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H-Allegro uses the Navier-Stokes equations to solve combustion.
( 1 ) ∂ ρ ∂ t + ∂ ρ U j ∂ x j = 0 {\displaystyle (1){\dfrac {\partial {\rho }}{\partial {t}}}+{\dfrac {\partial {{\rho }U_{j}}}{\partial {x_{j}}}}=0}
( 2 ) ∂ ρ U i ∂ t + ∂ ρ U i U j ∂ x j + ∂ P ∂ x i = ∂ τ i j ∂ x j + S i {\displaystyle (2){\dfrac {\partial {{\rho }U_{i}}}{\partial {t}}}+{\dfrac {\partial {\rho }U_{i}U_{j}}{\partial {x_{j}}}}+{\dfrac {\partial {P}}{\partial {x_{i}}}}={\dfrac {\partial {\tau _{ij}}}{\partial {x_{j}}}}+S_{i}}
( 3 ) ∂ ρ E ∂ t + ∂ ( P + ρ E ) U j ∂ x j = ∂ q j ∂ x j + ∂ τ i j U i ∂ x j + S 5 {\displaystyle (3){\dfrac {\partial {{\rho }E}}{\partial {t}}}+{\dfrac {\partial {(P+{\rho }E)}U_{j}}{\partial {x_{j}}}}={\dfrac {\partial {q_{j}}}{\partial {x_{j}}}}+{\dfrac {\partial {\tau _{ij}U_{i}}}{\partial {x_{j}}}}+S_{5}}
( 4 ) ∂ ρ Y k ∂ t + ∂ ρ Y k U j ∂ x j = ∂ q j k ∂ x j + S k {\displaystyle (4){\dfrac {\partial {{\rho }Y_{k}}}{\partial {t}}}+{\dfrac {\partial {{\rho }Y_{k}U_{j}}}{\partial {x_{j}}}}={\dfrac {\partial {q_{j}^{k}}}{\partial {x_{j}}}}+S_{k}}
