The hypotheses to simplify the Navier-Stokes equations are :
ρ D E D t = ∂ ρ E ∂ t + ∂ ρ u i E ∂ x i = ω T ˙ + ∂ ∂ x i ( λ ∂ T ∂ x i ) − ∂ ∂ x i ( ρ ∑ k = 1 N h s , k Y k V k , i ) + ∂ σ i j u i ∂ x j + Q ˙ + ρ ∑ k − 1 N Y k f k , i ( u i + V k , i ) {\displaystyle {\rho }{\dfrac {DE}{Dt}}={\dfrac {\partial {{\rho }E}}{\partial {t}}}+{\dfrac {\partial {{\rho }u_{i}E}}{\partial {x_{i}}}}={\dot {\omega _{T}}}+{\dfrac {\partial }{\partial {x_{i}}}}({\lambda }{\dfrac {\partial {T}}{\partial {x_{i}}}})-{\dfrac {\partial }{\partial {x_{i}}}}({\rho }{\sum }_{k=1}^{N}h_{s,k}Y_{k}V_{k,i})+{\dfrac {\partial {\sigma _{ij}}u_{i}}{\partial {x_{j}}}}+{\dot {Q}}+{\rho }{\sum }_{k-1}^{N}Y_{k}f_{k,i}(u_{i}+V_{k,i})}
The term ∂ ∂ x i ( ρ ∑ k = 1 N h s , k Y k V k , i ) {\displaystyle {\dfrac {\partial }{\partial {x_{i}}}}({\rho }{\sum }_{k=1}^{N}h_{s,k}Y_{k}V_{k,i})}
