## 3 Solution of Equations by a Galerkin Method
### Effect of the Radial Confinement \(\alpha_{ch}\)  
Here, it is interesting to see the effect of different radial confinement values \(\alpha_{ch}\) on the leaching-tubing-like system behavior. According to Equation (47), decreasing \(\alpha_{ch}\) corresponds to an increase in radial confinement and to a related increase in the dimensionless annular flow velocity \(u_{o}\). The leaching-tubing-like systems behavior with the variation of \(\alpha_{ch}\) from 1.10 to 20 is summarized in Table 2, based on \(r_{ann}\) = 0.50, and \(\varepsilon\) = 1124.9 (\(L\) = 200 m) and the remaining parameters shown in Table 1. Theoretical results of both \(u_{cr}\) and \(\omega_{cr}\) are shown in Figure 6. \(u_{cr}\) is a dimensionless critical instability flow velocity and indicates the onset of instability of the internal pipe. For example, Figure 7 shows the vibration and stability of the system with \(\alpha_{ch}\) = 1.20, for the first three modes. From Figure 7, we can see that when \(u_{cr}\) = 4.34 the first predicted instability of the system occurs, and the associated frequency \(\omega_{cr}\) = 0, which indicates the onset of divergence. Naturally, these results presented here are related to this particular parameters selection, and only indicate what the leaching-tubing-like systems behavior could be. As shown in Table 2, when \(\alpha_{ch}\approx\) 20, i.e., the leaching-tubing-like system is effectively unconfined, flutter instability occurs in the second mode. For 1.35 \(\leq\alpha_{ch}\) < 20, the decrease in \(\alpha_{ch}\) results in flutter instability in which the modes are sequentially lowered. However, for 1.32 \(\leq\alpha_{ch}\) < 1.35, flutter instability occurs in the third mode. A further decrease of \(\alpha_{ch}\) below \(\alpha_{ch}\) < 1.32 can still result in flutter instability in which the modes are sequentially lowered, until first-mode divergence (i.e., a static instability) arises for 1.10 \(\leq\alpha_{ch}\) < 1.27. The vanishing of Re(\(\omega\)) indicates the beginning of divergence, as clearly seen in Figure 6.  
With regard to the dimensionless critical flow velocity \(u_{cr}\), and with the very confined case (\(\alpha_{ch}\) = 1.10) as a reference, the increase of \(\alpha_{ch}\) leads initially to the rapid increase of \(u_{cr}\). This is followed by a steep and significant decrease of \(u_{cr}\) as \(\alpha_{ch}\) is increased continuously. Moreover, when \(\alpha_{ch}\) > 1.48, \(u_{cr}\) is less than 1.00. However, with a further increase of \(\alpha_{ch}\) above \(\alpha_{ch}\) > 2.00 approximately, \(u_{cr}\) remains almost unaffected. As mentioned above, when 1.10 \(\leq\alpha_{ch}\) < 1.27, a first-mode divergence arises, so \(\omega_{cr}\) is equal to 0. An increase of \(\alpha_{ch}\) to 1.27 \(\leq\alpha_{ch}\) < 1.35 causes a significant increase of \(\omega_{cr}\). However, when \(\alpha_{ch}\) > 1.35, decreased confinement causes a very steep reduction in \(\omega_{cr}\). When 1.35 \(\leq\alpha_{ch}\) < 4.46, \(\omega_{cr}\) remains almost unaffected with the increase of \(\alpha_{ch}\). When 4.46 \(\leq\alpha_{ch}\) < 20, the system instability state changes from first-order flutter to second-order flutter, \(\omega_{cr}\) steeply increases and remains almost unaffected with the increase of \(\alpha_{ch}\). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{Dimensional} & \multirow{2}{*}{\(D_{i}\) (m)} & \(D_{o}\) (m) & \(D_{ch}\) (m) & \(L\) (m) & \(L_{s}\) (m) & \(EI\) (N\(\cdot\)m\({}^{2}\)) & \(M_{p}\) (kg/m) & \\ \cline{2-9}  & 0.159 & 0.1778 & 0.298 & 1283 & 1085 & 3.47 \(\times\) 10\({}^{6}\) & 38.7 & \\ \hline \multirow{2}{*}{Dimensionless parameters} & \(\alpha\) & \(\alpha_{ch}\) & \(\varepsilon\) & \(\beta_{i}\) & \(\beta_{o}\) & \(h\) & \(\gamma\) & \(r_{ann}\) \\ \cline{2-9}  & 0.897 & 1.676 & 7216 & 0.239 & 0.297 & 1.479 & 2.019 \(\times\) 10\({}^{5}\) & 0.85 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Sample properties of real-life leaching tubings and associated dimensionless parameters [1].