# Numerical Analyses of Ultrasonic Atomization Utilizing Acoustic Effects of a Beam Diaphragm
## 4 Results and Discussions
with the hydrophilic diaphragm, atomization occurred, and the intensity of the signal-at-peak frequencies became smaller than that in the hydrophobic case. This tendency was qualitatively expressed by incorporating the interaction between the diaphragm and water film with viscosity effects. It was indicated through these results that one of the reasons for the reduction in the intensity of the signal in the experiment is that extra energy was dissipated in the water film due to the process of atomization. The quantitative relationship between atomization and energy dissipation by the interaction is to be investigated in further study. The conditions of atomization for the present experiments are discussed as follows. Based on the experiment in which the atomization was ceased, frequencies higher than 10.0 kHz, which are marked as \(\mathbf{e}\) and \(\mathbf{f}\) in Figures 4 and 10, appeared to have a close relationship to the conditions of atomization. Therefore, the diameters of the droplets formed by 10.0 kHz and 2.4 MHz of excitation frequencies were estimated by [PERSON]'s equation, which is expressed as a function of excitation frequency \(f\) as [8]  
\[d(f)=0.34\bigg{(}\frac{8\pi\sigma}{\rho f^{2}}\bigg{)}^{\frac{1}{3}}\]  
The diameters were 86.5 \(\upmu\)m for \(f\) = 10.0 kHz and 2.32 \(\upmu\)m for \(f\) = 2.4 MHz, respectively. The average size in the present experiment was 2.75 \(\upmu\)m and was close to the size for \(f\) = 2.4 MHz. This result indicates that atomization mainly occurred by a surface wave from the transducer. However, judging from the results of the hydrophobic cases in which no atomization occurred, the power transferred to the surface of the water film was not sufficient to initiate atomization by itself. It is hypothesized that the extra energy source that initiates atomization in hydrophilic cases is the interaction between the diaphragm and water film. As already mentioned in Section 1, [PERSON] et al. found that the acceleration threshold for atomization depends on the excitation frequency and surface tension for low-viscosity fluid and viscosity for viscous fluids [10]. For low-viscosity fluids, the acceleration threshold is expressed as  
\[\alpha_{th}(f_{atom})=0.261\bigg{(}\frac{\sigma}{\rho}\bigg{)}^{\frac{1}{3}}(2 \pi f_{atom})^{\frac{4}{3}} \tag{33}\]  
And for viscous fluids,  
\[\alpha_{th}(f_{atom})=1.306\ \
u^{\frac{1}{2}}(2\pi f_{atom})^{\frac{3}{2}} \tag{34}\]  
These acceleration thresholds \(\alpha_{th}\) were compared to maximum acceleration \(\alpha_{max}\) caused by the sub-harmonic waves in the present experiments with the following ratio. Figure 8:0  
By letting \(A_{0}\) be the amplitude of the oscillation, acceleration is expressed as  
Figure 8:1  
Therefore, the maximum amplitude is  
Figure 8:2  
In a case with frequency \(f\) = 10.0 kHz, which is marked as e in Figures 4 and 10,  
Figure 8:3where 10.0 \(\upmu\)m was used as amplitude \(A_{0}\) because the amplitude was estimated in the order of 10\({}^{1}\)\(\upmu\)m from the results of the present experiments. For the frequency \(f=6.2\) kHz, which is marked as **a** in Figures 4 and 10,  
Figure 8:4  
Although more precise measurements of the amplitude of the diaphragm should be conducted in a further study, these results indicate that oscillation by sub-harmonic wave with 10.0 kHz of frequency provides enough acceleration for the initiation of atomization, but the acceleration with 6.2 kHz of frequency is insufficient. This is one of the reasons why the intensity of the signal became lower, mainly in a frequency range higher than 10 kHz, in the experiment where the atomization ceased, as shown in Figure 8:5 In summary, the results of the experiments and numerical analyses in the present study showed the possibility that atomization occurred when the superposed accelerations of the primary wave from the ultrasonic transducer and the extra acceleration due to interaction between the oscillating diaphragm and water film exceeded the acceleration threshold for atomization.