Untitled - Aerobib - Universidad Politécnica de Madrid

3.4. ENERGY EQUATION 65
where the subscripts indicate components, and the repeated subscripts indicate, according
to Einstein’s rule, summation respect to them, from one to three.
As shown in chapter 2, the components τ ij of the stress tensor are expressed as
a function of the gas pressure p at the point, and of the velocity ¯v of the motion, in the
following form
( ∂vi
τ ij = −pδ ij + µ + ∂v )
j
+
(µ ′ − 2 )
∂x j ∂x i 3 µ (∇ · ¯v) δ ij , (3.16)
where δ ij is the Kronecker symbol and µ and µ ′ are the coefficients of viscosity and
volumetric viscosity of the mixture, respectively. These coefficients depend on the
state and composition of the mixture as indicated in the above mentioned chapter. In
particular, for the diluted monatomic gases µ ′ = 0 and in general, a good approximation
is obtained by assuming that it is also zero for all other cases.
In Eq. (3.14) the pressure and viscous stresses can be made explicit, by using
the expression
τ e = −pU + τ ev (3.17)
previously given in chapter 2. By taking this expression of τ e into Eq. (3.14), we
obtain
ρ D¯v
Dt = −∇p + ∇ · τ ev + ρ ¯F . (3.18)
This equation is identical to those obtained in Gas Dynamics for mixtures of uniform
composition with no chemical reaction. The variations in the composition of the
mixture act only through their influence on the values of the viscosity coefficients µ
and µ ′ .
3.4 Energy equation
This equation is obtained when expressing that the variations of the energy contained
in V , Fig.3.1, are due to the work done by the forces acting upon the element and the
heat received by the said element. Let us calculate, separately, the different terms in
this equation.
a) Energy.
Let u i be the internal energy per unit mass of species A i . The internal energy u
per unit mass of the mixture is
u = ∑ i
Y i u i , (3.19)