5.2 Constitutive Theory5 Relativistic Thermodynamics of Gases. 5 Relativistic Thermodynamics of Gases.

5.1 Thermodynamic Processes in Viscous, Heat-Conducting Gases 

The objective of thermodynamics of viscous, heat-conducting gases is the determination of the 14 fields

  equation1188

in all events tex2html_wrap_inline3331 . Both tex2html_wrap_inline4263 and tex2html_wrap_inline4265 are Lorentz tensors. The energy-momentum tensor is assumed symmetric so that it has 10 independent components.

For the determination of these fields we need field equations and these are formed by the conservation laws of particle number and energy-momentum, viz.

  equation1198

  equation1203

and by the equations of balance of fluxes

  equation1208

tex2html_wrap_inline4267 is the flux tensor - it is completely symmetric -, and tex2html_wrap_inline4269 is its production density. We assume

  equation1216

so that among the 15 equations (79Popup Equation, 80Popup Equation, 81Popup Equation) there are 14 independent ones, which is the appropriate number for 14 fields.

The components of tex2html_wrap_inline4263 and tex2html_wrap_inline4265 have the following interpretations

  equation1226

The motivation for the choice of equations (79Popup Equation, 80Popup Equation, 81Popup Equation), and in particular (81Popup Equation), stems from the kinetic theory of gases. Indeed tex2html_wrap_inline4263 and tex2html_wrap_inline4265 are the first two moments in the kinetic theory and tex2html_wrap_inline4279 and tex2html_wrap_inline4281 are the first two equations of transfer. Therefore it seems reasonable to take further equations from the equation of transfer for the third moment tex2html_wrap_inline4267 and these have the form (81Popup Equation). In the kinetic theory the two conditions (82Popup Equation) are satisfied.

The set of equations (79Popup Equation, 80Popup Equation, 81Popup Equation) must be supplemented by constitutive equations for the flux tensor tex2html_wrap_inline4267 and the flux production tex2html_wrap_inline4269 . The generic form of these relations in a viscous, heat-conducting gas reads

  equation1260

If the constitutive functions tex2html_wrap_inline4289 and tex2html_wrap_inline4291 are known, we may eliminate tex2html_wrap_inline4267 and tex2html_wrap_inline4295 between (79Popup Equation, 80Popup Equation, 81Popup Equation) and (84Popup Equation) and obtain a set of field equations for tex2html_wrap_inline4297, tex2html_wrap_inline4299 . Each solution is called a thermodynamic process .

It is clear upon reflection that this theory, based on (79Popup Equation, 80Popup Equation, 81Popup Equation) and (84Popup Equation), provides a special case of the generic structure explained in Chapter  2 .



5.2 Constitutive Theory5 Relativistic Thermodynamics of Gases. 5 Relativistic Thermodynamics of Gases.

image Speeds of Propagation in Classical and Relativistic Extended Thermodynamics
Ingo Müller
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