 & 0.62707 & 0.57333 & 0.55784 & 0.53236 \\ \hline \(\left(\overline{F_{B}}\right)_{\text{max}}\) & 0.68895 & 0.69707 & 0.70677 & 0.73453 & 0.74467 & 0.76373 \\ \hline \(\left(\Delta\overline{F}\right)_{\text{min}}\) & \(-0.06392\) & \(-0.06343\) & \(-0.05868\) & \(-0.05649\) & \(-0.05463\) & \(-0.04752\) \\ \hline \(\left(\Delta\overline{F}\right)_{\text{max}}\) & 0.01451 & 0.01608 & 0.00970 & 0.00963 & 0.01301 & 0.00756 \\ \hline \end{tabular}
\end{table} 
Table 9: Comparison of parameters for different end face relative axial load variation processes. the tube expands, minimally affecting the CIDZ range within the cavity. This increase in the equivalent CIDZ range within the tube extends the duration of the inertial forces in the cavity, while the essentially unchanged CIDZ range within the cavity has little impact on the oscillation phase difference between \(\overline{F_{A}}\) and \(\overline{F_{B}}\). Consequently, both \(\overline{F_{A}}\) and \(\overline{F_{B}}\) increase as \(l_{1}/d\) increases, while \(\Delta\overline{F}\) decreases as \(l_{1}/d\) increases.