Untitled - Aerobib - Universidad Politécnica de Madrid

2.3. VISCOSITIES 51
Here µ and µ ′ are called viscosity coefficient and bulk viscosity coefficient respectively,
and
3
8π √ 2 γ ij = 1 ( ∂vi
+ ∂v )
j
2 ∂x j ∂x i
are the components of deformation velocity tensor
τ vd ≡
⎛
⎜
⎝
(2.41)
⎞
γ 11 γ 12 γ 13
⎟
γ 21 γ 22 γ 23 ⎠ , (2.42)
γ 31 γ 32 γ 33
which determines the deformation suffered by a gas element as it moves. 22
The method of Chapman and Enskog allows calculation of the viscosity coefficient
of a gas by means of successive approximations. Expressions of the form
µ j = [µ] 1
f j µ (2.43)
are obtained, where [µ] 1
is the first approximation, j is the computed order of the approximation
and f j µ is a function which differs slightly from unity. When comparing,
for example, the first and third approximations for the Lennard-Jones potential it is
verified that error in the first is smaller than 0.8%. Thereby, as for diffusion coefficients,
the first approximation is usually enough.
In the case of a pure gas this approximation is given by 23
[µ] 1
= 5
16 √ N −1√ MRT
π σ 2 Ω (2,2)∗ (T ∗ ) . (2.44)
Here, as for diffusion, Ω (2,2)∗ is a function of reduced temperature T ∗ = kT/ε, whose
law of variation depends on the form of the molecular interaction potential. For example,
for rigid spheres Ω (2,2)∗ = 1.
For a Sutherland potential, Eq. (2.18), is
Ω (2,2)∗ = 1 + S µ
T , (2.45)
where S µ = g µ (δ) ε . In practice, this value is determined experimentally. When
k
Eq. (2.45) is substituted into Eq. (2.44) one obtains
22 See Ref. [15].
23 See Ref. [2], p. 527.
[µ] 1
= 5
16 √ π
N −1√ MR
σ 2 T 3 2
S µ + T . (2.46)