and the components have suggestive meaning as follows
At least this is how
n
through
are to be interpreted in the rest frame of the gas.
We have defined
and
m
is the molecular rest mass.
The decomposition (86) is not only popular because of its intuitive quality but also,
since it is now possible to characterize
equilibrium
as a process in which the stress deviator
, the heat flux
and the dynamic pressure
- the non-equilibrium part of the pressure - vanish.
The equilibrium pressure p is a function of n and e, the thermal equation of state . In thermodynamics it is often useful to replace the variables (n, e) by
because these two variables can be measured - at least in
principle. Also
and
T
are the natural variables of statistical thermodynamics which
provides the thermal equation of state in the form
. The transition between the new variables
and the old ones (n,
e) can be effected by the relations
where
and
here and below denote differentiation with respect to
and
respectively.
If we restrict attention to a linear theory in
,
, and
, we can satisfy the principle of relativity with linear
isotropic functions for
,
viz.
Note that
vanishes in equilibrium so that no entropy production occurs in
that state. The coefficients
C
and
B
in (89
,
90
) are functions of
e
and
n, or
and
T
. In fact, the entropy principle determines the
C
's fully in terms of the thermal equation of state
as follows
The
B
's in (90) are restricted by inequalities, viz.
All
B
's have the dimension
and we may consider them to be of the order of magnitude of the
collision frequency of the gas molecules.
In conclusion we may write the field equations in the form
where
must be inserted from (89
) and (91
). This set of equations represents the field equations of
extended thermodynamics. We conclude that extended thermodynamics
of viscous, heat-conducting gases is quite explicit - provided we
are given the thermal equation of state
- except for the coefficients
B
. These coefficients must be measured and we proceed to show
how.
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Speeds of Propagation in Classical and Relativistic
Extended Thermodynamics
Ingo Müller http://www.livingreviews.org/lrr-1999-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |