Transport Phenomena in Ionized Gases - Fundamentals of Ionized Gases - page 197

Then (4.10) leads to the following relationship between pressure and temperature

p D NT . (4.12)

Equation (4.12) is the equation of state for an ideal gas.

4.1.3
The Navier-Stokes Equation

Note that above in deriving the generalized Euler equation (4.6) for a one-compo-
nent system we ignored collisions between atoms, and collisions between atoms
or molecules of different components are taken into account in (4.7)-(4.9) through
typical times qs . We now include collisions between atoms in the balance equation

for the average atom momentum for a one-component system. Such collisions lead
to the frictional force between neighboring gas layers which are moving with dif-
ferent velocities. This effect is expressed through the gas viscosity coefficient

τ

,
which is taken into account as arising from the above-mentioned frictional force.
According to the definition of the viscosity coefficient

η

, if a gas (or liquid) is mov-
ing along the x -axis and varies in the z direction, the frictional force f per unit area
is given by

η

D η @

w x
@

f , (4.13)

z

where w x is the average gas velocity.
We include the gas viscosity in the expression for the pressure tensor, which
takes the form

P 0 α ,
where the latter term accounts for the gas viscosity. On the basis of the viscosity
definition and transferring to an arbitrary frame of reference, one can represent
the pressure tensor P 0 α

P α D p δ α C

, which is symmetric, in the form

@ .

w α
@ x C @

w
@

δ α @

w α
@

P 0 α D η

x α C

a

x

The factor a may be found as follows. The forces of viscous friction in a gas occur
because neighboring gas layers move with different velocities. If a gas could be
decelerated as a whole, this friction mechanism would have no effect and the force
due to the gas viscosity would vanish. Hence, the trace of the pressure tensor is
zero. This yields a

D 2/3, and the viscosity term in the pressure tensor can be

writtenintheform

@ .

w α
@ x C @

w
@

2

3 δ α @

w α
@

P 0 α D η

x α

(4.14)

x

With the viscosity part of the pressure tensor taken into account, (4.4) takes the
form of the Navier-Stokes equation [6-10]:
@ w
@

C ν Δ w C 3 r w C m ,

p

( w r ) w D r

t C

(4.15)