Thermodynamic processes in viscous, heat-conducting 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

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.

and by the equations of balance of fluxes

$AABC$ is the flux tensor – it is completely symmetric –, and $IAB$ is its production density. We assume

so that among the 15 equations (79

, 80

, 81

) 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 (79

, 80

, 81

), and in particular (81

), 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 (81

). In the kinetic theory the two conditions (82

) are satisfied.

The set of equations (79 , 80 , 81 ) 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

If the constitutive functions $ˆ A$ and $ˆ I$ are known, we may eliminate $ABC A$ and $BC I$ between (79

, 80

, 81

) and (84

) 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 (79 , 80 ,81 ) and (84 ), provides a special case of the generic structure explained in Section 2.