The flow of the air in a human pharynx can be described by the stationary Navier-Stokes equations, which are a set of nonlinear partial differential equations of second order prescribing mass and momentum conservation of a fluid

$$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\nabla \cdot \left(\rho u\right)=0,\\ \rho u\cdot \nabla u=-\nabla p+\nabla \cdot \left[\mu \left(\nabla u+{\left(\nabla u\right)}^{\text{T}}\right)-\frac{2}{3}\mu \left(\nabla \cdot u\right)I\right],\end{array}$$(1)where *u* is the fluid’s velocity vector, *p* the pressure, *ρ* the fluid’s density and *μ* its dynamic viscosity. *I* denotes the identity matrix. The values for viscosity and density are chosen to be *μ* = 18.82 · 10^{−6} Pa s and *ρ* = 1.15 kgm^{−3} matching the physical properties of air with a temperature of *T* = 34° C, which was measured as mean temperature value in the nasopharynx in 50 volunteers [9].

To deal with turbulence effects that may occur, the Navier-Stokes equations are averaged over time to get the Reynolds Averaged Navier-Stokes (RANS) equations

$$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\nabla \cdot \left(\rho U\right)=0,\\ \rho U\cdot \nabla U=-\nabla p+\nabla \cdot \left[\mu \left(\nabla U+{\left(\nabla U\right)}^{\text{T}}\right)-\frac{2}{3}\mu \left(\nabla \cdot U\right)I\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\nabla \cdot \left(\overline{\rho u\prime \times u\prime}\right),\end{array}$$(2)where *U* and *P* denote the time averaged values of *u* and *p*. The RANS equations introduce additional unknowns *τ* = −*ρ***u**′ × *u*′, which are called Reynolds stresses. The Reynolds stresses are the time averaged cross correlation of the turbulent fluctuations *u*′ multiplied by *ρ*. To deal with this closure problem of having more unknowns than equations, auxiliary equations are introduced that model the transport of turbulent parameters. The Reynolds stresses are therefore replaced by the so called turbulent viscosity

$${\mu}_{T}=\rho {C}_{\mu}{k}^{2}{\epsilon}^{-1}=\rho k{\omega}^{-1},$$(3)where *k* is the turbulent kinetic energy, *∊* is the dissipation, *ω* is the specific dissipation rate and *C*_{μ} = 0.09 is an experimentally determined constant. By using model equations for the kinetic energy *k* and either modeling equations for the dissipation *∊* or the specific dissipation rate *ω* solutions for the time averaged values for velocity *U* and *P* pressure can be obtained. The free stream behavior of a flow is quite different to the near-wall flow, hence the modeling equations for the turbulent parameters are incorrect in the near-wall regime. Therefore wall functions are used that estimate the near-wall flow based on a logarithmic profile. The *k* − *∊* turbulence model by Launder and Sharma [10] as well as the *k* − *ω* turbulence model by Wilcox [11] where used within this study. For all different simulation conditions and domains (with and without MAA) an inlet volume flow of *u*_{in} = 400 ml s^{−1} was chosen and zero pressure at the outlet. All other boundaries were chosen to be no-slip walls. The simulations were performed with the commercially available simulation environment COMSOL Multiphysics^{©}.

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