For the determination of the n fields u we need field equations - generally n of them - and these are based on the equations of balance of mechanics and thermodynamics. The generic form of these balance equations reads
The comma denotes partial differentiation with respect to
, and
is the
n
-vector of densities, while
is the
n
-vector of flux components. Thus
represents
n
four-fluxes, and
is the
n
-vector of productions.
Obviously the balance equations (1) are not field equations for the fields
, at least not in this form. They must be supplemented by
constitutive equations. These relate the four-fluxes
and the productions
to the fields
u
in a materially dependent manner. We write
and
denote the
constitutive functions
. Note that the constitutive quantities
and
at one event depend only on the values of
u
at that same event. In particular there is no dependence on
gradients and time derivatives of
.
If the constitutive functions
and
are explicitly known, we may eliminate
and
between the balance equations (1
) and the constitutive relations (2
) and obtain a set of explicit field equations for the fields
. These are quasilinear partial differential equations of first
order. Every solution of the field equations is called a
thermodynamic process
.
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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 |