# Numerical Analyses of Ultrasonic Atomization Utilizing Acoustic Effects of a Beam Diaphragm
## 1 Introduction  
Ultrasonic atomization is a process that creates small-scale particles from fluids by vibration with a frequency higher than the upper audible limit of hearing. Since it was first reported in the literature about a century ago [1], its application has spread into various fields, such as the fabrication of fine metallic powders in the material industry [2, 3], inhalation therapy in the field of medicine, solid-state batteries in the field of electronics, and fuel injection and cooling systems in the mechanical industry [4, 5]. Mechanisms of ultrasonic atomization have been discussed over a long period of time. [PERSON] [6] described the process of atomization as a result of a capillary wave at the surface of a liquid, based on equations of motion for oscillatory systems containing a liquid with a free surface. The hypothesis proposed in this theory was experimentally verified by consideration of the instability of the surface of a fluid, known as the Faraday wave. In a study by [PERSON] [7], a method for measuring the viscosity of a liquid from the wavelength of the capillary wave was developed, and an amplitude threshold for atomization was introduced. [PERSON] derived a formula for the size of droplets from the atomization based on the wavelength of the capillary wave, which was experimentally verified [8]. Another formula for droplet size, based on hydrodynamic instability, was proposed by [PERSON] and [PERSON] [9]. Understanding the mechanisms of atomization has progressed due to a lot of experimental studies. Among them, [PERSON] et al. focused on the acceleration of the capillary wave as the atomization threshold [10], while other studies by scientists, including [PERSON] et al. [2], [PERSON] [7] and [PERSON] et al. [11], focused on amplitude as the atomization threshold. Based on the balance of energy for capillary waves and the formation of droplets, [PERSON] [12] and [PERSON] and [PERSON] [13] proposed functions for the prediction of the droplets' diameter, which is of a similar form to [PERSON]'s formula [8]. Moreover, [PERSON] et al. proposed an equation for the distributions of droplet size [14]. In studies by [PERSON] et al. [15,16], attention was directed to the contact angle between sessile drops and an oscillatory substrate to further investigate the energy transfer for the onset of atomization. It was reported by [PERSON] et al. that the process of atomization is related not only to the capillary waves but also to cavitation, which is the formation of small cavities in liquid due to a reduction in static pressure [17, 18, 19]. The conditions for atomization with and without cavitation were recently discovered by [PERSON] [20]. A number of theoretical and numerical models have been proposed to investigate mechanisms of atomization. [PERSON] et al. proposed a theoretical model to describe the effect of Faraday instability on the process of atomization [21]. [PERSON] and [PERSON] developed a numerical solver of Navier-Stokes' equation for an axisymmetric problem to simulate vibration-induced droplet ejection [22]. [PERSON] et al. conducted numerical analyses using a finite element method with consideration of Faraday instability to predict critical vibrational amplitude for the onset of atomization [23,24]. A novel method of experiment to analyze the process of atomization utilizing acoustic signals from a beam diaphragm has been proposed by [PERSON] et al. [25,26]. In this method, the beam diaphragm is indirectly excited by an ultrasonic transducer through water as a transmission medium of the wave. The oscillating diaphragm emits amplified acoustic sounds with different frequencies depending on the state of atomization, which can be analyzed through the common procedure of fast Fourier transformation. The results of the experiments showed the necessary conditions for the onset and cessation of atomization that strengthened the hypotheses proposed until now. However, the physical meaning of the conditions has not been completely clarified.