Results of the 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 $A A$ , $AB A$ and $A h$ into their suggestive time-like and space-like components. We have

AA = nmU A AAB = t⟨AB⟩ + (p(n,e) + π)hAB + 1(U AqB + UBqA ) + e-U AU B (86 ) c2 c2 hA = hU A + ΦA,

and the components have suggestive meaning as follows

n : number density U A : velocity ⟨AB⟩ t : stress deviator p + π : pressure (87 ) qA : heat flux e : energy density h : entropy density A Φ : (non-convective) entropy flux.

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

We have defined $1 hAB = -2 UAU B − gAB c$ and $m$ is the molecular rest mass.

The decomposition (86 ) 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 $⟨AB⟩ t$ , the heat flux $qA$ and the dynamic pressure $π$ – 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

fugacity α and absolute temperature T,

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

where $˙$ and $′$ here and below denote differentiation with respect to $α$ and $ln T$ respectively.

If we restrict attention to a linear theory in $⟨AB ⟩ t$ , $A q$ , and $π$ , we can satisfy the principle of relativity with linear isotropic functions for $ABC A$ , $BC I$ viz.

2 AABC = (C01 + C π1π)U AU BU C + c6 (nm − C01 − C π1π )⋅ ⋅(gABU C + gBC UA + gCAU B) + C3(gABqC + gBC qA + gCAqB )− (89 ) − 6c2C3 (UAU BqC + U BU CqA + UC UAqB )+ ⟨AB ⟩ C ⟨BC⟩ A ⟨CA ⟩ B +C5 (t U + t U + t U ),

Note that $IBC$ vanishes in equilibrium so that no entropy production occurs in that state. The coefficients $C$ and $B$ in (89

, 90

) are functions of $e$ and $n$ , or $α$ and $T$ . In fact, the entropy principle determines the $C$ ’s fully in terms of the thermal equation of state $p = p(α, T)$ as follows

Γ ′ ∫ ˙p C01 = ---1- with Γ 1 = − 2c2T6 ---dT 2c2T T 7 ⌊ ⌋ − ¨p ˙p − ˙p′ Γ˙1 |⌈ ˙p − ˙p′ p′ − p′′ Γ ′1 − Γ 1 |⌉ 2 Γ˙ Γ ′ − Γ 5Γ Cπ1 = − -----⌊---1---1----1---3--2-⌋- c2T − ¨p ˙p − ˙p′ ˙Γ 1 |⌈p˙− p˙′ p′ − p′′ Γ ′1 − Γ 1|⌉ − p˙ − p′ 5Γ (91 ) 3 1 [ ] ˙p − ˙Γ 1 1 Γ 1 Γ 2 C3 = − 2T-[-------˙---] p˙′ − Γ 1 ′ p Γ 1 − Γ 1 -1-Γ-2 2 8∫ -1-˙ C5 = − 2T Γ with Γ 2 = 2c T T 3Γ 1dT. 1

The $B$ ’s in (90 ) are restricted by inequalities, viz.

All $B$ ’s have the dimension $1∕sec$ 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

where $BCA A$ must be inserted from (89

) and (91

). 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 $p = p(α,T )$ – except for the coefficients $B$ . These coefficients must be measured and we proceed to show how.