# An All-Mach Number HLLC-Based Scheme for Multi-Phase Flow with Surface Tension
## Appendix A Derivation of Thermal EOS  
The expression for the thermal EOS is derived as follows. The internal energy is first written as a function of pressure and specific volume \(v\):  
\[e(v,T)=\bigg{[}\frac{p(v,T)}{\gamma-1}+\frac{\gamma p_{\infty}}{\gamma-1} \bigg{]}v. \tag{1}\]  
By differentiating the internal energy with respect to (w.r.t.) the temperature at constant volume, the following relationship is obtained:  
\[\bigg{(}\frac{\partial e}{\partial T}\bigg{)}_{v}=\frac{v}{\gamma-1}\bigg{(} \frac{\partial p}{\partial v}\bigg{)}_{v}. \tag{2}\]  
The internal energy is also differentiated w.r.t. specific volume at constant temperature, yielding  
\[\bigg{(}\frac{\partial e}{\partial v}\bigg{)}_{T}=\frac{v}{\gamma-1}\bigg{(} \frac{\partial p}{\partial v}\bigg{)}_{T}+\frac{p(v,T)+\gamma p_{\infty}}{ \gamma-1}. \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \(\frac{R_{0}}{\Delta x}\) & \(\varepsilon_{r}(R_{min})\%\) & \(\varepsilon_{r}(T)\%\) & \(\varepsilon_{r}(R_{r})\%\) \\ \hline
20 & 4.90 & 2.19 & 0.001522 \\
40 & 2.43 & 0.91 & 0.001405 \\
80 & 1.23 & 0.49 & 0.001375 \\ \hline \end{tabular}
\end{table}
Table 2: Rayleigh–Plesset problem (curvilinear mesh (circular domain)): error in minimum radius (\(R_{min}\)), time period (\(T\)) and radius of rebound (\(R_{r}\)).  
[PERSON]'s rule relates the internal energy to the temperature, pressure and specific volume as follows:  
\[\left(\frac{\partial e}{\partial v}\right)_{T}=T\bigg{(}\frac{\partial p}{ \partial T}\bigg{)}_{v}-p. \tag{10}\]  
By substituting Equation (11) into Equation (10), the following expression is obtained:  
\[\left(\frac{\partial p}{\partial v}\right)_{T}=\frac{(\gamma-1)T}{v}\bigg{(} \frac{\partial p}{\partial T}\bigg{)}_{v}-\frac{\gamma(p(v,t)+p_{\infty})}{v}. \tag{12}\]  
Further, substituting for the specific heat capacity at constant volume, \(c_{v}=\left(\frac{\partial e}{\partial T}\right)_{v}\) into Equation (10), leads to  
\[\left(\frac{\partial p}{\partial T}\right)_{v}=\frac{(\gamma-1)c_{v}}{v}. \tag{13}\]  
Integrating with respect to T yields  
\[p(v,T)=\frac{(\gamma-1)c_{v}T}{v}+\beta(v). \tag{14}\]  
Next, differentiating the above with respect to the volume at constant temperature leads to  
\[\left(\frac{\partial p}{\partial v}\right)_{T}=-\frac{(\gamma-1)c_{v}T}{v^{2 }}+\frac{d\beta}{dv}, \tag{15}\]  
Substituting Equations (13) and (14) into (12) leads to the following expression:  
\[\left(\frac{\partial p}{\partial v}\right)_{T}=-\frac{(\gamma-1)c_{v}T}{v^{2 }}-\frac{\gamma}{v}(\beta(v)+p_{\infty}). \tag{16}\]  
Substituting Equation (16) into Equation (15) yields  
\[\frac{d\beta}{dv}+\frac{\gamma}{v}\beta(v)=-\frac{\gamma}{v}p_{\infty} \tag{17}\]  
Using an integrating factor such as \(e^{\int\frac{\gamma}{v}dv}\), an expression for \(\beta(v)\) is obtained as follows:  
\[\beta(v)=\frac{A}{v^{\gamma}}-p_{\infty}, \tag{18}\]  
where \(A\) denotes the constant of integration.  
Then, substituting Equation (18) into Equation (14), an expression for \(p(v,T)\) is derived as  
\[p(v,T)=\frac{(\gamma-1)c_{v}T}{v}+\frac{A}{v^{\gamma}}-p_{\infty}. \tag{19}\]  
Using a reference state \((p_{0},T_{0},v_{0})\), an expression for \(A\) is obtained as  
\[A=\bigg{[}p_{0}+p_{\infty}-\frac{(\gamma-1)c_{v}T_{0}}{v_{0}}\bigg{]}v_{0}^{ \gamma}. \tag{20}\]  
Re-expressing the above equation in terms of the reference density \(\rho_{0}\) yields  
\[A=[p_{0}+p_{\infty}-(\gamma-1)c_{v}T_{0}\rho_{0}]\rho_{0}^{-\gamma}. \tag{21}\]  
Finally, substituting Equation (21) into Equation (14) and re-expressing in terms of density, the equation for temperature reads  
\[T=\frac{1}{\rho c_{v}}\bigg{[}\frac{p+p_{\infty}-A\rho^{\gamma}}{\gamma-1} \bigg{]}. \tag{22}\]where the nomenclature has previously been defined.