Untitled - Aerobib - Universidad Politécnica de Madrid

3.8. STATIONARY, ONE-DIMENSIONAL MOTIONS 75
3.8 Stationary, one-dimensional motions
If the motion is not only one-dimensional but stationary, the preceding equations are
considerably simplified. This type of motion, as previously stated, is of special interest
for the study of the structure of combustion waves.
The stationary motions are characterized by the condition
∂ (·)
= 0. (3.63)
∂t
Therefore, the only independent variable is x.
Let us see how the equations in the preceding paragraph can be simplified when
condition (3.63) is introduced therein.
Continuity Equations
The continuity equation (3.49) reduces to
v dρ
dx + ρ dv
dx ≡
This equation can be integrated, thus obtaining
d (ρv) = 0. (3.64)
dx
ρv = m, (3.65)
where m is an integration constant which gives the mass flux per unit surface normal
to the direction of motion.
Similarly, equations (3.50) reduces to
d
dx (ρY iv + ρY i v di ) = w i , (i = 1, 2, . . . , l) . (3.66)
By introducing the flux of the different species, these equations reduce to a
very simple form. In fact, let m i = mε i be the flux of species A i through the unit
surface normal to the direction of motion, that is to say that ε i is the fraction of the
total flux corresponding to species A i . Since the velocity v i of species A i is
v i = v + v di , (3.67)
evidently the said flux is
mε i = ρY i (v + v di ) . (3.68)
Now, when this expression is compared with the left hand side of Eq. (3.66) it
is seen that this system can be written in the form
m dε i
dx = w i, (i = 1, 2, . . . , l). (3.69)