# Numerical Study on Single-Bubble Contraction-Rebound Characteristics in Cryogenic Fluids
## 2 Derivation and Verification of the Single-Bubble Collapse Model
### Derivation of the Single-Bubble Collapse Model  
The equation for bubble growth was first derived and used by [PERSON] [34] in the absence of surface tension and viscosity. Subsequently, this equation was improved by [PERSON] [35] and was applied to the problem of traveling cavitation bubbles. \[R\frac{d^{2}R}{dt^{2}}+\frac{3}{2}\bigg{(}\frac{dR}{dt}\bigg{)}=\frac{P_{v}(T _{v})-P_{\rm{inf}}}{\rho_{l}}-\frac{4\
u_{l}}{R}\frac{dR}{dt}-\frac{2S}{\rho _{l}R} \tag{1}\]  
where \(R\) is the bubble radius, \(T_{v}\) is the temperature inside the bubble, \(P_{v}\) is the pressure inside the bubble, \(\rho_{l}\) is the liquid density, \(P_{inf}\) is the operating pressure, \(\
u_{l}\) is the kinematic viscosity of the liquid and \(S\) is the surface tension of the fluid. In addition to the Rayleigh-Plesset equation, the noncondensable gas in the bubble was considered. In general, it was assumed that there is no mass transfer between the noncondensable gas and the liquid. When the initial partial pressure of the noncondensable gas is \(P_{G0}\), the partial pressure, \(P_{G}\), varies with the bubble radius, as shown in Equation (2). \[P_{G}=P_{G0}\bigg{(}\frac{T_{v}}{T_{\rm{inf}}}\bigg{)}\bigg{(}\frac{R_{0}}{R} \bigg{)}^{3k} \tag{2}\]  
where \(R_{0}\) is the initial bubble radius, \(T_{inf}\) is the operating temperature and \(k\) is the polytropic exponent; \(k=1\) for bubble growth and \(k=1.4\) for bubble collapse. In the process of bubble collapse, the contraction speed of the bubble interface is greater than the bubble growth rate. When the Mach number (\(1\,dR/dt\,1/c\)) is higher than \(0.3\), the influence of liquid compressibility should be considered [10]. For this point, the near-acoustic solutions of [PERSON] [36] and [PERSON] [37] modified the Rayleigh-Plesset equation, and the results are shown in Equation (3):  
\[R\frac{d^{2}R}{dt^{2}}\bigg{(}1-\frac{2}{c_{l0}}\frac{dR}{dt}\bigg{)}+\frac{3}{2} \bigg{(}\frac{dR}{dt}\bigg{)}^{2}\bigg{(}1-\frac{4}{3c_{l0}}\frac{dR}{dt} \bigg{)}=\frac{R}{\rho_{l0}c_{l0}}\frac{dP}{dt}+\frac{P_{v}-P_{\text{inf}}}{ \rho_{l0}} \tag{3}\]  
where \(c_{l0}\) and \(\rho_{l0}\) are the constant sound speed and the density of the liquid, respectively. The Rayleigh-Plesset equation adopted by the model could be modified by substituting Equations (1) and (2) into Equation (3):  
\[R\frac{d^{2}R}{dt^{2}}\Big{(}1-\frac{2}{c_{l0}}\frac{dR}{dt} \Big{)}+\frac{3}{2}\bigg{(}\frac{dR}{dt}\bigg{)}^{2}\Big{(}1-\frac{4}{3c_{l0}} \frac{dR}{dt}\Big{)}=\frac{R}{\rho_{l0}c_{l0}}\frac{d}{dt}\bigg{(}P_{v}(T_{v}) +P_{G_{0}}\Big{(}\frac{T_{v}}{T_{\text{inf}}}\Big{)}\Big{(}\frac{R_{0}}{R} \Big{)}^{3k}-\frac{2S}{R}-\frac{4\mu_{l}}{R}\frac{dR}{dt}\bigg{)}+ \tag{4}\] \[\frac{P_{v}(T_{v})+P_{G_{0}}\Big{(}\frac{T_{v}}{T_{\text{inf}}} \Big{)}\Big{(}\frac{R_{0}}{T_{\text{inf}}}\Big{)}^{3k}-\frac{2S}{R}-\frac{4\mu _{l}}{R}\frac{dR}{dt}-P_{\text{inf}}}{\rho_{l0}}\]  
where \(\mu_{l}\) is the dynamic viscosity of the liquid. In order to further consider the thermal effect in the bubble model, the thermal equilibrium equation was introduced. The vapor density \(\rho_{v}\) in the bubble varies greatly with the temperature. Additionally, the variation in temperature inside the bubble could not be ignored during the bubble growth-collapse process. The transient variation in vapor density in the bubble was also considered. In order to obtain the temperature inside the bubble, \(T_{B}\), and the temperature at the bubble boundary, \(T_{l}\), it was assumed that only heat conduction exists in the thermal boundary layer around the bubble. According to the thermal balance equation, Equation (5) was introduced:  
\[4\pi R^{2}k_{l}\bigg{(}\frac{dT_{l}}{dr}\bigg{)}_{r=R}=L\frac{d}{dt}\bigg{(} \frac{4}{3}\pi R^{3}\rho_{v}\bigg{)}=4\pi R^{2}L\rho_{v}\frac{dR}{dt}+\frac{4} {3}\pi R^{3}L\frac{d\rho_{v}}{dT_{l}}\bigg{(}\frac{dT_{l}}{dt}\bigg{)}_{r=R} \tag{5}\]  
where \(L\) is the latent heat of the fluid and \(k_{l}\) is the thermal conductivity of the liquid.