# Numerical Analyses of Ultrasonic Atomization Utilizing Acoustic Effects of a Beam Diaphragm
## 3 Numerical Analyses
### Modeling of Interaction Between Diaphragm and Water Film  
While the oscillating body that consists of the diaphragm and water film in the experiment was under steady-state vibration, the water film was just attached to the surface of the oscillating diaphragm without macroscopic change in its shape and thickness. Therefore, the displacement of material points in the water film is assumed to be small. Based on this assumption, an approximation of the Navier-Stokes equation is conducted to model the interaction between the diaphragm and water film as follows.  
In fluid dynamics, the principle of energy conservation is written as a Navier-Stokes equation. For an incompressible Newtonian fluid, the equation is expressed with \(\mathbf{v}\) representing the fluid's velocity, \(\rho\) as its mass density, and \(\mu\) as its viscosity [30]:  
\[\rho\frac{D\mathbf{v}}{Dt}=\rho\Big{\{}\frac{\partial\mathbf{v}}{\partial t}+( \boldsymbol{\
abla}\boldsymbol{\cdot}\mathbf{v})\mathbf{v}\Big{\}}=- \boldsymbol{\
abla}\mathbf{p}+\mu\boldsymbol{\
abla}^{2}\mathbf{v}+\rho \mathbf{f_{g}} \tag{25}\]  
where \(\mathbf{p}\) and \(\mathbf{f_{g}}\) are pressure and external acceleration, such as gravity, respectively.  
In this equation, the time rate of change in fluid velocity \(\mathbf{v}\) is expressed as a material derivative. When the displacement of the material points is small, to the extent that the Eulerian derivative approximately equals the Lagrange derivative, as expressed as \(\frac{D\mathbf{v}}{Dt}\approx\frac{\partial\mathbf{v}}{\partial t}\), the Equation (25) can be rewritten as follows:  
\[\rho\frac{d^{2}\mathbf{u}}{dt^{2}}-\mu\boldsymbol{\
abla}^{2}\frac{d\mathbf{u }}{dt}-\rho\mathbf{f_{g}}+\boldsymbol{\
abla}\mathbf{p}=\mathbf{0} \tag{26}\]  
where \(\mathbf{u}\) is a displacement vector of the material points.  
This equation can be translated to the following weak form in the same way that Equation (5) is derived.  
\[\int_{\Omega}\mathbf{W}^{T}\rho\frac{d^{2}\mathbf{u}}{dt^{2}}d\Omega+\int_{ \Omega}\mathbf{W}^{T}\mu\boldsymbol{\
abla}^{2}\frac{d\mathbf{u}}{dt}d\Omega+ \int_{\Omega}\mathbf{W}^{T}\rho\mathbf{f_{g}}d\Omega=\mathbf{0}\;\square\, \forall\mathbf{W}^{T} \tag{27}\]  
After the finite element discretization for the equation above, a system of equations is obtained as follows.  
\[\sum_{\epsilon=1}^{n}\mathbf{W}_{d}^{T}\bigg{(}\int_{\Omega}\mathbf{N}^{T}\rho \mathbf{N}\frac{d^{2}\mathbf{u}_{d}}{dt^{2}}d\Omega+\int_{\Omega}\mathbf{N}^{ T}\boldsymbol{\
abla}_{s}^{T}\mu\mathbf{\Gamma}\boldsymbol{\
abla}_{s}\mathbf{N} \frac{d\mathbf{u}_{d}}{dt}d\Omega+\int_{\Omega}\mathbf{N}^{T}\rho\mathbf{f_{g} }d\Omega\bigg{)}=\mathbf{0} \tag{28}\]  
Then, the following equation in a similar form is obtained, as already derived for the case of solid dynamics in Equation (13).  
\[\mathbf{M}_{f}\frac{d^{2}\mathbf{u}_{d}}{dt^{2}}+\mathbf{C}_{f}\frac{d\mathbf{ u}_{d}}{dt}+\mathbf{F}_{f}=\mathbf{0} \tag{29}\]  
where  
\[\mathbf{M}_{f}=\sum_{\epsilon=1}^{n}\int_{\Omega}\mathbf{N}^{T}\rho\mathbf{N}d\Omega \tag{30}\]  
\[\mathbf{C}_{f}=\sum_{\epsilon=1}^{n}\int_{\Omega}\mathbf{N}^{T}\boldsymbol{ \
abla}_{s}^{T}\mu\mathbf{\Gamma}\boldsymbol{\
abla}_{s}\mathbf{N}d\Omega \tag{31}\]  
\[\mathbf{F}_{f}=\sum_{\epsilon=1}^{n}\int_{\Omega}\mathbf{N}^{T}\rho\mathbf{f_{ g}}d\Omega \tag{32}\]  
It is important to note that this system of equations can be solved by a finite element solver for solid mechanics, with simple efforts on a user subroutine for the viscosity matrix \(\mathbf{C}_{\text{f}}\). Therefore, responses of the diaphragm and water film, which were composed of an oscillating body, were simultaneously solved with this formulation in the present study.