## 2 Instability of Shock Wave-Sub-Millimeter Liquid Droplet Interfaces
### The Influence of Weber Number and Initial Diameter on the R-T Instability  
Under low Weber number conditions, initial perturbations manifested on the surfaces of the sub-millimeter liquid droplets prior to their fragmentation. These perturbations primarily underwent small-scale development, rendering their observation challenging. Nevertheless, with the ongoing interaction between the high-speed airflow and the droplet, the R-T instability underwent progressive evolution, culminating in the aggregation of large-scale R-T waves. The fundamental reason for the formation of one or more bag-like structures during bag breakup (\(We=19\)) and multiple breakup (\(We=42\)) was the continued piercing of the R-T instability waves, as depicted in Figure 1. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(Ma\)} & \(\mathbf{p_{2}/}\) & \multirow{2}{*}{\(u_{g}\)/(m/s)} & \multirow{2}{*}{\(\rho_{g}\)/(kg/m\({}^{3}\))} & \multirow{2}{*}{\(T_{2}\)/(K)} & \multirow{2}{*}{\(We\)} \\  & (Pa) & & & & \\ \hline
1.09 & 123,701 & 48.83 & 1.38 & 315.52 & 19 \\
1.13 & 138,337 & 69.34 & 1.47 & 323.12 & 42 \\
1.27 & 174,391 & 136.58 & 1.76 & 349.44 & 196 \\
1.49 & 247,241 & 231.75 & 2.22 & 391.82 & 713 \\
2.12 & 514,480 & 466.5 & 3.4 & 533.64 & 4272 \\ \hline \hline \end{tabular} In the table, \(p_{2}\), \(u_{g}\), \(\rho_{g}\), and \(T_{2}\) represent the post-shock airflow pressure, velocity, density, and temperature, respectively.  
\end{table} 
Table 1: Experimental operating conditions. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Properties**} & \multicolumn{1}{c}{**Symbol**} & \multicolumn{1}{c}{**Value**} \\ \hline Water surface tension & \(\sigma\) & 0.0728 N/m \\ Water density & \(\rho_{l}\) & 998 kg/m\({}^{3}\) \\ Water viscosity & \(\mu_{l}\) & \(1.003\times 10^{-3}\) Pa/s \\ Air temperature & \(T_{1}\) & 298 K \\ Air viscosity & \(\mu_{g}\) & \(1.79\times 10^{-5}\) Pa/s \\ Air pressure & \(p_{1}\) & 101.325 kpa \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Pre-shock air and water droplet properties. Following the passage of a shock wave over a sub-millimeter liquid droplet, the droplet was immediately exposed to high-speed post-shock airflow. Assuming exponential growth (\(e^{it}\)) in the surface amplitude of the droplet, the dispersion equation for the R-T instability waves can be expressed as follows [28; 29; 30; 31]:  
\[n_{RT}=-k_{RT}{}^{2}\frac{\mu_{l}+\mu_{g}}{\rho_{l}+\rho_{g}}+(k_{RT}\frac{\rho_ {l}-\rho_{g}}{\rho_{l}+\rho_{g}}a-k_{RT}{}^{3}\frac{\sigma}{\rho_{l}+\rho_{g}}+ k_{RT}{}^{4}(\frac{\mu_{l}+\mu_{g}}{\rho_{l}+\rho_{g}}){}^{2}){}^{0.5}, \tag{2}\]  
where \(n_{RT}\) is the growth rate of the R-T instability waves, \(k_{RT}\) is the wavenumber of the R-T instability waves, \(\mu_{l}\) and \(\mu_{g}\), respectively, represent the liquid viscosity and gas viscosity, and \(a\) is the acceleration of the liquid droplet's center of mass, which is expressed as follows:  
\[a=\frac{\Delta l_{2}-\Delta l_{1}}{\Delta t^{2}}, \tag{3}\]  
where \(\Delta l_{1}\) and \(\Delta l_{2}\) represent the displacements of the sub-millimeter liquid droplets within the time intervals shown in Figure 2.  
Figure 3 illustrates how the growth rate of a sub-millimeter liquid droplet (with an initial diameter of 0.42 mm) in R-T instability waves changed as a function of the wavenumber. Initially, the growth rate of R-T instability waves increased with the wavenumber, followed by an overall decrease. This occurred due to the aerodynamic breakup of the sub-millimeter liquid droplets, resulting in the continuous detachment of sub-droplets from the main droplet. This resulted in droplet sizes that were insufficient to support the growth of the R-T instability waves. As the Weber number parameter increased, the extremum shifted toward the upper right. This indicated that the surface R-T instability waves of the sub-millimeter liquid droplets tended to develop in the direction of the increasing wavenumbers, decreasing scales, and increasing growth rates.