# Numerical Investigations of Micro-Scale Diffusion Combustion: A Brief Review
## 2 Governing Equations  
The main governing equations used for the numerical simulations are the conservation equations for a continuous, compressible, multi-component and thermally-perfect gas mixture for laminar flows. It is noted that the numerical simulations for combustion at micro scales do not concern turbulence as the flow is usually laminar due to the small scales of the problems. The conservation equations of total mass, momentum, species mass fractions and energy are presented below, respectively [9].  
\[\frac{\partial\rho}{\partial t}+\
abla(\rho\textbf{u})=0 \tag{1}\]  
\[\rho\left[\frac{\partial\textbf{u}}{\partial t}+\
abla\cdot\textbf{u}\textbf{ u}\right]=-\
abla p+\
abla\tau+\rho\textbf{g} \tag{2}\]  
\[\frac{\partial}{\partial t}(\rho Y_{k})+\
abla(\rho Y_{k}\textbf{u})=-\
abla( \rho Y_{k}\textbf{V}_{k})+\hat{R}_{k}\qquad k=1,\ldots,N_{C} \tag{3}\]  
\[\rho C_{P}\frac{\partial T}{\partial t}+\rho C_{P}\textbf{u}\
abla T=-\
abla \textbf{q}-\rho\sum_{k=1}^{N_{C}}C_{P,k}Y_{k}\textbf{V}_{k}-\sum_{k=1}^{N_{C }}h_{k}\hat{\Omega}_{k} \tag{4}\]  
Here, \(t\), \(\rho\), \(\textbf{u}\), \(\tau\), \(\textbf{o}\) and \(\textbf{g}\) represent time, the mixture density, the pressure, the mixture velocity vector, the stress tensor, the fluid stress tensor, and the acceleration vector due to gravity, respectively.  
\(N_{\mathcal{C}}\) is the total number of species, \(T\) is the temperature, and \(Y_{k}\), \(\mathbf{V_{k}}\), \(\hat{R}_{k}\), \(h_{k}\) and \(\hat{\Omega}_{k}\) are the mass fraction, the diffusion velocity, the formation rate, the enthalpy and the net production rate of the \(k\)th species, respectively. \(C_{p}\) represents the specific heat coefficients. The gas mixture is assumed to be an ideal gas and its density is calculated through the equation of state.  
The diffusion velocities of species \(k\) are calculated from [PERSON]'s law and the thermal diffusion effect:  
\[\mathbf{V_{k}}=-\frac{D_{k}}{Y_{k}}\
abla Y_{k}-\frac{D_{k}\Theta_{k}}{X_{k}} \frac{1}{T}\
abla T \tag{5}\]  
where, \(X_{k}\), \(\Theta_{k}\), and \(D_{k}\) are the mole fraction, the thermal diffusion and the mixture-averaged diffusion coefficient of species \(k\).  
The heat flux vector, \(\mathbf{q}\), is defined as:  
\[\mathbf{q}=-\lambda\
abla T+\mathbf{q}_{rad} \tag{6}\]  
where \(\lambda\) is the mixture thermal conductivity, \(\mathbf{q}_{rad}\) is the radiative heat transfer. However, in some studies the radiative heat transfer is neglected, due to the lower impact at micro scale [2].  
The radiative heat transfer can be modeled by assuming the optically thin radiation hypothesis, in which the self-absorption of radiation is neglected. The radiative heat transfer contribution in the energy equation is described as:  
\[\
abla\mathbf{q}_{rad}=-4\sigma a_{P}(T^{4}-T^{4}_{env}) \tag{7}\]  
where, \(T_{env}\) is the environment temperature, \(\sigma\) is the Stefan-Boltzmann constant and \(a_{P}\) is the Planck mean absorption coefficient.  
The initial numerical studies of micro diffusion combustion did not consider the wall thermal effects and, as a consequence, the numerical results close to the flame edge shifted away from those of the experiments, showing the important impact of the wall quenching in that region, and the importance of the thermal interactions between the solid and the fluid. For this reason, the numerical studies started to incorporate the energy equation for the solid region, using conjugate heat transfer condition for the fluid-solid interaction.