## 4 Conclusions  
In this study, we conducted experimental investigations into the aerodynamic breakup of sub-millimeter liquid droplets with initial diameters of 0.42 mm under varied shock wave conditions. It quantitatively assessed the development of Rayleigh-Taylor and Kelvin-Helmholtz instabilities at different Weber numbers and initial diameters. Furthermore, three-dimensional numerical simulations, based on experimental parameters, were conducted to examine the evolution of the sub-millimeter liquid droplets and the surrounding flow fields in a supersonic post-shock gas velocity environment. The analysis centered on the deformation characteristics of these droplets and the gas-liquid interface evolution during aerodynamic breakup. The present work yielded the following conclusions:  
For constant initial diameters, the growth rates of the R-T and K-H instability waves on the sub-millimeter liquid droplets' surfaces initially increased and then decreased with varying wavenumbers. Moreover, with an increase in the Weber number, both types of instability waves exhibited inclinations towards higher growth rates and smaller scales. The differences in the growth rate extremes were notably more pronounced for the K-H instability compared to the R-T instability.  
At a constant Weber number, decreasing the initial diameter of sub-millimeter liquid droplets led to gradual increases in both the maximum growth rate and the wavenumbers of the Rayleigh-Taylor and Kelvin-Helmholtz instability waves. Additionally, as the Weber number increased, the disparities between the maximum growth rates and the wavenumbers of the R-T instability waves in the sub-millimeter liquid droplets with varying initial diameters decreased, while the K-H instability waves exhibited an increasing trend.  
The numerical simulations revealed that, under the supersonic gas velocities behind a Mach 2.12 shock, the robust shear forces at the gas-liquid interface induced the formation of K-H instability waves. During this phase, the sub-millimeter liquid droplets underwent aerodynamic fragmentation, adhering to the SIE mechanism and comprising the following two sequential phases: flattening and shear stripping. The flattening phase deformation arose from the pressure gradients between the high-pressure regions ahead of the sub-millimeter liquid droplets and the low-pressure zones encircling their equators. In the shear stripping phase, fragmentation occurred due to the continual development of vortex structures on the leeward sides, which resulted from the flow separations at the equatorial planes.  
Conceptualization, Y.W. and R.D.; data curation, Y.Z.; formal analysis, Y.W.; investigation, Y.Z.; methodology, S.L.; supervision, R.D.; writing--original draft, Y.W.; and  
Figure 16: Evolution of the non-dimensional axial widths and transverse widths of the sub-millimeter liquid droplets for an _Ma_ value of 2.12.  
writing--review and editing, R.D. All authors have read and agreed to the published version of the manuscript.  
This study was funded by the Zhejiang Provincial Basic Public Welfare Research Program Project (grant no. LGG19A020002).  
Not applicable.  
**Data Availability Statement:** Data is contained within the article.  
The authors declare no conflict of interest.