## 2 Methodology
A computer with a dual-core processor at a speed of 3.0 GHz and 8 GB of RAM was used. The parametric nonlinear numerical method and further details were described extensively by [PERSON] [8] and [PERSON] [21]. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Name** & **Equations** & **Number** \\ \hline Navier-Stokes & & \\ Laminar Flow & \(\rho\frac{\partial u}{\partial t}-\
abla\bullet\mu\left(\
abla u+(\
abla u)^{T} \right)+\rho u\bullet\
abla u+\
abla p=0\) & (1) \\ Stefan-Maxwell & & \\ Diffusion and Convection & \(\
abla\left[-\rho\omega_{i}\sum_{j=1}^{N}D_{ij}^{eff}\frac{M}{M_{j}}\left( \
abla\omega_{j}+\omega_{j}\frac{\
abla M}{M}\right)+\rho\omega_{i}u\right]\) & (2) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Governing equations. Figure 1: (**A**) Concept of the geometries of bipolar plates. (**B**) Design of the geometries in AutoCAD (CAD). (**C**) Three-dimensional (3D) view in Multiphysics COMSOL®v.3.4. (**D**) Figure mesh. (**E**) Isometry and components of the PEM fuel cell. (**F**) Simulation of variables according to the parameters. In (A,E,F), we can see the shape of the channels. \begin{table}
\begin{tabular}{l c} \hline \hline
**Name** & **Equations** & **Number** \\ \hline Stefan–Maxwell & & \\ Diffusion Coefficient & \(D_{ij}=k\frac{T^{1.75}}{p\left(v_{i}^{1/3}+v_{j}^{1/3}\right)^{2}}\left[\frac{ 1}{M_{i}}+\frac{1}{M_{j}}\right]\) & (3) \\ Stefan–Maxwell & & \\ Effective Diffusion & \(D_{ij}^{eff}=D_{ij}\epsilon^{1.5}\) & (4) \\ Total Molar Mass & \(M=\sum_{i=1}^{n}M_{i}\omega_{i}\) & (5) \\ Electrical Current Density & \(i_{c}=-S_{a}\delta i_{0}\frac{\omega_{j}}{\omega_{j,0}}e\left(\frac{\eta F}{2 RT }\right)\) & (6) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Scalar equations. The equations are written in indicial-tensor notation with: \(\rho\), density kg/m\({}^{3}\); t, time \(s\); \(\mu\) dynamic viscosity of oxygen = 2 \(\times\) 10\({}^{-5}\) Pa-s (5); \(u\), velocity vector m/s; p, pressure Pa; \(\omega_{i}\) weight fraction of the ith species; with an inlet mass fraction \(O_{2}\) = 0.1447, a mass fraction of \(H_{2}O\) = 0.3789, and a mass fraction of \(N_{2}\) = 0.4764; \(D_{eff,ji}\), the effective binary diffusion coefficient (see Equation (4) in [22; 23; 24]); \(M\), the total molar mass of the mixture g/mol (see Equation (5) in [19; 20; 21; 25; 26; 27]); \(M_{j}\), the molecular weight of species \(j\), g/mol; \(M\)O\({}_{2}\), the molecular weight of \(O_{2}\) = 32 g/mol; \(M_{H2O}\), the molecular weight of \(H_{2}O\) = 18 g/mol; \(M_{N2}\), the molecular weight of \(N_{2}\) = 28 g/mol; \(D_{ij}\), the binary diffusion coefficient for species \(i\) and \(j\); \(k\), the Maxwell diffusion constant = 3.16 \(\times\) 10\({}^{-8}\) Pa\(\cdot\)m\({}^{2}\)/s; T, temperature = 353 \({}^{\circ}\)K; \(v_{i}\), the molar diffusion volume of the jth species, cm\({}^{3}\)/mol; \(\
u_{O2}\), the molar diffusion volume of \(O_{2}\) = 16.6 \(\times\) 10\({}^{-6}\) cm\({}^{3}\)/mol; \(\
u H_{2}O\), molar diffusion volume of \(H_{2}O\) = 12.7 \(\times\) 10\({}^{-6}\) cm\({}^{3}\)/mol; \(\
u N_{2}\), the molar diffusion volume of \(N_{2}\) = 17.9 \(\times\) 10\({}^{-6}\) cm\({}^{3}\)/mol; for \(M_{j}\), \(MO_{2}\), \(MH_{2}O\), and \(MN_{2}\), see Table 1; \(\varepsilon\), porosity = 0.5; \(i_{c}\), cathode current; \(S_{a}\) specific surface area = 1 \(\times\) 10\({}^{7}\)m\({}^{2}\)/m\({}^{3}\); \(\delta\), active layer thickness = 10 um; \(i_{0}\), exchange current density = 1.06 \(\times\) 10\({}^{-6}\) mA/cm\({}^{2}\); F, [PERSON]'s constant = 96485 C/mol, R, gas constant = 8.314 J/(mol\(\cdot\)K); \(\eta\) overpotential between 0.2 and 0.82 V; \(t_{H_{2}O}\) electro-osmotic drag = 3; gdl, gas diffusion layer thickness = 0.2 mm. Five different air flow field geometries adjacent to a PEMFC oxygen cathode were compared by numerical simulations. Table 5 details the dimensions of the flow channels, and Figure 2 shows the geometries of Designs 1-5.