## 2 Methodology
### Fluid-Structure Interaction Equations  
The ANSYS finite element (FE) package is used to solve the fluid structure interaction simulation. While the simulation is in process, the solver simultaneously uses a mixed discretization with Lagrangian-Eulerian (ALE) formulas. The Lagrangian and Eulerian formulations have been used for deformable walls and fluid domains, respectively. Details of the FSI formulations are available elsewhere [11]. The visco-elastic model draws on the Kelvin-Voigt visco-elastic model shown in Figure 2.  
Figure 1: Geometric representation of alveolar; (**a**) model geometry [12]; (**b**) closed form; (**c**) opened form. All dimensions are based on \(\upmu\)m.  
Figure 2: The visco-elastic Kelvin–Voigt model.  
The Kelvin-Voigt model has both elasticity and viscosity properties simultaneously. A purely viscous damper and a purely elastic spring are connected in parallel in the Kelvin-Voigt model (see Figure 2).  
\[\sigma_{\text{Total}}=\sigma_{\text{D}}+\sigma_{\text{S}} \tag{2}\]  
\[\varepsilon_{\text{Total}}=\varepsilon_{\text{D}}=\varepsilon_{\text{S}} \tag{3}\]  
\[\sigma_{ij}^{\text{s}}=\text{E}\epsilon_{ij}+\eta\frac{\partial\varepsilon_{ ij}}{\partial t} \tag{4}\]  
\[\varepsilon_{ij}=\frac{1}{2}(\frac{\partial d_{i}}{\partial z_{j}}+\frac{ \partial d_{j}}{\partial z_{i}}) \tag{5}\]  
In Equation (4), \(\eta\) is the viscosity. The boundary conditions at the interface regions between the two domains must be satisfied by coupling the fluid and structure domains at FSI modeling technique as shown below.  
\[d_{i}^{\text{f}}=d_{i}^{\text{s}} \tag{6}\]  
\[\mathbf{n}_{j}\sigma_{ij}^{\text{s}}=\mathbf{n}_{j}\sigma_{ij}^{\text{f}} \tag{7}\]  
\[\sigma_{ij}^{\text{f}}=-p\delta_{ij}+\mu(\frac{\partial v_{i}}{\partial z_{j} }+\frac{\partial v_{j}}{\partial z_{i}}) \tag{8}\]  
\[\mathbf{n}_{i}v_{i}=\mathbf{n}_{i}\frac{\partial d_{i}^{\text{s}}}{\partial \mathbf{t}} \tag{9}\]  
In Equation (4) the fluid and structure nodal displacements are shown by \(d_{i}^{\text{f}}\) and \(d_{i}^{\text{s}}\), respectively. In addition, \(\mathbf{n}_{j}\) is the interface normal vector and \(\sigma_{ij}\) is the Kronecker delta function. The fluid and solid stress tensors are shown by \(\sigma_{ij}^{\text{s}}\) and \(\sigma_{ij}\), respectively. The fluid velocity continuity condition in Equation (9) needs that the structure and fluid domains have equal nodal velocities at the movable fluid-structure boundary. The boundary conditions at the interface regions of fluid and structure domains must be satisfied by coupling the fluid and structure domains with the FSI technique [11]. For this, we set up a pair of coupled systems consisting of a Transient Structural system and a Fluid Flow (CFX) system for performing the two-way FSI analysis in ANSYS Workbench 16.  
In this research, we utilize an independent discretization model of the fluid and structure domains. Our structure meshes are suppressed during the CFX system set up by Transient Structural system, and consists of multi-block and body-fitted hexahedral elements. While the fluid meshes set up by the CFX system contain 8304 total elements and 4561 nodes, there are 7064 total elements and 25,341 nodes in the solid model. The FSI technique defines the interface between the structure domain in the Transient Structural system and the fluid domain in the CFX system. Data of solid and fluid conditions are interchanged across this interface during the implementation of the simulation.  
To ensure valid results, mesh independency for the obtained results was investigated by resolving the airflow field for a finer mesh with 1,104,000 cells for the 18 th generation. There was no noticeable difference between the fine and course meshes in terms of flow field and deposition of 1-5 \(\upmu\)m diameter particles. Specifically, the results were observed to vary approximately 0.7% and 0.2% for 1 and 5 \(\upmu\)m diameter particles, respectively.