## 1 Introduction  
The deformation and breakup of droplets under the action of shock waves are classical multiphase hydrodynamic phenomena, which hold significant practical applications and research value across various fields. As hypersonic vehicles continue to operate at increasingly high speeds, the demand for high-performance propulsion systems has surged. The combustion chamber of these systems, such as the scramjet, is afflicted by complex interactions between shock waves and fuel droplets, ultimately influencing engine performance [1]. Therefore, conducting an in-depth study of the breakup behavior under shock wave exposure and deciphering its underlying physical mechanism is critical for realizing its engineering potential. Since the 1950s, researchers have theoretically, experimentally, and numerically studied the droplet deformation and breakup process. [PERSON] [2] examined the deformation and breakup behavior of individual droplets subjected to gas flow, emphasizing the crucial role of the Weber number (_We_) in droplet deformation and breakup. Subsequent experiments have revealed a wide range of droplet breakup modes. [PERSON] et al. [3] and [PERSON] et al. [4] summarized the various droplet breakup modes and classified them into five categories: vibrational, bag, bag-and-stamen, stripping, and catastrophic breakup. According to [PERSON] et al. [5; 6; 7; 8], the primary factor affecting is Rayleigh-Taylor instability (R-T instability) when \(We<10^{2}\). In contrast, when \(We>10^{3}\), Kelvin-Helmholtz instability (K-H instability) emerges as a crucial factor determining droplet breakup. When \(10^{2}<We<10^{3}\), droplet breakup results from the combination of R-T instability and K-H instability. [PERSON] [9] proposed a method for determining aerodynamic drag based on the body velocity-relaxation parameter behind a shock wave. In some typical flow-free-body interaction scenarios that consider droplet deformation, the analytical form of the motion function was discovered. [PERSON] et al. [10] investigated the relationship between millimeter-sized droplet displacement in the gas flow direction and lateral deformation and time under high-speed gas flow induction with high-speed shadowgraph systems. They discovered that an increase in interfacial tension prevents droplet deformation and breakup, while gas-liquid density ratios have the opposite effect. [PERSON] et al. [11] employed high-speed schlieren systems to capture the deformation and breakup characteristics of submillimeter droplets under supersonic conditions. Additionally, they analyzed the spatial-temporal relationships of droplet motion parameters. [PERSON] et al. [12] recorded droplet deformation behavior induced by shock waves using high-speed shadowgraph systems. They observed that, in cases where there is a wide range of interacting inertia, the rate of droplet deformation is reduced due to the additional viscous forces provided by viscosity. As computational fluid dynamics and numerical simulation methods have advanced, it has become possible to obtain the pressure distribution and velocity distribution around liquid droplets. [PERSON] et al. [13] conducted two-dimensional numerical simulations of droplet deformation and breakup modes using the Lattice Boltzmann method. They found that increasing the Reynolds or Weber number resulted in greater droplet deformation and breakup. [PERSON] [14], [PERSON] [15,16], and [PERSON] [17] successively demonstrated that the pressure difference between the equatorial and anterior(posterior) stationary points is the main driver of droplet flattening. Thus the uneven pressure distribution on the droplet surface thereby leads to a difference in the deformation rate between the windward and leeward sides. [PERSON] et al. [18] performed numerical simulations to study the evolution of low-pressure regions inside droplets after their collision with shock waves. [PERSON] et al. [19] used the volume of fluid (VOF) method to resolve the phase interface and the large eddy simulation (LES) model to describe turbulence.