## 2 Problem Formulation
### Derivation of the Motion Equation of Theoretical Model
For \(L_{s}<x\leq L\), there is a purely hydrostatic pressure distribution along the unconfined portion of the cantilever pipe. For \(0\leq x<L_{s}\), an annular fluid element (i.e., the external flow of length \(\delta x\)) (Figure 5a) is considered, and a force balance per unit length is obtained as  
Figure 3:1  
where \(A_{ch}=\pi({D_{ch}}^{2}-{D_{o}}^{2})/4\) represents the cross-sectional flow area of the annular passage; \(F_{f}\) represents the total wall-friction force acting on the fluid element. It is assumed that an equal wall shear stress acts on both annular passage surfaces, the total wall-friction force \(F_{f}\) can be expressed as  
Figure 3:2  
in which \(S_{tot}=\pi(D_{ch}+D_{o})\) is the total wetted area of the annular flow per unit length, and \(S_{o}\) is the outside wetted pipe perimeter. Combining Equations (23) and (24), multiplied by \((A_{o}/A_{ch})\), the resultant equation is rearranged as  
Figure 3:3  
where \(D_{h}=4A_{ch}/S_{tot}=(D_{ch}-D_{o})\) is the hydraulic diameter of the external flow in the annular passage. Integration of Equation (25), where for the confined portion external fluid pressure at \(x=0\) is the reference pressure (\(p_{o}\,\rvert_{x\,=\,0}=0\)), with respect to x gives  
Figure 3:4for \(0\leq x<L_{s}\). For the unconfined part, i.e., \(L_{s}<x\leq L\), the external fluid pressure distribution is hydrostatic  
Figure 3:5  
which upon integration results in  
Figure 3:6  
in which \(C_{1}\) is an integration constant determined in the following description. It is assumed that \(x_{1}=L_{s}{}^{-}\) is the axial location just inside the annular passage, and \(x_{2}=L_{s}{}^{+}\) is the location just outside the annular passage. Based on Bernoulli's equation, an energy balance of the fluid is obtained from \(x_{1}\) to \(x_{2}\)  
Figure 3:7  
where the quantity \(h_{e}\), with \(K_{1}=1\), is the head-loss associated with sudden enlargement of the external flow in the annular passage into the surrounding fluid at \(x=L_{s}{}^{+}\)[19]. Combination Equations (28) and (29) yields  
Figure 3:8  
in which \(\mid\mid F_{L}\mid\mid\) = 0.5\(\rho\)/\(C\)/\(D_{o}\)/\(D_{o}\)/\(D_{o}\)/\(D_{o}\)/\(2\). Through the above analysis, the resulting fluid pressure gradient and fluid pressure distribution over the whole cantilevered pipe, for \(0\leq x\leq L\), are, respectively  
Figure 3:9  
and  
\[Q=-\frac{\partial}{\partial x}\left(EI\frac{\partial^{2}w}{\partial x^{2}}\right) \tag{3}\]0  
where \(\delta_{D}(x-L_{s})\) is the Dirac delta function. Substituting Equations (16) and (21) in (10), and subsequently substituting (10) into (8), the force balance on the cantilevered pipe is obtained in the \(x\)-direction, namely  
\[Q=-\frac{\partial}{\partial x}\left(EI\frac{\partial^{2}w}{\partial x^{2}}\right) \tag{3}\]1  
Substituting Equations (15), (16), (19), and (22) in (11), and subsequently substituting the result in (1), the motion equation is obtained in the \(z\)-direction,  
\[Q=-\frac{\partial}{\partial x}\left(EI\frac{\partial^{2}w}{\partial x^{2}}\right) \tag{3}\]2  
Additionally, the dimensionless (complex) frequency of vibration, \(\omega\), is related to the dimensional one, \(\Omega\), by  
\[Q=-\frac{\partial}{\partial x}\left(EI\frac{\partial^{2}w}{\partial x^{2}}\right) \tag{3}\]3  
It is noted that the external and internal fluid pressures in dimensionless form at the free end have the relationship  
\[Q=-\frac{\partial}{\partial x}\left(EI\frac{\partial^{2}w}{\partial x^{2}}\right) \tag{3}\]4  
where  
\[Q=-\frac{\partial}{\partial x}\left(EI\frac{\partial^{2}w}{\partial x^{2}}\right) \tag{3}\]5  
with \(h_{a}\) given in Equation (37), and \(K_{1}\) as shown in Equation (29). Finally, the dimensionless flow velocities are expressed as  
\[Q=-\frac{\partial}{\partial x}\left(EI\frac{\partial^{2}w}{\partial x^{2}}\right) \tag{3}\]6  
where \(\alpha\) and \(\alpha_{ch}\) are defined in (41).