## 4 Conclusions  
A fluid-solid-heat coupling model for WLRSBs considering the deflection of the propeller shaft was built, and the proposed model was used to investigate the fluid-solid-heat performance of the JWR system. Based on the Euler method, a general transfer equation in polar coordinates was employed to describe the heat convection in a water film and the heat conduction in rubber bushing and a journal. The segmented mixed forces, including the hydrodynamic and contact forces, and the superposition method were bridged to evaluate the deflection of the stern shaft. Additionally, the FSH coupling performances obtained with two propeller shaft cases, i.e., aligned shaft case and misaligned shaft case, were compared systematically. The conclusions are as follows:  
Figure 14: Distributions of contact pressure in the axial section obtained with the aligned shaft case and the misaligned shaft case, respectively: (**a**) water film pressure; (**b**) contact pressure; (**c**) edge effect of the aligned shaft and the misaligned shaft case.  
1. The temperature distribution and the load capacity obtained by the proposed model agree well with the experimental data given in the literature, thus validating that the presented model can be used to predict the FSH coupling performances of hydrodynamic bearings.
2. The asymmetric tribological performance distributions along the axial direction occur due to the propeller shaft deflection, including the hydrodynamic force, asperity contact, and temperature. The maximum asperity contact and temperature rise of the misaligned shaft are obviously larger than those of the aligned shaft. Furthermore, the temperature yielded by the water outlet is larger than that yielded by the water inlet for both the misaligned and aligned shaft cases.
3. Owing to the WLRSB having a larger length-to-diameter ratio compared with the general journal bearing, the misalignment caused by the deflection of the propeller shaft affects the FSH performance of the JWR system significantly, especially under a relatively heavy load condition. Furthermore, the aligned shaft case was found to be a proper substitute for a misaligned shaft case when the load capacity is smaller than 4 kN for the current simulation.  
This study develops a comprehensive steady FSH coupling model for a water-lubricated rubber bearing operated at the stern shaft system, in which shaft deflection is caused by the self-weight of the propeller that is integrated. The present mathematical model conducts toward a guideline to help design appropriate parameters for increasing the reliability of water-lubricated rubber bearings. In addition, future works are needed to integrate rotor dynamic characteristics into the present model and to study the interaction between the FSH coupling performance and the high inertia components of a propeller in specific applications.  
Conceptualization, Y.H.; funding acquisition, Y.H.; methodology, T.T. and G.X.; resources, Y.H.; investigation, T.T.; format analysis, T.T.; software, G.X.; validation, H.J.; data curation, H.J.; writing--original draft preparation, T.T.; writing--review and editing, T.T. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Science Foundation of China (51605053 and 51475051) and the Fundamental Research Funds for the Central Universities Self-determined Project of the State Key (106112017 CDJXY110007).  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
The authors declare no conflict of interest.