## 5 Conclusions  
A linear theoretical model has been formulated for a hanging vertical cantilevered pipe which is subjected concurrently to two dependent axial flows. It should be pointed out that after the external flow exits from the outside annular passage, fluid is conveyed upwards as the internal flow in the cantilever pipe. The motion equation of the system is solved by means of a Galerkin method, and eigenfunctions of Euler-Bernoulli beam are used as comparison functions. Theoretical predictions were obtained for a long leaching-tubing-like system with parameters related to the salt cavern energy storage. A theoretical study into the effect of radial confinement (i.e., the degree of radial confinement of the annular passage) has shown that, the system loses stability in flutter and even in divergence regardless of length, as the radial confinement degree and consequently the annular flow velocity is increased. When \(\alpha_{ch}\) is greater than a certain value, \(u_{\sigma}\) decreases as \(\alpha_{ch}\) increases, even to a small value. Considering that the flow velocity in industrial applications is relatively large, the value of \(\alpha_{ch}\) should not be too large. Additionally, only when values of \(\alpha_{ch}\) exceed a certain parameter-dependent threshold does the radial confinement has no appreciable effect on the pipe system. The confinement length \(r_{ann}\) (i.e., confined-flow length), was shown to have a destabilizing or stabilizing effect on the pipe system, depending on value ranges of \(r_{ann}\). The effect is a clear and significant stabilization, especially when the annular flow extends over most portion of the cantilevered pipe. Figure 10: Leaching-tubing-like systems, theoretical variation of \(U_{cr}\) and \(\Omega_{cr}\) with the increase of the pipe system length \(L\). The results are based on the instability of the first prediction. (**a**) Leaching-tubing with \(\alpha_{ch}=1.676\) and \(r_{ann}=0.50\). (**b**) Leaching-tubing with \(\alpha_{ch}=1.676\) and \(r_{ann}=0.85\). Figure 9: Leaching-tubing-like systems, theoretical variation of \(U_{cr}\) and \(\Omega_{cr}\) with the increase of the pipe system length \(L\). The results are based on the instability of the first prediction. (**a**) Leaching-tubing with \(\alpha_{ch}=1.676\) and \(r_{ann}=0.125\). (**b**) Leaching-tubing with \(\alpha_{ch}=1.676\) and \(r_{ann}=0.25\). Furthermore, investigations have been conducted in dimensional terms on the effect of the pipe system length \(L\), for the leaching-tubing-like system. The pipe system length \(L\) was shown to have a stabilizing effect on the pipe system. Increasing the pipe system length \(L\) results in an increase of dimensional critical flow velocity \(U_{cr}\), while it results in a decrease of the associated dimensional frequency \(\Omega_{cr}\). For a long enough system, the frequency \(\Omega_{cr}\) almost vanishes, which indicates a divergence is about to begin. The next step is to conduct experiments to check the validity of the linear theoretical model, based on the device described in [23], and further research results are deferred to another paper. [PERSON] conceived the study, derived formulas, interpreted the results and wrote the paper under the guidance of [PERSON] and [PERSON]. [PERSON] and [PERSON] helped in the numerical calculation. [PERSON] and [PERSON] reviewed the study plan, edited the manuscript, and corrected the grammar mistakes. [PERSON] participated in data analysis. This research was funded by National Natural Science Foundation of China grant number 51874273, 51874274, 51774266 and by National Key Research and Development Program of China grant number 2018 YFC0808401. And The APC was funded by 51874274. The authors would like to be sincerely grateful to [PERSON] from University of Nevada, USA, for his English help and thoughtful review of this paper. The authors would like to gratefully acknowledge the financial support from National Natural Science Foundation of China (nos.