## 2 Instability of Shock Wave-Sub-Millimeter Liquid Droplet Interfaces
### The Influence of Weber Number and Initial Diameter on K-H Instability  
When air flows over the surface of a sub-millimeter liquid droplet, a tangential velocity difference between the gas-liquid phases induces shear forces that generate Kelvin-Helmholtz instability waves at the interface. These waves continue to develop over time. Figure 5 depicts the evolution of K-H instability waves. With an increasing Weber number, R-T instability is suppressed and the K-H instability progressively intensifies. Since K-H instability primarily affects the equatorial plane of a sub-millimeter liquid droplet and exerts a more pronounced influence compared to R-T instability, the process involved droplet shear and stripping, followed by rapid downstream travel. This was distinct from the bag and multiple breakup mechanisms observed at low Weber numbers. Figure 3: Relationship between the growth rate of the R-T instability waves and the wavenumber. The growth process of a K-H instability wave directly affects the aerodynamic breakup of a liquid droplet [20]. While investigating the theory of interface instability, [PERSON] [28], derived the following dispersion equation for planar K-H instability:  
\[n_{KH}=-{k_{KH}}^{2}\frac{\mu_{l}+\mu_{g}}{\rho_{l}+\rho_{g}}+(k_{KH}\frac{\rho_ {l}-\rho_{g}}{\rho_{l}+\rho_{g}}a+\frac{{k_{KH}}^{2}\rho_{l}\rho_{g}\big{(}u_{ gs}-u_{ls}\big{)}}{\left(\rho_{l}+\rho_{g}\right)^{2}}-{k_{KH}}^{3}\frac{\sigma}{ \rho_{l}+\rho_{g}}+{k_{KH}}^{4}(\frac{\mu_{l}+\mu_{g}}{\rho_{l}+\rho_{g}})^{2})^ {0.5}, \tag{4}\]  
Figure 4: Relationship between the R–T instability wave growth rate and the wavenumber for sub-millimeter droplets with different initial diameters. (**a**) We = 19. (**b**) We = 196. (**c**) We = 713. Figure 5: Growth and development process of K–H instability. where \(n_{KH}\) is the growth rate of the K-H instability waves, \(k_{KH}\) is the wavenumber of the K-H instability waves, and \(u_{gs}\) and \(u_{ls}\) are the surface air velocity and fluid motion velocity of the droplet, respectively. Figure 6 illustrates the relationship between the growth rate of K-H instability waves in 0.42 mm liquid droplets and wavenumber variations. Similar to the R-T instability waves, the growth rate of K-H instability waves initially increased and then decreased. Notably, the extremum of the K-H instability wave growth rate and the wavenumber variations were significantly more pronounced than they were in the R-T instability shown in Figure 3 (reaching a value of \(10^{6}\)). Moreover, as the Weber number increased, both the extremum of the K-H instability wave growth rate and the wavenumber steadily rose. This indicated that at low Weber numbers, K-H instability exhibited minimal growth, whereas under high Weber number conditions, the influence of K-H instability on sub-millimeter liquid droplet breakup surpassed that of R-T instability. In such cases, K-H instability predominated in sub-millimeter liquid droplet breakup. Figure 7 compares the growth rates of the K-H instability waves with the wavenumbers for sub-millimeter liquid droplets with various initial diameters but under constant Weber number conditions. As the initial diameters of the sub-millimeter liquid droplets decreased, the K-H instability wave curve showed similarities to the R-T instability wave curve. Both the peak growth rates and the corresponding wavenumbers increased gradually. This trend implied that, under constant Weber conditions, decreasing the droplet diameter led to the development of K-H waves on the droplets' surfaces at smaller scales and higher growth rates. This corresponded to the experimental observation that K-H instability waves on sub-millimeter liquid droplet surfaces had smaller sizes and led to faster droplet breakup. K-H instability wave growth is primarily associated with the velocity difference between the gas and liquid phases. With an increasing velocity difference, K-H instability develops rapidly.