Untitled - Aerobib - Universidad Politécnica de Madrid

2.3. VISCOSITIES 53
Its value depends on the easiness of energy transfer between external and internal
degrees of freedom of the molecules during collisions. Such facility is characterized
by a relaxation time for each internal degree of freedom. The value for µ ′ is expressed
as a function of these relaxation times. Generally, it is sufficiently approximate to
assume µ ′ = 0. Additional information on this problem can be found in Ref. [2],
pp. 501 and 710.
Viscosity coefficient of a mixture of gases is a complicated function of molar
masses, mass fractions and viscosity coefficients of the species as well as of the interaction
potentials between different species. 25 For binary mixture, for example, the
first approximation [µ] 1
is given by
where
1
[µ] 1
= X µ + Y µ
1 + Z µ
, (2.49)
X µ = X2 1
+ 2X 1X 2
+ X2 2
,
µ 1 µ 12 µ
[
2
]
Y µ = 3 X 2
5 A∗ 1 M 1
12 + 2X 1X 2 (M 1 + M 2 ) 2 µ 2 12
+ X2 2 M 2
,
µ 1 M 2 µ 12 4M 1 M 2 µ 1 µ 2 µ 2 M 1
[ X
2
1 M 1
(2.50)
Z µ = 3 5 A∗ 12
M 2
(
(M 1 + M 2 ) 2 (
µ12
+ 2X 1 X 2 + µ ) )
12
− 1
4M 1 M 2 µ 1 µ 2
]
+ X2 2 M 2
.
M 1
In these expressions, µ 1 and µ 2 are the first approximations of the viscosity coefficients
for both species. µ 12 is given by
( )
µ 12 = 5
N −1 M1 M 2
√2RT
16 √ M 1 + M 2
π σ12 2 Ω(2,2)∗ 12 (T12 ∗ ) , (2.51)
ε 12 and σ 12 are the constants of the interaction potential between molecules of both
species. A ∗ is a function of reduced temperature T ∗ 12 = kT
ε 12
, whose value is given in
Table 2.3 for the Lennard-Jones potential.
25 See Ref. [2], p. 529.
T12 ∗ 0.3 0.5 0.75 1 1.25 1.5 2
A ∗ 1.046 1.093 1.105 1.103 1.099 1.097 1.094
T12 ∗ 3 4 5 10 50 100 400
A ∗ 1.095 1.098 1.101 1.11 1.13 1.138 1.154
Table 2.3: The coefficient A ∗ as a function of T ∗ 12.