## 2 Materials and Methods
### Finite Element Method  
#### 2.2.1 Coupling Relationship between the Tuning Fork and Fluid  
The structure of the tuning fork is shown in Figure 1a. The length of the tuning fork arm is \(L\), the width of the tuning fork arm is \(e\), the thickness of the tuning fork arm is \(h\), and the spacing between the tuning fork arms is \(g\). The bottom of the tuning fork is fixed, thus the displacement is zero. Figure 1b is a schematic diagram of the tuning fork performing first-order in-plane anti-phase flexural vibration. This paper mainly studies the relationship between the resonant frequency and quality factor of this vibration mode with respect to fluid density and viscosity. The material of the tuning fork is quartz, with a density of 2650 kg/m\({}^{3}\), and other material property parameters of quartz are defined by the elastic matrix (the conversion matrix between stress and strain, related to material parameters such as Young's modulus and Poisson's ratio), the coupling matrix (the conversion matrix between charge and stress), and the relative dielectric constant. The governing equations for fluid motion in the finite element method are defined by the linear Navier-Stokes equations, including the momentum equation, the continuity equation, and the constitutive relation equation:  
\[\mathrm{i}\omega\mathrm{\rho_{D}}\mathrm{u_{t}}=\
abla\cdot\sigma \tag{4}\]  
\[\mathrm{i}\omega\mathrm{\rho_{t}}+\
abla\cdot(\mathrm{\rho_{0}}\mathrm{u_{t} })=0 \tag{5}\]  
\[\mathrm{\rho_{t}}=\mathrm{\rho_{0}}\big{(}\mathrm{\beta_{T}}\mathrm{p_{t}}- \mathrm{\alpha_{P}}\mathrm{T_{t}}\big{)} \tag{6}\]  
\[\sigma=-\mathrm{p_{t}}\mathrm{I}+\eta\Big{(}\
abla\mathrm{u_{t}}+(\
abla \mathrm{u_{t}})^{\mathrm{T}}\Big{)}-\big{(}\frac{2}{3}\mathrm{\eta_{D}}- \mathrm{\eta_{B}}\big{)}(\
abla\cdot\mathrm{u_{t}})\mathrm{I} \tag{7}\]where \(\rho_{0}\) represents the initial density of the fluid in the absence of disturbance, \(u_{t}=u_{0}+u\) represents the total acoustic velocity field, \(u_{0}\) represents the initial background acoustic velocity field, and \(u\) represents the acoustic velocity field generated by tuning fork's vibration. \(T_{t}=T_{0}+T\) indicates the total temperature, where \(T_{0}\) indicates the initial background temperature and \(T\) indicates the temperature change caused by the tuning fork's vibration. \(p_{t}=p_{0}+p\) represents the total acoustic pressure, where \(p_{0}\) represents the initial background pressure and \(p\) represents the acoustic pressure caused by the tuning fork's vibration. \(p_{t}\) represents the density of the fluid during the propagation of the acoustic wave; \(\alpha_{p}\) represents the coefficient of thermal expansion at constant pressure; \(\eta\) represents the dynamic viscosity; and \(\eta_{B}\) represents the volume viscosity, which is related to the momentum loss caused by the expansion and compression of the fluid. \(\beta_{T}\) represents the isothermal compression rate; \(\sigma\) represents the stress; and \(I\) represents the unit diagonal matrix, \(i^{2}=-1\). Fluid in finite element method is compressible fluid, and density, \(\rho_{0}\), and dynamic viscosity \(\eta\) are related to temperature and pressure. \(\rho_{0}\) and \(\eta\) are set to constants in simulation, so that, \(\rho_{0}\) and \(\eta\) are not affected by temperature and pressure. When the tuning fork and the fluid are in coupled vibration, their displacements are continuous, satisfying the following relationship:  
\[c_{fluid}=\frac{\partial x}{\partial t} \tag{8}\]  
where \(x\) and \(c_{fluid}\) represent the displacement of the tuning fork and the velocity of the fluid, respectively, and their relationship in the frequency domain is \(c_{fluid}=i\omega x\). This coupling relationship shows that the normal stress on the boundary between the tuning fork and the fluid is also continuous. To sum up, the equations of fluid motion in the analytical method and the finite element method are both linear Navier-Stokes equations.