# The Development and Analysis of a Multistage Spraying Method for Liquids in an Ultrasonic Field
## 1 Introduction
The apparent correlation between the median droplet diameter and the wavelength of capillary waves is considered to confirm the capillary wave theory. Cavitation bubbles are assumed to be generated within a liquid film when exposed to ultrasound, provided that the liquid film on the vibrating surface has a certain minimum thickness. High-intensity pressure surges are generated upon the implosive collapse of these cavities, especially with the implosion of cavities near the liquid surface. These pressure surges initiate the disintegration of the liquid film and cause the direct release of droplets. In this case, the size of the shaped droplets is also proportional to the size of the cavitation bubbles generated, i.e., the frequency of ultrasonic impact. [PERSON] and [PERSON] [17] combined these two theories and proposed the conjunctive theory. According to this theory, periodic pressure surges from cavitation disturbances generate and interact with capillary waves of finite amplitude and destroy and excite them with the generation of droplets. It should also be noted that cavitation shock (which is essentially a statistically random phenomenon) results in non-uniform random disintegration, and the observed non-uniformity of droplet distribution confirms this theory. Since most droplets are formed through capillary wave generation, the droplet diameter can be estimated fairly accurately using [PERSON]'s equation:  
\[D_{1}=0.34\lambda=0.34\sqrt[3]{\frac{8\pi\sigma}{\rho f^{2}}} \tag{1}\]  
The use of this equation is justified by the fact that the diameter of aerosol droplets is determined by the frequency of ultrasonic oscillations applied to the liquid film. As the frequency increases, the diameter of the generated droplets decreases [18; 19]. Clearly, the spraying productivity decreases proportionally, which is described by the following well-known equation [20]:  
\[P_{1}=2\frac{f}{\lambda^{2}}\pi\frac{D_{1}{}^{3}}{6} \tag{2}\]It follows from the presented expressions that the productivity is inversely proportional to the spraying frequency. In practice, the dependence on frequency is not linear, which results in a more significant decrease in spraying productivity. So, a number of researchers have shown that with an increase in vibration frequency from 22 kHz to 80 kHz, the median droplet diameter decreases from 93 \(\upmu\)m to 30 \(\upmu\)m (by threefold). But the spraying productivity upon practical implementation of the process decreases by more than 12-fold (from 3 mL/s to 0.25 mL/s) [21]. Thus, the key challenge of the ultrasonic spraying technique in a liquid layer is the fundamental limitation of productivity with increasing dispersity of the generated aerosol. Since all existing ultrasonic spraying techniques do not facilitate the generation of finely dispersed aerosols with high productivity, this significantly limits the application scopes for solving many problems of contemporary high-tech industries. In this connection, a need arises to develop new ways of implementation of ultrasonic spraying, providing an increase in the productivity of generating a finely dispersed liquid phase.