## 2 Numerical Models
### Governing Equation
\begin{array}{l}\delta_{RE}\big{(}\theta_{j},z_{k}\big{)}=\sum\limits_{ \xi}\sum\limits_{\eta}G_{RE}\big{(}\theta_{j},z_{k},\theta_{\xi},z_{\eta}\big{)} \times\big{[}p_{h}\big{(}\theta_{\xi},z_{\eta}\big{)}+p_{c}\big{(}\theta_{\xi}, z_{\eta}\big{)}\big{]}\\ \delta_{RT}\big{(}\theta_{j},z_{k}\big{)}=\sum\limits_{\xi}\sum\limits_{\eta} \sum\limits_{\zeta}G_{RT}\big{(}\theta_{j},z_{k},\theta_{\xi},z_{\eta},r_{\zeta }\big{)}\times\Delta T\big{(}\theta_{\xi},z_{\eta},r_{\zeta}\big{)}\\ \delta_{JT}\big{(}\theta_{j},\Delta T\big{)}=\alpha_{I}\Delta T_{j}r_{j}(1+ \varepsilon_{j}\cos\big{(}\theta_{j}-\varphi\big{)})\end{array} \right. \tag{12}\]  
where the node \((\theta_{\xi},z_{\eta},r_{\zeta})\) is shown in Figure 4; \(\Delta T(\theta_{\xi},z_{\eta},r_{\zeta})\) is the temperature rise of the rubber bushing at the nodal of \((\theta_{\xi},z_{\eta},r_{\zeta})\), and \(G_{RE}(\theta_{j},y_{k},\theta_{\xi},y_{\eta})\) and \(G_{RT}(\theta_{j},z_{k},\theta_{\xi},z_{\eta},r_{\zeta})\) represent the influence functions of the elastic and thermal deformations, respectively. The calculated method of the former is presented by [PERSON] et al. [10], and that of the latter can be referred to in Ref. [34]. In the thermal analysis, the journal, water film, and rubber bushing are treated as a heat conduction coupling system (JWR system for simplicity), as shown in Figure 4, using the Euler method. Based on this, the thermal behavior of the water film, propeller shaft, and rubber bushing can be described by a general transfer equation in polar coordinates, which is given by:  
\[\rho C_{P}\bigg{[}v_{r}\frac{\partial T}{\partial r}+v_{\theta}\frac{\partial T }{\partial\theta}+v_{z}\frac{\partial T}{\partial z}\bigg{]}=\frac{\partial}{ \partial r}\bigg{(}k_{r}\frac{\partial T}{\partial r}\bigg{)}+\frac{\partial} {r\partial\theta}\bigg{(}k_{\theta}\frac{\partial T}{r\partial\theta}\bigg{)} +\frac{\partial}{\partial z}\bigg{(}k_{z}\frac{\partial T}{\partial z}\bigg{)} +\Phi \tag{13}\]  
where \(\rho\) is the density of the rubber bushing \((\rho_{R})\), journal \((\rho_{I})\), or lubricant \((\rho_{W})\); \(C_{P}\) is the specific heat of the bushing \((C_{PR})\), journal \((C_{P})\), or water \((C_{PW})\); \(k_{r}(k_{rR},k_{rR},k_{rW})\), \(k_{\theta}(k_{RR},k_{R\theta},k_{R\theta})\), and \(k_{z}(k_{zR},k_{zR},k_{zR})\) denote the thermal conductivities in \(r\), \(\theta\), and \(z\) directions, respectively; \(v_{r}\), \(v_{\theta}\), and \(v_{z}\) denote the velocity at which the rotation speed corresponds to the journal and the flow velocity corresponds to the water in \(r\), \(\theta\), and \(z\) directions, respectively. One study [29] pointed out that the thermal behaviors of water-lubricated bearings depend significantly on the intensity of the enforced axial flow. In the present study, we assume the axial flow is sufficiently low, and therefore, the effect of the axial flow on the thermal behavior of the water-lubricated bearing is neglected in the current simulation. Subsequently, the flow velocity in the \(\theta\) direction, \(v_{W\theta}\), and in the \(z\) direction, \(v_{Wz}\), is given as follows:  
\[\left\{ \begin{array}{l}v_{W\theta}=u+\frac{u\theta_{\xi}}{\eta}\int_{0}^{h} \frac{1}{\eta}\mathrm{d}\zeta-\phi_{\theta}\frac{\partial p_{h}}{\partial x} \int_{0}^{h}\frac{z_{m}-\zeta}{\eta}\mathrm{d}\zeta\\ v_{Wz}=-\phi_{z}\frac{\partial p_{h}}{\partial z}\int_{0}^{h}\frac{z_{m}-\zeta} {\eta}\mathrm{d}\zeta\end{array} \right. \tag{14}\]  
where \(u\) is the linear velocity of the journal, \(u=\omega R_{1}\). Furthermore, the parameter \(z_{m}\) can be evaluated by \(z_{m}=I_{R}/I_{R}\). Figure 4: Meshed journal, water film, and rubber bushing (JWR) system in polar coordinates. The heat source in the JWR system is composed of the viscous shear heat of the water film, \(\Phi_{W}\), and the friction heat, \(\Phi_{C}\), generated from the asperity contact occurring at the journal-rubber interface, which can be described by the following formulas:  
\[\left\{