The compressible Euler equations for gas dynamics are (mass, momentum, internal energy) ∂ ρ ∂ t + ∇ ⋅ ( ρ u ) = 0 , {\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \left(\rho \mathbf {u} \right)=0\,,} ∂ ( ρ u ) ∂ t + ∂ ∂ x ( ρ u 2 + p ) + ∂ ∂ y ( ρ u v ) = 0 , {\displaystyle {\frac {\partial (\rho u)}{\partial t}}+{\frac {\partial }{\partial x}}\left(\rho u^{2}+p\right)+{\frac {\partial }{\partial y}}\left(\rho uv\right)=0\,,} ∂ ( ρ v ) ∂ t + ∂ ∂ x ( ρ u v ) + ∂ ∂ y ( ρ v 2 + p ) = 0 , {\displaystyle {\frac {\partial (\rho v)}{\partial t}}+{\frac {\partial }{\partial x}}\left(\rho uv\right)+{\frac {\partial }{\partial y}}\left(\rho v^{2}+p\right)=0\,,} ∂ E ∂ t + ∂ ∂ x ( u ( E + p ) ) + ∂ ∂ y ( v ( E + p ) ) = 0 . {\displaystyle {\frac {\partial E}{\partial t}}+{\frac {\partial }{\partial x}}\left(u(E+p)\right)+{\frac {\partial }{\partial y}}\left(v(E+p)\right)=0\,.} A suitable equation of state used to close the system is given by E = p γ − 1 + ρ 2 ( u 2 + v 2 ) , {\displaystyle E={\frac {p}{\gamma -1}}+{\frac {\rho }{2}}\left(u^{2}+v^{2}\right)\;,} where typically γ = 1.4 {\displaystyle \gamma =1.4} for a monoatomic gas.
for a monoatomic gas.