Transport Phenomena in Ionized Gases - Fundamentals of Ionized Gases - page 203

The viscosity coefficient is the proportionality factor for the relationship between

the frictional force acting on a unit area of a moving gas and the gradient of the
mean gas velocity in the direction perpendicular to the surface of a gas element. If
the mean gas velocity w is parallel to the x -axis and varies in the z direction (see
Figure 4.1), the frictional force is proportional to

η

@ w x / @ z and acts on an xy surface

in the gas. Thus, the force f per unit area is

D η @

w x
@

f (4.28)

z

and this definition of the viscosity coefficient for gases is also valid for liquids.
Assume that transport processes in a gas are determined by elastic collisions of
gas atoms and obtain the dependence of kinetic coefficients on the cross section
of elastic scattering of atoms and other parameters of a gaseous system, starting
from the diffusion coefficient. The diffusive flux is the difference between fluxes
in opposite directions. Each of these, in order of magnitude, is N k v ,where N k

is the number density of atoms of a given species and v is the thermal velocity

of atoms. Thus, the net flux behaves as j

N k is the difference
between the number densities of atoms which determine oppositely directed fluxes
of atoms. Atoms that reach a given point without collisions have distances from it
of the order of the mean free path

Δ N k v ,where

Δ

) 1 ,where is a typical cross section

for elastic collisions of atoms and N is the total number density of atoms. Hence
Δ

λ ( N σ σ

N k .Comparingthis
estimation with the definition of the diffusion coefficient (4.25), we obtain for the
diffusion coefficient

N k λ r N k , and the diffusive flux behaves as j λ v r

p T

D v λ

σ p m .

(4.29)

N

Here T is the gas temperature and m is the mass of atoms of a given species,
assumed to be of the same order of magnitude as the masses of other particles
composing the gas. In this analysis we do not need to account for the sign of the

Figure 4.1 The geometry of fluxes in the definition of the viscosity coefficient.