## 2 Problem Formulation  
### Derivation of the Motion Equation of Theoretical Model  
The derivation of the linear theoretical model is carried out as follows. Take a small element of length \(\delta x\) of the cantilevered pipe into consideration, as shown in Figure 4a, under the action of fluid-related and structural forces and moments. Force balances in the \(x\)- and \(z\)-directions renders, respectively  
\[\frac{\partial F_{T}}{\partial x}-\frac{\partial}{\partial x}\bigg{(}Q\frac{ \partial w}{\partial x}\bigg{)}+M_{p}g-F_{it}+F_{et}-(F_{in}+F_{en})\frac{ \partial w}{\partial x}=0 \tag{1}\]  
\[\frac{\partial Q}{\partial x}+\frac{\partial}{\partial x}\bigg{(}F_{T}\frac{ \partial w}{\partial x}\bigg{)}+F_{in}+F_{en}+(F_{et}-F_{it})\frac{\partial w }{\partial x}-M_{p}\frac{\partial^{2}w}{\partial t^{2}}=0 \tag{2}\]  
where \(w\) is the lateral deflection; \(F_{T}\) is the axial tension; \(M_{p}\) is the mass per unit length of the cantilever pipe; \(Q\) is the transverse shear force in the cantilevered pipe; \(g\) is the gravitational acceleration; \(F_{in}\) and \(F_{it}\) are the normal and tangential hydrodynamic forces because of the internal flow \(U_{i}\), respectively; \(F_{en}\) and \(F_{et}\) are the normal and tangential hydrodynamic forces because of the external flow, respectively. Figure 3: Excessive bending of the inner tubing at a Chinese salt cavern storage [3]. As approximated by the Euler-Bernoulli (E-B) beam theory, one obtains the following relationship  
\[Q=-\frac{\partial}{\partial x}\left(EI\frac{\partial^{2}w}{\partial x^{2}}\right) \tag{3}\]  
where \(EI\) is the flexural rigidity. Internal dissipation in the cantilevered pipe material is relatively small compared to dissipation by the surrounding fluid so it can be neglected [13]. Substituting Equation (3) into Equations (1) and (2), and neglecting second-order terms, one finds the following expressions in the \(x\)- and \(z\)-directions, respectively  
\[\frac{\partial F_{T}}{\partial x}-\left(F_{it}+F_{in}\frac{\partial w}{  
Substituting Equations (6) and (7) in (4) and (5) renders the following equations in the \(x\)- and \(z\)-directions, respectively  
\[\frac{\partial F_{T}}{\partial x}+\left(M_{f}g-A_{f}\frac{\partial p_{i}}{ \partial x}\right)+\left(F_{et}-F_{en}\frac{\partial w}{\partial x}\right)+M_{ p}g=0 \tag{8}\]  
\[EI\frac{\partial\frac{\partial w}{\partial x}}{\partial x}-\frac{\partial}{ \partial x}\left(F_{t}\frac{\partial w}{\partial x}\right)+\left[M_{f}\left( \frac{\partial^{2}w}{\partial x}-2U_{i}\frac{\partial^{2}w}{\partial x\partial i }+U_{i}^{2}\frac{\partial^{2}w}{\partial x^{2}}\right)+A_{f}\frac{\partial}{ \partial x}\left(p_{i}\frac{\partial w}{\partial x}\right)\right]-\left(F_{en }+F_{et}\frac{\partial w}{\partial x}\right)+M_{p}\frac{\partial^{2}w}{ \partial x^{2}}=0 \tag{9}\]  
The field of flow outside the cantilevered pipe can be simplified because the cantilever pipe displacements \(w\) are considered to be small according to E-B beam theory. More specifically, the external flow is modelled as the superposition of a perturbation potential because of the cantilever pipe vibrations in the mean axial flow [15,16]. Hence, the effects of fluid viscosity are considered to be different from this potential flow, and the viscosity-related forces are added separately to the system. When linear investigations of fluid-structure interaction issues are carried out, this is a common approach. There are more details in [5,15]. The resultant force of all external hydrodynamic forces acting on the cantilevered pipe (Figure 4a) can be broken down into \(F_{en}\) and \(F_{et}\) in the directions perpendicular and tangential to the cantilever pipe centerline, respectively.