# Innovative Acoustic-Hydraulic Method for High-Performance Fine Liquid Atomization
## 3 Mathematical Model of the Acoustic-Hydraulic Nozzle
### Ultrasonic Exposure: Acoustic Cavitation Threshold  
In the cavitation zone of the structure, between the end of the emitter and the outlet, conditions for the development of cavitation were created. Even with a low intensity of ultrasonic exposure, bubbles filled with water vapor appear in the liquid in this area. The size of the bubbles depends significantly on the intensity of the sound field:  
\[D_{c}\sim\frac{2}{\omega}\sqrt{\frac{I}{\rho c}}. \tag{6}\]  
This expression is obtained from the following considerations. The amplitude of the displacement of liquid particles in a sound wave \(A\) is related to the ultrasound intensity \(I\) [W/m\({}^{2}\)], the speed of sound \(c\), and the frequency \(\omega\) by the known relationship: \(I=\frac{A^{2}\omega^{2}c}{2}\)[17]. When a wave passes through a liquid in the unloading phase, discontinuities of size \(A\) are formed in it, and in the compression phase, gas bubbles with a diameter \(D_{c}\) are formed at the site of these discontinuities. The diameter of the cavitation bubble corresponds to the size of the gap \(A\). From Equation (6), it follows that the higher the ultrasound intensity, the larger the diameter of the bubbles formed. In this case, with the development of cavitation, the wave resistance \(W_{l}=\rho\cdot c\) decreases significantly in accordance with the increase in the volume fraction of vapors in the liquid [17], which means that the diameter of the bubbles also increases. We consider the following qualitative physical picture of the development of cavitation. At a minimum intensity of ultrasound of \(I_{\min}\) in the liquid volume, cavitation bubbles in the amount of \(N_{0}\) are formed on existing inhomogeneities (cavitation nuclei), which, in accordance with Equation (6), will have a minimum size of \(\sim\)\(l_{\min}\)\({}^{1/2}\). Their number does not increase with increasing intensity, but their size, and, accordingly, volume gradually increase. The mode of developed cavitation occurs when a certain bubble size \(D_{cr}\) is reached. In this case, the volume of bubbles \(W_{g}\) becomes comparable to the volume of liquid \(W_{w}\), and the continuity of the medium is disrupted. In the mode of developed cavitation, as confirmed experimentally, the cavitation coefficient K = \(W_{g}/W_{w}\) remains unchanged and corresponds to approximately 0.2 \(\div\) 0.3 [18]. This limit is probably determined by the strength properties of the liquid. The volume of liquid in our case is determined as \(W_{w}=S_{1}\cdot L\). If the ultrasound intensity increases further, the size of the bubbles will continue to increase and their number will decrease. We denote the intensity corresponding to the mode of developed cavitation as \(l_{cr}\). We will define this regime as critical and find the conditions for its occurrence. With a further increase in intensity at a constant total volume of bubbles, their number \(N\) decreases in accordance with a power law with an exponent of 3/2. This means that the number of bubbles drops quickly at first and then more gradually. We can conditionally select an intensity value \(I_{\text{max}}\) such that the change in the number of bubbles relative to the change in intensity is small. That is, a further increase in intensity practically does not affect the number of bubbles in the liquid. It can be shown that the bubble diameter in this regime will be approximately e times greater than \(D_{cr}\), and the intensity \(I_{\text{max}}\sim\text{e}^{2}\,I_{cr}\). The above considerations correspond to the case of the absence of excess hydrostatic pressure in the liquid. The size of the bubbles, of course, also depends on the pressure in the liquid.