## 2 Materials and Methods  
### Analytical Model  
Assuming that the cross section of the tuning fork arm is uniform along the length, and the length of the tuning fork arm is much larger than the width and thickness, according to the Euler-Bernoulli cantilever-beam theory, the flexural vibration equation of the tuning fork arm during vibration in an incompressible fluid is as follows [12]:  
\[\text{EI}\frac{\partial^{4}}{\partial\text{x}^{4}}\text{W}(\text{x},\omega)- \omega^{2}\rho_{\text{b}}\text{elW}(\text{x},\omega)=\text{F}_{\text{ex}}( \text{x},\omega)+\text{F}_{\text{hydro}}(\text{x},\omega) \tag{1}\]  
where E represents [PERSON]'s modulus; I represents the moment of inertia; \(\omega\) represents the resonant angular frequency of the tuning fork when it vibrates in the fluid; \(\rho_{\text{b}}\) represents the density of the tuning fork material; e represents the width of the tuning fork arm;1 represents the thickness of the tuning fork; \(\mathrm{W}(\mathrm{x},\omega)\) represents the displacement function, where \(\mathrm{x}\) represents the position coordinate along the length direction of the tuning fork arm; \(\mathrm{F}_{\mathrm{ex}}(\mathrm{x},\omega)\) represents the external driving force of the tuning fork; and \(\mathrm{F}_{\mathrm{hydro}}(\mathrm{x},\omega)\) represents the force of the fluid on the tuning fork. \(\mathrm{F}_{\mathrm{hydro}}(\mathrm{x},\omega)\) = \(\frac{\pi}{4}\rho\omega^{2}\mathrm{I}^{2}\Gamma_{\mathrm{hydro}}(\omega) \mathrm{W}(\mathrm{x},\omega)\) is obtained by solving the linear Navier-Stokes equation, and \(\Gamma_{\mathrm{hydro}}\ =\ \Gamma_{\mathrm{hydro}}^{\mathrm{r}}(\omega)\ +\mathrm{j} \Gamma_{\mathrm{hydro}}^{\mathrm{i}}(\omega)\) represents hydrodynamic function. When the tuning fork is immersed in the fluid with a small viscous effect, Formula (1) is solved to obtain the resonant frequency of the tuning fork when it is performing first-order in-plane anti-phase flexural vibration [12],  
\[\omega=\omega_{\mathrm{vac}}[1+\frac{\pi\mathrm{\rho_{P}}\mathrm{e}}{4\mathrm{ \rho_{D}}\mathrm{I}}\Gamma_{\mathrm{hydro}}^{\mathrm{r}}]^{-1/2} \tag{2}\]  
where \(\omega_{\mathrm{vac}}=2\pi\mathrm{f}_{\mathrm{vac}}\) represents the resonant angular frequency of the tuning fork when it vibrates in vacuum, \(\mathrm{f}_{\mathrm{vac,n}}=\frac{\mathrm{e}}{4\pi\sqrt{3}}\frac{\mathrm{E}}{ \mathrm{\rho_{D}}}(\frac{\mathrm{C}_{1}}{\mathrm{L}})^{2}\), \(\mathrm{C}_{1}=1.875\) is the first positive root of the equation \(1+\mathrm{cos}(\mathrm{C}_{n})\mathrm{cosh}(\mathrm{C}_{n})=0\), \(\Gamma_{\mathrm{hydro}}^{\mathrm{r}}=1+\frac{4}{\sqrt{2\mathrm{Re}}}\) represents the real part of the hydrodynamic function [16], and \(\mathrm{Re}=\frac{\mathrm{\rho_{D}}\mathrm{I}^{2}\omega}{4\mathrm{\eta_{D}}}\) represents the Reynolds number.  
By substituting Formula (2) into Formula (1), the fluid viscosity quality factor of the tuning fork during the first-order flexural vibration can be obtained:  
\[\mathrm{Q}=\frac{\frac{4\mathrm{\rho_{D}}\mathrm{e}}{\mathrm{\pi\mathrm{\rho_{ D}}\mathrm{I}}}+\Gamma_{\mathrm{hydro}}^{\mathrm{r}}}{\Gamma_{\mathrm{hydro}}^{ \mathrm{i}}} \tag{3}\]  
where \(\Gamma_{\mathrm{hydro}}^{\mathrm{i}}=\frac{4}{\sqrt{2\mathrm{Re}}}+\frac{2}{ \mathrm{Re}}\) represents the imaginary part of the hydrodynamic function [16]. The quality factor of the tuning fork in Formula (3) is related to the resonant frequency of the tuning fork in Formula (2), and Formula (2) is derived under the assumption of small viscous effects of the fluid, which implies that the resonant frequency and quality factor in Formula (2) and Formula (3) are only applicable when the viscous effect of the fluid is small [17; 18].