## 1 Introduction  
The research into the interaction between shock waves generated by hypersonic airflow and liquid droplets is involved in various fields, including liquid atomization, multiphase combustion, and discrete multiphase flow. Meanwhile, such issues also have practical applications in daily life and engineering. For example, during high-speed aircraft flights through thunderstorms, investigations on the erosive effects of raindrops on the aircrafts' surface coatings have shown reduced aircraft corrosion [1]. In natural disaster prevention, the dispersion of inert liquid substances under shock wave influences has ensured their wide distribution, effectively preventing fires and their spread. Moreover, due to the complex shock waves present in a combustion chamber when a hypersonic aircraft is in operation, the fragmentation and atomization of liquid droplets after interactions with shock waves affect the subsequent combustion stages and determine an engine's performance [2]. Consequently, the investigation of liquid droplet deformation and fragmentation under shock wave influences holds substantial academic and engineering significance. The deformation and aerodynamic breakup of liquid droplets under shock wave influence represents a classic phenomenon in multiphase fluid dynamics. Initially, research was primarily based on shock tube experiments and focused on the deformation and breakup of Newtonian fluid droplets in low-velocity flow conditions. The goal was to characterize various droplet breakup mechanisms, measure breakup times, and quantitatively assess the impacts of a shock wave's intensity and viscosity ratio, among other parameters [3; 4; 5; 6; 7; 8; 9; 10]. Theoretically, the Weber number (_We_), which represents the ratio of disruptive aerodynamic and restorative surface tensions (\(W_{e}=\rho_{g}{u_{g}}^{2}d_{o}/\sigma\)), and the Ohnesorge number (_O_\(h\)), which indicates the ratio of the liquid viscosity and surface tension (\(O_{h}=\mu_{I}/\sqrt{\sigma d_{o}\rho_{I}}\)), serve as pivotal parameters for describing the aerodynamic breakup of liquid droplets. It is commonly accepted that when _O_\(h<0.1\), the influence of liquid viscosity can be neglected so that the aerodynamic breakup process of liquid droplets is practically controlled by the Weber number [6]. [PERSON] and [PERSON] [7] classified five modes of liquid droplet aerodynamic breakup based on Weber numbers, which included vibrational breakup, bag breakup, multiple breakups, sheet-stripping breakup, and catastrophic breakup. Subsequently, as experimental techniques advanced, [PERSON] [11; 12; 13] utilized laser-induced fluorescence (LIF) to address the challenge of liquid droplet obscuration during observations. Their research revealed that at a high Weber number, the aerodynamic breakup of liquid droplets results from shear-induced interface detachment, thus refuting the conventional catastrophic breakup mechanism. Based on these findings they introduced two novel droplet breakup mechanisms: RTP (Rayleigh-Taylor piercing) and SIE (shear-induced entrainment). While experimental research on the interactions between shock waves and liquid droplets has continued to advance, it is noteworthy that liquid droplets often have relatively large initial diameters. For instance, [PERSON] [14] examined the breakup process of 2.7 mm water droplets under shock waves with Mach numbers ranging from 1.109 to 1.34. However, there remains a lack of systematic research on the aerodynamic breakup of smaller-scale liquid droplets, such as sub-millimeter-sized droplets, under the impact of a shock wave. In recent years, [PERSON] et al. [15] shifted their research focus to explore the impact of shock waves on sub-millimeter liquid droplets. They devised a spatiotemporal evolution model based on high-speed schlieren imaging for various modes of sub-millimeter liquid droplet fragmentation. More recently, [PERSON] et al.