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50 CHAPTER 2. TRANSPORT PHENOMENA IN GAS MIXTURES
d σ
x
3
P
f
n
f
3
f 1
x
x 2
1
f 2
Figure 2.5: Schematic diagram showing the components of the stress ¯f = ¯n · τ e.
the gas. 19 Such transfer is produced by the thermal agitation of the molecules. 20 If the
state of gas is uniform, the velocity distribution of thermal agitation is a Maxwell’s
distribution. In such case transfer of momentum is normal to dσ and independent
from the orientation ¯n. The corresponding state of stresses is a pure compression,
defined by a scalar: the pressure p of the gas. Tensor of Eq. (2.33) reduces as follows
τ e = −pU, (2.36)
where U is the unit tensor whose components are Kronecker’s δ ij
U ≡
⎛
⎜
⎝
1 0 0
0 1 0
0 0 1
Stress ¯f acting upon an element dσ in Fig. 2.5, is
⎞
⎟
⎠ . (2.37)
¯f = −p¯n. (2.38)
When the state of motion of the gas is not uniform other stresses produce, which add
to those of pressure, Eq. (2.38). Such stresses are given by viscous stress tensor
⎛
⎞
τ 11 τ 12 τ 13
⎜
⎟
τ ev ≡ ⎝ τ 21 τ 22 τ 23 ⎠ . (2.39)
τ 31 τ 32 τ 33
The Chapman-Enskog method described in the introduction to the present chapter
shows that components τ ij of the viscous stress tensor have the form 21
( ) 2
τ ij = 2µγ ij −
3 µ − µ′ (∇ · ¯v) δ ij . (2.40)
19 Within dense gases stresses are partially due to transfer of momentum and partially to the forces of
molecular interaction whose effect is not negligible.
20 If ¯v is the gas velocity at a point and ¯v j the velocity of a molecule at the neighborhood of such point,
velocity ¯v a of thermal agitation is, by definition, ¯v a = ¯v j − ¯v.
21 Actually Enskog and Chapman’s theory was developed for monotonic dilute gases, where µ ′ = 0.