Untitled - Aerobib - Universidad Politécnica de Madrid

3.5. GENERAL EQUATIONS 69
obtained for the said term
∇ · (¯v · τ e ) = −∇ · (p¯v) + ∇ · (¯v · τ ev ) . (3.36)
Now, if the first term of the right hand side of this equation is expanded and
combined with the continuity equation (3.6), the following is obtained
∇ · (¯v · τ e ) = ∂p
∂t − ρ D ( ) p
+ ∇ · (¯v · τ ev ) . (3.37)
Dt ρ
Substituting this equation into Eq. (3.25), taking into account Eq. (3.35), it finally
gives
ρ D Dt
(h + 1 2 v2 )
= ∂p
∂t + ∇ · (¯v · τ ev)
+ ρ ¯F · ¯v + ∇ · (λ∇T ) − ∇ ·
(
ρ ∑ i
Y i h i¯v di
)
,
(3.38)
where Eq. (3.33) has been used to eliminate ¯q.
3.5 General equations
As a summary of preceding paragraphs we shall once more write the general system
of equations that govern the transformation of a mixture of reactant gases in motion,
that is to say the general equations of Aerothermochemistry.
Continuity Equation
For the mixture:
For the species:
Dρ
+ ρ∇ · ¯v = 0. (3.6)
Dt
ρ DY i
Dt + ∇ · (ρY i¯v di ) = w i , (i = 1, 2, . . . , l). (3.7)
Equation of Motion
ρ D¯v
Dt = −∇p + ∇ · τ ev + ρ ¯F . (3.18)