###### Abstract  
A linear theoretical model is established for the dynamics of a hanging vertical cantilevered pipe which is subjected concurrently to internal and reverse external axial flows. Such pipe systems may have instability by flutter (amplified oscillations) or static divergence (buckling). The pipe system under consideration is a slender flexible cantilevered pipe hanging concentrically within an inflexible external pipe of larger diameter. From the clamped end to the free end, fluid is injected through the annular passage between the external pipe and the cantilevered pipe. When exiting the annular passage, the fluid discharges in the counter direction along the cantilevered pipe. The inflexible external pipe has a variable length and it can cover a portion of the length of the cantilevered pipe. This pipe system has been applied in the solution mining and in the salt cavern underground energy storage industry. The planar motion equation of the system is solved by means of a Galerkin method, and Euler-Bernoulli beam eigenfunctions are used as comparison functions. Calculations are conducted to quantify the effects of different confinement conditions (i.e., the radial confinement degree of the annular passage and the confined-flow length) on the cantilevered pipe stability, for a long leaching-tubing-like system. For a long system, an increase in the radial confinement degree of the annular passage and the confined-flow length gives rise to a series of flutter and divergence. Additionally, the effect of the cantilevered pipe length is studied. Increasing the cantilevered pipe length results in an increase of the critical flow velocity while a decrease of the associated critical frequency. For a long enough system, the critical frequency almost disappears. s salt cavern; leaching tubing; flutter instability; flow-induced vibration; internal and reverse external axial flows Xinbo Ge 1,2,*, 2,*stagnant fluid in the closed cavity is then forced through the cantilevered pipe, as a bounded reverse internal flow, discharging from the closed cavity upwards along the cantilevered pipe. Comparing the pipe configuration shown in Figure 1a with that given in Figure 1b, it is clear that the fluid flowing directions are opposite from each other. Contrary to the direction of fluid circulation illustrated in Figure 1a, fluid [1] is injected as internal flow into the closed cavity through the cantilever pipe and discharges through the annular passage, as shown in Figure 1b. The pipe configuration shown in Figure 1 has been applied in solution mining and in the salt cavern underground energy storage industry. Figure 2 shows the schematic view of pipe configurations applied in the salt cavern underground energy storage. Figure 2a,b correspond to Figure 1a,b, respectively. In fact, the pipe failure or excessive bending caused by flow-induced vibrations is a well-known issue in the salt cavern storage industry [2; 3]. For example, excessive bending of the inner tubing in a Chinese salt cavern storage is shown in Figure 3. This paper focuses on the pipe model shown in Figure 1a, and the motivation is desired to potentially improve the design of slender cantilever pipes in order to avoid issues cause by flow-induced vibration. For several decades, a few researchers have been done the work on conveying-fluid pipes subjected simultaneously to both internal and external axial flows. Most of the work done involves concurrent flows, i.e., internal and external axial flows in the same direction. In [4], [PERSON] and [PERSON] investigated the buckling instability of pipes subjected to internal and external axial fluid flow. In [5], [PERSON] et al. investigated the dynamics and stability of both clamped-clamped cylindrical tubular beams conveying fluid, which simultaneously are subjected to independent axial external flows. They also studied the dynamics of cantilevered pipes fitted with a tapered nozzle at the free end. They provided theoretical and experimental results to support their model.