# Spatiotemporal Evolution of Gas in Transmission Fluid under Acoustic Cavitation Conditions
## 2 Methods
### Cavitation Types and Conditions  
To obtain an expression for the sound velocity of the mixture fluid, Equation (1) is rewritten as:  
\[\frac{\partial\mathbf{U}}{\partial t}+\mathbf{B}\frac{\partial\mathbf{U}}{ \partial x}=\mathbf{M} \tag{22}\]  
\[\mathbf{B}=\begin{bmatrix}0&1\\ \gamma^{2}-u^{2}&2u\end{bmatrix} \tag{23}\]  
\[\gamma=\sqrt{\frac{\partial P}{\partial\rho}} \tag{24}\]  
where \(\mathbf{B}\) is the Jacobian matrix of conservation hydrodynamic equations.  
Two eigenvalues are obtained from Jacobian Matrix \(\mathbf{B}\):  
\[\lambda_{1,2}=u\pm\gamma \tag{25}\]  
To calculate the speed of sound, the equation of the bulk modulus of elasticity of the mixture is introduced:  
\[E=-V\frac{\mathrm{d}P}{\mathrm{d}V} \tag{26}\]  
where \(V\) is the fluid volume of the mixture.  
According to the expression of sound velocity in Equation (24), the derivative of Equation (4) with respect to pressure \(P\) is obtained, and the relationship between the speed of sound and the gas void fraction is obtained:  
\[\frac{1}{\rho\gamma^{2}}=\frac{\alpha_{v}}{E_{v}}+\frac{\alpha_{g}}{E_{g}}+ \frac{1-\alpha_{v}-\alpha_{g}}{E_{l}}-\rho(\frac{1}{\rho_{v}}-\frac{1}{\rho_{ l}})\frac{\mathrm{d}\mu_{v}}{\mathrm{d}P} \tag{27}\]  
When the local pressure inside the transmission tube is lower than the saturated vapor pressure, vapor cavitation occurs. Considering the case of pure vapor cavitation (undissolved air content is zero), Equation (26) is simplified to:  
\[\frac{1}{\rho\gamma^{2}}=\frac{\alpha_{v}}{E_{v}}+\frac{1-\alpha_{v}}{E_{l}}- \rho(\frac{1}{\rho_{g}}-\frac{1}{\rho_{l}})\frac{\mathrm{d}\mu_{v}}{\mathrm{d }P} \tag{28}\]  
Since only a small portion of the transmission fluid undergoes vapor cavitation, the heat generated by the phase change of the liquid can be ignored here. On the premise that the cavitation zone is an isothermal flow, the pressure is the saturated vapor pressure at that temperature, so \(E_{v}\) and \(E_{l}\) are constants. The mass fraction of vapor is given, and the derivative of \(\mu_{v}\) with respect to \(P\) tends to infinity, i.e., \(\gamma=0\).  
When the transmission fluid pressure is greater than the saturated vapor pressure and less than the air separation pressure, air cavitation will occur, and in a high-pressure environment, the air separation pressure will increase accordingly. Meanwhile, considering the case of air pseudo-cavitation (the air content decreases as the fluid pressure increases), Equation (27) is simplified as follows:  
\[\frac{1}{\rho\gamma^{2}}=\frac{\alpha_{g}}{E_{g}}+\frac{1-\alpha_{g}}{E_{l}} \tag{29}\]  
A simple pure liquid flow will emerge when the transmission fluid pressure is greater than the air separation pressure:  
\[\gamma=\gamma_{l} \tag{30}\]  
Expressions for the speed of sound for pure liquid flow, air cavitation, and vapor cavitation are derived from Equations (28)-(30):\[\gamma=\left\{\begin{aligned} \gamma_{I}&P\geq P_{j}\\ \sqrt{\frac{E_{p}\rho_{I}\gamma_{I}\ ^{2}}{\rho[a\rho_{T}\gamma_{I}\ ^{2}+(1-\alpha)E_{S}]}}&P_{\
u}\leq P<P_{j}\\ 0&P<P_{\
u}\end{aligned}\right. \tag{31}\]