Finally, the energy equation: D e D t = − − p v ∇ ∇ ⋅ ⋅ u can be further simplified in convective form by changing variable from specific energy to the specific entropy: in fact the first law of thermodynamics in local form can be written: D e D t = T D s D t − − p D v D t by substituting the material derivative of the internal energy, the energy equation becomes: T D s D t + p ρ ρ 2 ( D ρ ρ D t + ρ ρ ∇ ∇ ⋅ ⋅ u ) = 0 now the term between parenthesis is identically zero according to the conservation of mass, then the Euler energy equation becomes simply: D s D t = 0