Viscosity of Multicomponent Gas Mixtures
The Chapman-Enskog theory has been extended to predict the
viscosity of multicomponent (nonpolar) gas mixtures at low density.
For most of the time, Wilke’s following formula is utilized:
n
xiηi
η mix = ∑
n
i=1
∑ x j Φ ij
j=1
2

 ηi   M j  
M i 
1 
 
1 + 
 
Φ ij =
1+

η 
M j 
Mi  

8 
j 




where n is the number of chemical species in the mixture; xi and
xj are the mole fractions of species i and j; ηi and ηj are the
viscosities of species i and j at the system temperature and
pressure; Mi and Mj are corresponding molecular weights.
When i=j, φij=1.
−1 / 2
1/ 2
1/ 4
What is the main difference between viscosities of
gases and liquids?
In the case of liquids, we are interested in their
capability of yielding to stress instead of their resistance
to shearing stress (that is the case for gases).
Viscosity of Liquids-I
Based on the assumption that viscous flow takes place by
movement of particles past other particles, mobility of an
individual liquid particle can be predicted. Einstein’s approach:
D = BKBT
This relationship shows the mobility (B, mean velocity/force acting
on the particle) of a particle under the influence of an external
force is related the diffusion coefficient D.
Diffusion and fluidity are both activated processes. That is, they
both require a minimum activation energy for the initiation of the
process.
Diffusion and fluidity α exp(-∆G‡/RT)
Viscosity of Liquids-II
Diffusion and fluidity α exp(-∆G‡/RT)
Fluidity is the inverse of viscosity
Therefore, viscosity α exp(+∆G‡/RT)
±
 ∆ G vis

η = A exp 

 RT 
Where η= viscosity, A=constant, T=absolute temperature, K, R =
gas constant, and ∆G‡vis = activation energy of viscosity.
The viscosity of liquids decreases with increasing
temperature while the viscosity of gases increases.
Molecular Liquids vs. non molecular liquids
A ≅
∆G
±
vis
N 0h
_
V
≅ 0 .41 ∆ E
vap
Viscosity of Liquid Metals and Alloys
Viscosity of Liquid Metals and Alloys (Chapman Model)
Chapman derived a relationship between the viscosity, an energy
parameter ε and a separation distance δ. Then by assuming that all
liquid metals obey the same function φ(r ) he concluded that all
substances with this φ(r ) should have a reduced viscosity η*, which
is a function of the reduced temperature T* and volume V*where
the functional relation is given by
  δ  12  δ  6  δ= interatomic distance in the closeΦ ( r ) = 4 ε    −    packed crystal at 0°K, Å
 r   ε= energy parameter characteristic of
  r 
specific metal
η * (V * ) 2 = f ( T * )
η =
*
T
*
V
*
ηδ
2
N0
MRT
K T
= B
ε
= 1 / nδ 3
No= Avogadro’s number
M= atomic weight
R= gas constant
T= absolute temperature, K
KB= Boltzmann’s constant
n= number of atoms per unit volume
Viscosity of Liquid Metals and Alloys-II