## 2 Methodology
When the thermodynamic conditions of the fluid are close to the critical point, the pressure and density become very sensitive to disturbances, and the ideal gas equation of state is no longer applicable. Therefore, the real gas equation of state that takes into account the true physical effects becomes especially important when studying the behavior of a fluid approaching its critical point. In order to validate the model for nitrogen, we carried out an equation of state verification of nitrogen using MD and compared the results with the modified Soave-Redich-Kwong (SRK) equation of state. Here, we provide a brief review of the SRK equation. The SRK equation of state is a commonly used semi-empirical state equation and its correctness has been verified by researchers in the past [20; 21; 22]. [PERSON] [23] introduced the eccentricity factor to modify the Redich-Kwong equation of state and obtained the three-parameter SRK equation, which has the following form:  
\[p=\frac{RT}{V-b}-\frac{a\alpha}{V(V+b)} \tag{10}\]  
where \(p\) is the pressure, \(R\) is the universal gas constant, \(V\) is the molar volume and \(a\), \(b\) and \(\alpha\) are the adjustable parameters, defined as follows:  
\[a=0.42747\frac{R^{2}T_{c}^{2}}{p_{c}} \tag{11}\]  
\[b=0.08664\frac{RT_{c}}{p_{c}} \tag{12}\]  
\[\alpha=(1+(0.48508+1.55171\omega-0.15613\omega^{2})(1-T_{r}^{0.5}))^{2} \tag{13}\]  
where \(T_{c}\) is the critical temperature and \(P_{c}\) is the critical pressure. \(\omega\) is the eccentric factor of molecules. \(T_{r}=T/T_{c}\) is the reduced temperature. For nitrogen, the critical temperature and critical pressure are 126 K and 3.4 MPa, respectively. We followed [PERSON]'s method [24] to obtain the growth rate of the liquid spray. It is well-known that the inviscid jet model was firstly proposed by [PERSON] [25] using linear stability theory. In his theory, the development model of the perturbation \(\eta(z,t)\) is as follows:  
\[\eta(z,t)=\eta_{0}\sin(kz+\phi)e^{\alpha t} \tag{14}\]  
where \(z\), \(t\), \(k\) and \(\phi\) are the position, time, wavenumber and phase, respectively. \(\eta_{0}\) is the initial perturbation amplitude, \(\alpha\) the spatial growth rate. The jet profile radius \(r_{i}(z)\) obtained by the simulation can be expressed in the following form:  
\[r_{i}(z)=\frac{d}{2}+\eta_{0}\sin(kz+\phi_{i})e^{\alpha z/U} \tag{15}\]  
where \(d\) is the jet diameter. After the dimension is reduced by \(d\) and [PERSON]'s time \(t_{R}=\sqrt{(\rho d^{3})/(8\gamma)}\), with \(\rho\) being the density and \(\gamma\) the surface tension, Equation (15) turns to:  
\[\overline{r}_{i}(\overline{z})=\frac{1}{2}+\overline{\eta}_{0}\sin(\overline {k}\overline{z}+\phi_{i})e^{\overline{\alpha}/\overline{U}} \tag{16}\] \begin{table}
\begin{tabular}{c c c c c} \hline
**Interaction** & \(\sigma\) (\(\hat{\mathbf{A}}\)) & \(\varepsilon\) (\(\mathbf{eV}\)) & \(\alpha\) & \(\beta\) \\ \hline N-Au & 4.718 & \(7.31\times 10^{-5}\) & 0.14 & 0.10 \\ N-N & 3.789 & \(6.15\times 10^{-3}\) & - & - \\ Au-Au & 2.640 & 0.4432 & - & - \\ \hline \end{tabular}
\end{table} 
Table 1: Lennard-Jones (L-) potential parameters used for the force routine. where \(\overline{\eta}_{0}=\eta_{0}/d\), \(\overline{k}=kd\), \(\overline{z}=z/d\), \(\overline{\alpha}=\alpha t_{R}\) and \(\overline{U}=Ut_{R}/d\). The standard deviation of surface fluctuations can be expressed as:  
\[\overline{\sigma}(\overline{z})=\frac{1}{\sqrt{N}}\sqrt{\sum_{i=1}^{N}\left( \overline{r}_{i}(\overline{z})-<\overline{r}_{i}(\overline{z})>\right)^{2}} \tag{17}\]  
where \(<\overline{r}_{i}(\overline{z})>\) is the average of \(\overline{r}_{i}(\overline{z})\) at each position. It is assumed that the jet profiles are not correlated in time, and each phase obeys a uniform distribution on [0, 2\(\pi\)].