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68 CHAPTER 3. GENERAL EQUATIONS
It has been shown in chapter 2 that ¯q is given by the expression
¯q = −λ∇T + ρ ∑ i
∑∑
Y j D T i
(¯v di − ¯v dj )
i j M i M j D ij
Y i h i¯v di + RT
∑ Y i
. (3.32)
M j
In this expression the first term of the right hand side gives the heat flux through conduction
due to temperature gradient. This is the only existing term in Gas Dynamics.
The second term gives the enthalpy flux of the various species through diffusion. The
third term gives the heat flux due to the Dufour effect of reciprocal effect of the thermal
diffusion in Onsager’s sens. 11 Such an effect is zero when all the molecules have
the same mass since, then, D T i is zero. In general, the Dufour effect can be neglected
and thereby ¯q reduces to the expression
j
¯q = −λ∇T + ρ ∑ i
Y i h i¯v di . (3.33)
Taken this expression of ¯q into Eq. (3.30), the following final expression for
the variation of internal energy of the mixture is obtained
(
ρ Du
Dt = −p∇ · ¯v + Φ + ∇ · (λ∇T ) − ∇ · ρ ∑ )
Y i h i¯v di . (3.34)
i
from:
This equation shows that the time variation of the internal energy originates
a) The work −p∇ · ¯v done by pressure to compress the gas.
b) The work Φ dissipated by the viscous forces to produce the deformation.
c) The heat ∇ · (λ∇T ) received through conduction.
d) The enthalpy flux −∇ · (ρ ∑ i Y ih i¯v di ) of the species through diffusion.
Forms a), b) and c) are the same as those that exist in Gas Dynamics when there is no
change in composition.
In many cases it is convenient to work with Eq. (3.25) which includes kinetic
energy, 12 bringing forth in this equation the enthalpy h of the mixture
h = u + p ρ . (3.35)
For this purpose we shall start by making the pressure explicit in the first term
of the right hand side of Eq. (3.25) by means of Eq. (3.17). Thus, the following is
11 See Ref. [5], p. 118.
12 See chapters 4 and 5.