Untitled - Aerobib - Universidad Politécnica de Madrid

72 CHAPTER 3. GENERAL EQUATIONS
3.7 One-dimensional motions
As an application we shall see the form taken by the general equations in the case of
one dimensional motions. These equations, in particular those relative to stationary
motions, will be widely used in the study of the detonations and flames. 18
It will be assumed that:
a) No mass forces exist.
b) The coefficient of volumetric viscosity µ ′ is zero.
c) The effects of thermal diffusion and radiation can be neglected.
We shall adopt a cartesian rectangular system with the x axis parallel to the
direction of motion. Then, the only independent variables of the motion are coordinate
x and time t, and the only velocity component different from zero is that parallel to
the x axis, which will be designated as v.
The substantial derivative (3.1) reduces in this case to
D (·)
Dt
= ∂ (·)
∂t
+ v ∂ (·)
∂x . (3.48)
Continuity equations
The continuity equations for the mixture reduces to
∂ρ
∂t + v ∂ρ
∂x + ρ ∂v = 0. (3.49)
∂x
Similarly, the continuity equations (3.7) for the various species, reduces to
ρ ∂Y i
∂t + ρv ∂Y i
∂x + ∂
∂x (ρY iv di ) = w i , (i = 1, 2, . . . , l). (3.50)
Equations of motion
In the vectorial Eq. (3.18) the only component not identically zero is that parallel to
x axis. Furthermore, the only component different from zero in the viscous stress
tensor τ ev is
τ xx = 4 3 µ ∂v
∂x , (3.51)
corresponding to viscous stress that acts upon the plane normal to the motion and
parallel to the x axis. This stress is subtracted from the pressure. Therefore, Eq. (3.18)
18 See chapters 5 and 6.