## 3 The Hydrodynamic Forces and Introduction into the Model and Solution  
The hydrodynamic forces around the driveshaft are calculated by integrating the pressure distribution over the driveshaft-stator surface. This section establishes all the pressures and fluid forces in the driveshaft-stator gap; they are integrated into the system's governing equation. With rotation, the flow in the driveshaft becomes helical as depicted in Figure 3a. The streamline propeller is strongly affected by the speed of the driveshaft and after rotation increases the shear stresses within the incompressible viscous fluid as the tangential boundary layer grows. A complete derivation of the hydrodynamic forces around the rotating shaft has been published in [11], however, the aspect of fluid motion around the eccentric mass is considered in this work to enhance the previous expression of fluid  
Figure 3: Schematic coordinate frame of (**a**) the whirling streamline of the driveshaft, (**b**) Lumped mass point. pressures and fluid forces exclusively. The viscous fluid in contact with the driveshaft-disc surface spins at a uniform angular speed \(\omega\). The laminar flow around the rotating system in a cylindrical polar coordinates system \(\left(r,\overline{\theta},z\right)\) is reduced as  
\[\frac{\partial u}{\partial t}=-\frac{1}{\rho}\frac{\partial P}{\partial\overline {r}}+v\left[\
abla^{2}u-\frac{u}{r^{2}}-\frac{2}{r^{2}}\frac{\partial v}{ \partial\overline{\theta}}\right] \tag{12}\]  
\[\frac{\partial v}{\partial t}=-\frac{1}{\rho r}\frac{\partial P}{\partial \overline{\theta}}+v\left[\
abla^{2}v-\frac{v}{r^{2}}+\frac{2}{r^{2}}\frac{ \partial u}{\partial\overline{\theta}}\right] \tag{13}\]  
where the Laplacian operator \(\
abla^{2}=\frac{\partial^{2}}{\partial\overline{r}^{2}}+\frac{1}{r}\frac{ \partial}{\partial\overline{r}}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial \overline{\theta}^{2}}\), \(P_{\theta}\) and \(v\) denotes the pressure, density and dynamic viscosity of the fluid. The analysis includes the estimation of the stream function \(\Psi(r,\overline{\theta},t)\) by assuming that the flow velocity component \(u\) and \(v\) along the radial and circumferential coordinate \(r\) and \(\overline{\theta}\), respectively, are  
\[ \begin{array}{l}u(r,\overline{\theta},t)=\frac{1}{r}\frac{\partial\Psi(r, \overline{\theta},t)}{\partial\overline{\theta}}\\ v(r,\overline{\theta},t)=-\frac{\partial\Psi(r,\overline{\theta},t)}{ \partial r}\end{array}  \tag{14}\]  
Let the acceleration potential \(\Lambda\) in which gradient gives the acceleration vector equals  
\[\Lambda(r,\overline{\theta},t)=v\
abla^{2}\Psi(r,\overline{\theta},t)-\frac{ \partial\Psi(r,\overline{\theta},t)}{\partial t} \tag{15}\]  
The linearised expression of Equation (16) is obtained by substituting Equations (14) and (15) into Equations (12) and (13), in frame coordinates, \(r\) and \(\overline{\theta}\) gives the Navier-Stokes equation in a non-inertial reference frame expressed as  
\[ \begin{array}{l}\frac{\partial u}{\partial t}=-\frac{1}{\rho}\frac{ \partial p}{\partial t}+\frac{1}{r}\frac{\partial\Lambda}{\partial\overline{ \theta}}+\frac{\partial u}{\partial t}\\ \frac{\partial v}{\partial t}=-\frac{1}{\rho r}\frac{\partial p}{\partial \overline{\theta}}-\frac{\partial\Lambda}{\partial\overline{\theta}}+\frac{ \partial v}{\partial t}\end{array}  \tag{16}\]  
This Equation (16) can be resolved by eliminating the pressure term and then writing it as the sum of two-stream functions such that \(\Psi(r,\overline{\theta},t)=\Psi_{1}(r,\overline{\theta},t)+\Psi_{2}(r, \overline{\theta},t)\) where  
\[ \begin{array}{l}\
abla^{2}\Psi_{1}(r,\overline{\theta},t)=0\\ \left(\
abla^{2}-\frac{1}{\rho}\frac{\partial}{\partial t}\right)\Psi_{2}(r, \overline{\theta},t)=0\end{array}  \tag{17}\]  
The solution of Equation (17) may be written in the form [9]  
\[ \begin{array}{l}\Psi_{1}(r,\overline{\theta},t)=f_{1}(r)\sin\overline{ \theta}e^{i\Omega t}\\ \Psi_{2}(r,\overline{\theta},t)=f_{2}(r)\sin\overline{\theta}e^{i\Omega t} \end{array}