5.5 Specific Results for a 5 Relativistic Thermodynamics of Gases. 5.3 Results of the Constitutive

5.4 The Laws of Navier-Stokes and Fourier 

It is instructive to identify the classical constitutive relations of Navier-Stokes and Fourier of TIP within the scheme of extended thermodynamics. They are obtained from (93Popup Equation, 94Popup Equation, 95Popup Equation) by the first step of the so-called Maxwell iteration which proceeds as follows: The tex2html_wrap_inline4405 iterate

displaymath1514

results from

tabular1530

  equation1574

where tex2html_wrap_inline4423 are equilibrium values.

A little calculation provides the first iterates for dynamic pressure, stress deviator and heat flux in the form

  equation1580

  equation1587

  equation1594

with

displaymath1601

These are the relativistic analogues of the classical phenomenological equations of Navier-Stokes and Fourier. tex2html_wrap_inline4425, tex2html_wrap_inline4427 and tex2html_wrap_inline4429 are the bulk viscosity, the shear viscosity and the thermal conductivity respectively; all three of these transport coefficients are non-negative by the entropy inequality.

The only essential difference between the equations (97Popup Equation, 98Popup Equation, 99Popup Equation) and the non-relativistic phenomenological equations is the acceleration term in (99Popup Equation). This contribution to the Fourier law was first derived by Eckart, the founder of thermodynamics of irreversible processes . It implies that the temperature is not generally homogeneous in equilibrium. Thus for instance equilibrium of a gas in a gravitational field implies a temperature gradient, a result that antedates even Eckart.

We have emphasized that the field equations of extended thermodynamics should provide finite speeds. Below in Section  5.6 we shall give the values of the speeds for non-degenerate gases. In contrast TIP leads to parabolic equations whose fastest characteristic speeds are always infinite. Indeed, if the phenomenological equations (97Popup Equation, 98Popup Equation, 99Popup Equation) are introduced into the conservation laws (93Popup Equation, 94Popup Equation) of particle number, energy and momentum, we obtain a closed system of parabolic equations for n, tex2html_wrap_inline4433 and e . This unwelcome feature results from the Maxwell iteration; it persists to arbitrarily high iterates.



5.5 Specific Results for a 5 Relativistic Thermodynamics of Gases. 5.3 Results of the Constitutive

image Speeds of Propagation in Classical and Relativistic Extended Thermodynamics
Ingo Müller
http://www.livingreviews.org/lrr-1999-1
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