## 2 Structure and Theoretical Analysis of the Piezoceramic Vibrating Mesh Atomizer
### Theoretical Analysis  
As indicated by the results in a previous study, when driven by an AC signal, the micro-tapered aperture experiences constant variations in its volume and tapered angle as well as its forward and reverse flow resistance. Such periodic fluctuation of the micro-tapered aperture's volume, combined with the difference between the forward and reverse flow resistance, create a pumping effect, which causes the one-way flow of liquid and thus results in atomization. [PERSON] et al. [32] compared the numerical calculation of tapered flow with the results obtained in [33]. This indicated that the characteristics of flow resistance in micron-sized divergence/convergence were similar to those for normal sizes. For this reason, the atomization process of the liquid was considered as the working process of a valveless piezoelectric pump [34]. In another way, the variation of the micro-tapered aperture volume is considered to be changed with a piezoelectric vibrator volume in the tapered flow tube valveless piezoelectric pump, and the micro-tapered aperture is processed as the tapered flow tube of the tapered flow tube valveless piezoelectric pump. Driven by a periodic signal, the micro-tapered aperture experiences dynamic variations, and subsequently forms a micron-sized tapered flow tube valveless piezoelectric pump.  
For the purpose of this study, as shown in Figure 2a, the liquid flow from the inlet to outlet was defined as forward flow, and the liquid flow from the outlet to inlet as reverse flow is also denoted. According to this definition, the forward and reverse flow resistances of the micro-tapered aperture are represented as \(\xi\)+ and \(\xi\)-, respectively. Considering that the tapered angle of the micro-tapered aperture and its forward-reverse flow resistance varies constantly, their relationship must be expressed  
Figure 1: Illustration of the atomizer structure with a piezoceramic vibrating mesh atomizer [31].  
in the form of mean flow resistance. The forward mean flow resistance and reverse mean flow resistance of the micro-tapered aperture are shown as follows:  
\[\overline{\xi(\chi)}_{+}=\frac{\int\limits_{\chi\rightarrow\infty}\xi(x)_{+}d \chi}{\chi} \tag{1}\]  
\[\overline{\xi(\chi)}_{-}=\frac{\int\limits_{\chi\rightarrow\infty}\xi(x)_{-}d \chi}{\chi} \tag{2}\]  
where and respectively represent the forward and reverse transient flow resistance of the micro-tapered aperture.  
According to the previous study [31], the variation of the micro-tapered aperture volume can be expressed as:  
\[\begin{array}{ll}\Delta V_{d}&=\iint\limits_{\Omega}[2z(f_{yy}+f_{xx}+f_{y} ^{2}f_{xx}\\ &-2f_{x}f_{y}f_{xy}+f_{x}^{2}f_{yy})m^{-1}]dV\end{array} \tag{3}\]  
Thus, the atomization rate incurred by the pumping effect of the micro-tapered aperture can be expressed as:  
\[Q=\Delta V_{d}fn\frac{\overline{\xi(\chi)}_{+}-\overline{\xi(\chi)}_{-}}{2+ \overline{\xi(\chi)}_{+}+\overline{\xi(\chi)}_{-}} \tag{4}\]  
where \(\Delta V_{d}\) represents the variation of the micro-tapered aperture volume in a given period; f represents the driving frequency of the piezoelectric vibrator; and n represents the number of micro-tapered apertures on an atomizing film. As seen from the above formula, the variation of the micro-tapered aperture volume, driving frequency, and the flow resistances of the micro-tapered apertures all exert some influence on the atomization rate.