## 3 Mathematical Modeling of the Inviscid Fluid System
### The Hydrodynamic Force under Lateral Excitation Force  
The derivation of the hydrodynamic forces operating on a vertical container of depth h and width 2 W under sinusoidal lateral excitation is established under translational and pitching excitations. The drill string of length L is partially submerged in the reservoir filled with an inviscid fluid, as shown in Figure 4b. Rising to a dynamic imbalance, the drill string's center of mass is then deflected, which causes the harmonic excitation of the (x and y) axes by:  
\[x(t)=X_{0}\cos\Omega t\text{ and }y(t)=Y_{0}\sin\Omega t \tag{16}\]  
where \(X_{0}\) and \(Y_{0}\) are the excitation amplitudes, and \(\Omega\) is a cyclical frequency in cycles per unit of time much lower than its fundamental frequency (\(\omega\gg\Omega\)) so that the liquid oscillates at exactly the excitation frequency. The following equations represent the linearized form of the fluid field equations for irrotational, inviscid, incompressible flow motion under the assumption that the submerged shaft oscillations are moderated [14]:  
\[\
abla^{2}\widetilde{\Phi}=0,\text{ within the domain of fluids} \tag{17}\]  
\[\
abla^{2}=\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{ \partial r}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}+\frac{ \partial^{2}}{\partial z^{2}} \tag{18}\]  
Where \(\widetilde{\Phi}=\widetilde{\Phi}(r,\theta,\text{z},t)\) (\(\widetilde{\Phi}=\widetilde{\Phi}(r,\theta,\text{z},t)\)) is related to the flow disturbances and depends on Eulerian coordinates and time. The presence of the drill string and the boundary conditions at the container boundaries are:  
\[\left.\frac{\partial\widetilde{\Phi}}{\partial r}\right|_{r=W}=0,\text{ \ \ \ \ }\left.\frac{\partial\widetilde{\Phi}}{\partial z}\right|_{z=h}=0,\text{ \ \ \ \ }\left.\frac{\partial \widetilde{\Phi}}{\partial\theta}\right|_{\theta=0,\pi/2}=0, \tag{19}\]  
Figure 4: (**a**) Rectangular tank subject to harmonic excitation; (**b**) fluid profile around the shaft. The combined free surface condition obtained from the kinematic condition is:  
\[\text{g}\eta-\frac{\partial\widetilde{\Phi}}{\partial t}+\tilde{x}cos\theta{=0}, \text{ at z }{=}\eta(r,\theta,t) \tag{20}\]  
The vertical velocity of a fluid particle resting on the free surface, \(\text{z }{=}\eta(r,\theta,t)\), must be equated to the vertical velocity of the free surface itself to give the linearized dynamic free surface condition:  
\[-\frac{\partial\widetilde{\Phi}}{\partial z}=\frac{\partial\eta}{\partial t} \quad\text{at z }{=}\eta(r,\theta,t) \tag{21}\]  
where \(\eta(r,\theta,t)\) is the fluid surface's height from the unaltered free surface. Once the kinematic free surface Condition (21) and the dynamic Circumstance (20) with respect to time have been differentiated, the outcomes are obtained:  
\[\frac{\partial^{2}\widetilde{\Phi}}{\partial t^{2}}+\text{g}\frac{\partial \widetilde{\Phi}}{\partial z}=\tilde{x}rcos\theta \tag{22}\]  
The function \(\widetilde{\Phi}\) satisfies the following conditions on the fluid boundary: \(\partial\widetilde{\Phi}/\partial\eta=0\), at the container walls and the free surface. \(\dot{\eta}(r,\theta,t)\) is the vertical velocity of the free surface. The typical solution of the continuity Equation (18) subject to boundary Condition (19) can be expressed as follows:  
\[\widetilde{\Phi}(r,\theta,\text{z},t)=\sum_{n=1}^{\infty}\left[C_{1n}(t)\cos \theta+D_{1n}(t)\sin\theta\right]J_{1}(k_{1n}r)\frac{\cosh(k_{1n}(z+h))}{\cosh (k_{1n}h)}] \tag{23}\]  
where \(C_{1n}\) and \(D_{1n}\) are time-dependent functions determined from the free-surface initial Condition (20), \(J_{1}(. )\) is the Bessel function of the first order, \(k_{1n}=\zeta_{1n}/\text{W}\) are roots of \(\partial J_{1}(k_{1n}r)/\partial r|_{r=W}=0\), and \(r\) is a Fourier-Bessel series expansion given in the following form:  
\[r=\sum_{n=1}^{\infty}F_{n}J_{1}(k_{1n}r) \tag{24}\]  
where \(F_{n}=2W/(k_{1n}^{2}W^{2}-1)J_{1}(k_{1n}W)\).