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
A A : particle flux vector (78 ) AAB : energy-momentum tensor
in all events D x. Both A A and AB A 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.

AA,A = 0 (79 )
AB A ,B = 0 (80 )

and by the equations of balance of fluxes

ABC AB A ,C = I . (81 )
AABC is the flux tensor – it is completely symmetric –, and IAB is its production density. We assume
IAA = 0 and AABB = c2AA (82 )
so that among the 15 equations (79View Equation, 80View Equation, 81View Equation) there are 14 independent ones, which is the appropriate number for 14 fields.

The components of AA and AAB have the following interpretations

A0 : c ⋅ rest mass density, Aa : flux of rest mass, A00 : energy density, 0a (83 ) A : 1∕c ⋅ energy flux, Aa0 : c ⋅ momentum density, Aab : momentum flux.
The motivation for the choice of equations (79View Equation, 80View Equation, 81View Equation), and in particular (81View Equation), stems from the kinetic theory of gases. Indeed AA and AAB are the first two moments in the kinetic theory and AA = 0 ,A and AB A ,B = 0 are the first two equations of transfer. Therefore it seems reasonable to take further equations from the equation of transfer for the third moment AABC and these have the form (81View Equation). In the kinetic theory the two conditions (82View Equation) are satisfied.

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

ABC ˆABC M MN A = A (A ,A ) (84 ) IAB = ˆIAB (AM ,AMN ).
If the constitutive functions ˆ A and ˆ I are known, we may eliminate ABC A and BC I between (79View Equation, 80View Equation, 81View Equation) and (84View Equation) and obtain a set of field equations for M A, MN A. Each solution is called a thermodynamic process.

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


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