The laws of Navier–Stokes and Fourier

5.4 The laws of Navier–Stokes and Fourier

It is instructive to identify the classical constitutive relations of Navier–Stokes and Fourier of TIP within the scheme of extended thermodynamics. They are obtained from (93

, 94

, 95

) by the first step of the so-called Maxwell iteration which proceeds as follows: The $nth$ iterate

(n) ( (n) (n) (n) ) I AB or π ,tAB, q A

results from

$A A ,A$ =	0
$(n−A1)AB ,B$ =	0
$(n−1) A ABC ,C$ =	$(n) I AB$

with the

initiation

agreement

$(E) A A,A$ =	0
$(E)AB A ,B$ =	0
$(EA )ABC ,C$ =	$(1I)AB$

where $(E ) A$ are equilibrium values.

A little calculation provides the first iterates for dynamic pressure, stress deviator and heat flux in the form

with

⌊ ′ ˙ ⌋ | − ¨p p˙− p˙ Γ 1 | ⌈ ˙p − ˙p′ p′ − p′′ Γ ′1 − Γ 1⌉ 1 ˙Γ 1 Γ ′1 − Γ 1 5Γ 2 λ = -----π-----[------------3′-]---- 2T B 1 − ¨p p˙− p˙ p˙− p˙′ p′ − p′′ --1--- μ = 2T B3 Γ 1[ ] ˙p − Γ˙ ′ 1′ κ = ---1-----p-Γ 1-−-Γ1- 2T 2Bˆ4 ˙p

These are the relativistic analogues of the classical phenomenological equations of Navier–Stokes and Fourier. $λ$ , $μ$ and $κ$ are the bulk viscosity, the shear viscosity and the thermal conductivity respectively; all three of these transport coefficients are non-negative by the entropy inequality.

The only essential difference between the equations (97 , 98 , 99 ) and the non-relativistic phenomenological equations is the acceleration term in (99 ). This contribution to the Fourier law was first derived by Eckart, the founder of thermodynamics of irreversible processes. It implies that the temperature is not generally homogeneous in equilibrium. Thus for instance equilibrium of a gas in a gravitational field implies a temperature gradient, a result that antedates even Eckart.

We have emphasized that the field equations of extended thermodynamics should provide finite speeds. Below in Section 5.6 we shall give the values of the speeds for non-degenerate gases. In contrast TIP leads to parabolic equations whose fastest characteristic speeds are always infinite. Indeed, if the phenomenological equations (97 , 98 , 99 ) are introduced into the conservation laws (93 , 94 ) of particle number, energy and momentum, we obtain a closed system of parabolic equations for $n$ , $UA$ and $e$ . This unwelcome feature results from the Maxwell iteration; it persists to arbitrarily high iterates.