# Spatiotemporal Evolution of Gas in Transmission Fluid under Acoustic Cavitation Conditions
## 2 Methods  
### Conservative Homogeneous Flow Model  
Considering the mass conservation of the transmission fluid before and after compression in the transmission tube and the effect of the flow on the shear stress of the tube wall, the compressible gas-liquid two-phase flow is modeled using the mass and the momentum conservation equations. The density \(\rho\) and the momentum \(\rho u\) are used as conservation variables to obtain a set of conservation-type fluid dynamics equations and expressed in the form of a matrix as follows:  
\[\frac{\partial\mathbf{U}}{\partial t}+\frac{\partial\mathbf{F}(\mathbf{U})}{ \partial x}=\mathbf{M} \tag{1}\]  
\[\mathbf{U}=\begin{bmatrix}\rho\\ \rho u\end{bmatrix},\mathbf{F}(\mathbf{U})=\begin{bmatrix}\rho u\\ \rho u^{2}+P\end{bmatrix},\mathbf{M}=\begin{bmatrix}0\\ -\frac{\ell\rho u|u|}{2\Delta}\end{bmatrix} \tag{2}\]  
where \(\rho\), \(u\), \(P\), \(t\), and \(f\) are the mixture fluid density, transmission fluid flow velocity, transmission fluid pressure, time variable, and transmission fluid flow resistance coefficient, respectively. The flow is laminar, \(f\) = 64/\(Re\), if the Reynolds number \(Re\) \(\leq\) 2000 and turbulent, \(f\) = 0.316/\(Re\)[25], if the Reynolds number \(Re\) > 2000.  
The expression for the gas void fraction is introduced to study the change in gas content of the mixture fluid during compression and expansion:  
\[\alpha=\frac{\rho_{l}-\rho}{\rho_{l}-\rho_{\chi}} \tag{3}\]  
where \(\chi\) is the air (\(g\)) or vapor (\(v\)), \(\alpha\) is the gas void fraction, and \(\rho_{l}\) is the transmission fluid density.  
The density of the homogeneous mixture can be expressed as a function of homogeneous bubbly and stratified flows according to the flow type:  
\[\rho=\alpha\rho_{g}+(1-\alpha)\rho_{l} \tag{4}\]  
\[\rho=\frac{\rho_{v}\rho_{l}}{\alpha\rho_{l}+(1-\alpha)\rho_{v}} \tag{5}\]  
The state equation of an ideal gas is given by the experimental law of gases and Avogadro's law:  
\[P=\rho_{\chi}r_{\chi}T \tag{6}\]  
where \(r\) and \(T\) are the gas constant and temperature, respectively.  
For positive pressure flow, usually in the cavitation zone, the temperature \(T\) is a function of the pressure \(P\), \(T\) = \(T(P)\). In Equation (3), the transmission fluid density is a function of pressure \(P\) and temperature \(T\), \(\rho_{l}\) = \(\rho_{l}\)(\(T\),\(P\)).  
Neglecting the heat exchange at the boundary, the energy conservation equation for the one-dimensional flow of a compressible fluid can be expressed as:  
\[\frac{\mathrm{d}h}{\mathrm{d}t}-\frac{1}{\rho}\frac{\mathrm{d}P}{\mathrm{d}t}=Q \tag{7}\]  
where \(h\) and \(Q\) denote the enthalpy per unit mass of the system and viscous power dissipation per unit mass in the system, respectively.  
To express the change of temperature, entropy \(s\), and the first law of thermodynamics are introduced:  
\[q\mathrm{d}s=0 \tag{8}\]  
\[T\mathrm{d}s=\mathrm{d}h-\frac{1}{\rho}\mathrm{d}P \tag{9}\]Considering the temperature change caused by the compression effect when the transmission fluid flows with pure liquid [25]:  
\[\mathrm{d}h=C_{p}\mathrm{d}T+\frac{1}{\rho}(1-\varphi T)\mathrm{d}P \tag{10}\]  
\[\mathrm{d}P=\gamma_{T}\,^{2}\mathrm{d}\rho+\varphi E_{T}\mathrm{d}T \tag{11}\]  
where \(C_{p}\) is the constant pressure specific heat capacity, \(\gamma_{T}\) is the isothermal speed of sound, \(\varphi\) is the coefficient of thermal expansion, \(E_{T}\) is the isothermal bulk modulus of elasticity, and \(q\) is the mass flow rate of the mixture.  
Under isentropic conditions, Equations (9)-(11) relate the pressure and temperature of the liquid in any process to obtain the following relationship:  
\[\frac{\mathrm{d}P}{\mathrm{d}T}=\frac{C_{p}E_{T}}{\varphi T\gamma_{T}\,^{2}} \tag{12}\]