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11.2. DIMENSIONLESS PARAMETERS OF AEROTHERMOCHEMISTRY 263
b) Continuity equation.
c) Momentum equation.
d) Energy equation.
( )
1
ρv
2 v2 + c p T − qε
e) Diffusion equation.
ρv dε = w. (11.2)
dx
p + ρv 2 − 4 3 µ dv = i. (11.3)
dx
D
v
dY
dx
− λ dT
dx − 4 dv
µv = me. (11.4)
3 dx
− Y + ε = 0. (11.5)
Be l 0 , ρ 0 , v 0 , w 0 , p 0 , µ 0 , c p0 , T 0 , λ 0 and D 0 characteristic values of the corresponding
quantities and let us change the variables of the preceding system as follows
x = l 0 x ′ , ρ = ρ 0 ρ ′ , v = v 0 v ′ , w = w 0 w ′ , p = p 0 p ′ ,
µ = µ 0 µ ′ , c p = c p0 c ′ p, T = T 0 T ′ , λ = λ 0 λ ′ , D = D 0 D ′ .
(11.6)
Thus all the “prime” quantities are dimensionless. For ε and Y this change is
not necessary since both are already dimensionless and their variation field has been
selected from zero to one.
When these new variables (11.6) are introduced into the preceding system it
may be written as follows:
a) Equation of mass conservation.
ρ ′ v ′ =
( m
ρ 0 v 0
)
. (11.7)
b) Continuity equation. ( )
ρ0 v 0
ρ ′ w ′ dε
l 0 w 0 dx = w′ . (11.8)
c) Momentum equation.
( )
p0
ρ 0 v0
2 p ′ + ρ ′ v ′ 2 4 −
3
d) Energy equation.
ρ ′ v ′ [ 1
2
( v
2
0
)
v ′ 2 + c
′
c p0 T p T ′ −
0
(
− 4 3
(
µ0
) (
µ ′ dv′
ρ 0 v 0 l 0 dx ′ =
( ) ] (
q
ε −
c p0 T 0
µ 0 v0
2 )
µ ′ v ′ dv′
l 0 ρ 0 v 0 c p0 T 0
i
ρ 0 v 2 0
)
. (11.9)
)
λ 0 T 0
λ ′ dT ′
l 0 ρ 0 v 0 c p0 T 0 dx ′
( )
dx = me
. (11.10)
ρ 0 v 0 c p0 T 0