5.4 The Laws of Navier-Stokes 5 Relativistic Thermodynamics of Gases. 5.2 Constitutive Theory

5.3 Results of the Constitutive Theory 

No matter how much a person may be conditioned to think relativistically, he will appreciate the decomposition of the four-tensors tex2html_wrap_inline4263, tex2html_wrap_inline4265 and tex2html_wrap_inline3425 into their suggestive time-like and space-like components. We have

  equation1340

and the components have suggestive meaning as follows

  equation1351

At least this is how n through tex2html_wrap_inline4327 are to be interpreted in the rest frame of the gas.

We have defined tex2html_wrap_inline4329 and m is the molecular rest mass.

The decomposition (86Popup Equation) 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 tex2html_wrap_inline4333, the heat flux tex2html_wrap_inline4335 and the dynamic pressure tex2html_wrap_inline3395 - 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

displaymath1373

because these two variables can be measured - at least in principle. Also tex2html_wrap_inline3779 and T are the natural variables of statistical thermodynamics which provides the thermal equation of state in the form tex2html_wrap_inline4351 . The transition between the new variables tex2html_wrap_inline4353 and the old ones (n, e) can be effected by the relations

  equation1377

where tex2html_wrap_inline4357 and tex2html_wrap_inline4359 here and below denote differentiation with respect to tex2html_wrap_inline3779 and tex2html_wrap_inline4363 respectively.

If we restrict attention to a linear theory in tex2html_wrap_inline4333, tex2html_wrap_inline4335, and tex2html_wrap_inline3395, we can satisfy the principle of relativity with linear isotropic functions for tex2html_wrap_inline4267, tex2html_wrap_inline4295 viz.

  equation1389

  equation1409

Note that tex2html_wrap_inline4295 vanishes in equilibrium so that no entropy production occurs in that state. The coefficients C and B in (89Popup Equation, 90Popup Equation) are functions of e and n, or tex2html_wrap_inline3779 and T . In fact, the entropy principle determines the C 's fully in terms of the thermal equation of state tex2html_wrap_inline4351 as follows

  equation1423

The B 's in (90Popup Equation) are restricted by inequalities, viz.

  equation1478

All B 's have the dimension tex2html_wrap_inline4397 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

  equation1483

  equation1487

  equation1495

where tex2html_wrap_inline4399 must be inserted from (89Popup Equation) and (91Popup Equation). 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 tex2html_wrap_inline4351 - except for the coefficients B . These coefficients must be measured and we proceed to show how.



5.4 The Laws of Navier-Stokes 5 Relativistic Thermodynamics of Gases. 5.2 Constitutive Theory

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