## 4 Conclusions  
Firstly, the existing analytical model of the relationship between tuning-fork resonance characteristics and fluid characteristics based on [PERSON]'s cantilever-beam theory was analyzed. It has the limitation of being applicable to the small viscous effects of fluid. In this paper, the finite element method was used to establish the simulation model to simulate the coupling vibration law of the tuning fork and fluid, and the influence of the fluid viscous effect on the resonant characteristics of tuning fork was considered. The comparison of the finite element numerical results, analytical results, and experimental measurement results showed that the finite element simulation results and experimental measurement results had a higher agreement, and the relative difference rate of resonant frequency between the two was less than 2%, and the relative difference rate of quality factor was less than 4%. This indicates that, compared with the approximate analytical method, the finite element method can consider the influence of the fluid viscous effect, so it is more in line with the actual situation than the analytical method. Finally, based on the finite element method, the sensitivity of tuning-fork resonance characteristics to fluid characteristic parameters (density and viscosity) was calculated. The results show that the sensitivity of tuning-fork resonant frequency to fluid density and viscosity increases with the increase in fluid density and viscosity, and the sensitivity of tuning fork quality factor to fluid viscosity generally increases with the increase in fluid viscosity. However, the sensitivity of quality factor to fluid density does not change monotonically. Compared with experiments, the established finite element simulation model can be used, conveniently, to analyze the sensitivity change in tuning fork to fluid density and viscosity when the fluid density and viscosity change in a large range, which can greatly save time and economic cost. It provides a theoretical reference for further optimization of the design of the tuning fork sensor to improve the accuracy of measuring fluid characteristic parameters. Next, the finite element simulation model can be optimized into the tuning fork's vibration in multiphase flow, so as to study the relationship between tuning fork's resonance parameters and multiphase flow's characteristic parameters.  
Conceptualization, F.J., D.C. and X.H.; methodology, F.J., Y.D. and M.T.; software, F.J.; validation, F.J.; formal analysis, F.J., D.C. and X.H.; investigation, F.J. and M.Z.; data curation, F.J.; writing--original draft preparation, F.J.; writing--review and editing, D.C. and Y.Z.; visualization, F.J. supervision, D.C.; project administration, D.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China, grant number 42074174 and funded by Chongqing Natural Science Foundation, grant number CSTB2023 NSCQ-MSX0968.  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
The authors declare no conflicts of interest.