# 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 derivative of the temperature at the bubble boundary, \(T_{l}\), with respect to time could be derived from Equation (5):  
\[\frac{dT_{l}}{dt}=\bigg{(}k_{l}\frac{dT_{l}}{dr}-L\rho_{v}\frac{dR}{dt} \bigg{)}/\bigg{(}\frac{1}{3}RL\frac{d\rho_{v}}{dT}\bigg{)} \tag{6}\]  
In Equation (6), the temperature gradient at the bubble boundary, \(dT_{l}/dr\), was unknown. The temperature gradient at the bubble boundary could be obtained by solving the energy equation in spherical coordinates for the moving boundary. The one-dimensional energy equation is as follows. \[\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial r}=\alpha_{l}\bigg{(} \frac{\partial^{2}T}{\partial r^{2}}+\frac  
the bubble boundary could be derived from the assumption of discontinuity at the bubble interface. The liquid temperature is constant at an infinite distance. \[\left\{ \begin{array}{l}T(r,0)=T_{\text{inf}}\\ T(R,t)=T_{l}\\ T(R_{\text{inf}},t)=T_{\text{inf}}\end{array} \right. \tag{10}\]  
By the external discretization of bubbles, Equations (4) and (9) could be solved using the fourth-order Runge-Kutta method. Photos and a schematic diagram of the bubble collapse-rebound process are shown in Figure 1. It was assumed that the liquid beyond the 5 mm bubble boundary is not affected by the temperature and pressure change inside the bubble [1], where the temperature and pressure are equal to \(T_{\text{inf}}\) and \(P_{\text{inf}}\), respectively. Based on the bubble radius and the temperature at the bubble boundary, the temperature and pressure inside the bubble could be obtained. At first, it was assumed that there is no noncondensable gas inside the bubble. During bubble collapse, the bubble boundary contracts inward rapidly. Inside the bubble, there exists vapor compression and phase transformation simultaneously. Taking the boundary layer at the bubble interface as the control volume, the heat flowing into the boundary layer is positive and the heat flowing out is negative. The latent heat from phase transformation flows into the boundary layer, and the conducted heat flows out of the boundary layer. Assuming that the mass in the boundary layer is constant and equals \(m\), the energy balance equation of the thermal boundary layer is shown in Equation (11). \[T_{l}^{(i+1)}-T_{l}^{(i)}=\frac{4\pi\Big{(}R^{(i)}\Big{)}^{2}\Big{[}\Delta nM \Big{(}R^{(i)}-R^{(i+1)}\Big{)}L-k_{l}\frac{dT_{l}}{dr}\Big{]}}{m} \tag{11}\]  
where \(\Delta n\) is the variation in the amount of vapor substances and the superscript (_i_) indicates the _i_-th time step. The total amount of vapor in the bubble could be obtained from Equation (12)  
\[n^{(i+1)}=n^{(i)}+\Delta n \tag{12}\]  
If only the phase transition process was being considered, the bubble radius \(R^{\prime}\) could be obtained using the Clapeyron Equation (13). \[\frac{4}{3}\pi(R\prime)^{3}P_{\text{p}}^{(i)}=n^{(i+1)}\mathbb{R}T_{\text{p}}^ {(i)} \tag{13}\]  
where \(\mathbb{R}\) is the molar gas constant which has a value of \(8.3145\,\mathrm{J}/(\mathrm{mol\cdot K})\). At last, the pressure and temperature inside the bubble could be solved with the polytropic compression/expansion Equation (14). In the equation, \(k\) is the polytropic exponent. Generally, the growth and rebound of bubbles are deemed to be isothermal  
Figure 1: Single-bubble experimental photos [38] (**a**) and collapse model schematic diagram (**b**). expansion processes with \(k=1\). The collapse process is an adiabatic compression with \(k=1.4\). \[\left\{ \begin{array}{l}P_{\triangledown}^{(i+1)}=P_{\triangledown}^{(i)}{ \left(\frac{R\prime}{R^{(i+1)}}\right)^{3k}}\\ T_{\triangledown}^{(i+1)}=T_{\triangledown}^{(i)}{\left(\frac{R\prime}{R^{(i +1)}}\right)^{3(k-1)}}\end{array} \right. \tag{14}\]  
For bubbles containing noncondensable gas, the solution process could be simplified. The variation in the partial pressure of the noncondensable gas could be solved directly according to the polytropic process Equation (15). Then, the temperature of the noncondensable gas was obtained.