344 16 Constitutive Relations
is made. Then the entropy production is given by
−
ρ
m
T
(
δs
δt
)( 2 )
irrev
=pνμ∇νvμ+
ρ
m
kBTΦμν
[(
δaμν
δt
)
irrev
+ 2 κ∇μvκaκν
]
=
[
pνμ+ 2 κ
ρ
m
kBTΦμκaκν
]
∇μvν+
ρ
m
kBTΦμν
(
δaμν
δt
)
irrev
.
(16.133)
WithΦμνand(
√
2 ρmkBT)−^1 (pνμ + 2 κmρkBTΦμκaκν)chosen as fluxes, and
(δaδμνt )irrevand
√
2 ∇νvμ as forces, as suggested in [171], the constitutive laws for
the second rank tensors now are
−Φμν=τa
(
δaμν
δt
)
irrev
+τap
√
2 ∇νvμ,
−
(√
2
ρ
m
kBT
)− 1 (
pνμ+ 2 κ
ρ
m
kBTΦμκaκν
)
=τpa
(
δaμν
δt
)
irrev
+τp
√
2 ∇νvμ.
(16.134)
As before, the quantitiesτ..are relaxation time coefficients where the subscriptsa
andprefer to “alignment” and “pressure”. The non-diagonal coefficients obey the
Onsager symmetry relation, cf. (16.72),τap=τpa. Positive entropy production is
guaranteed by the inequalitiesτa>0,τp>0,τaτp>τap^2.
Use of the first of the (16.133)in(16.69) yields the inhomogeneous relaxation
equation
daμν
dt
− 2 εμλκωλaκν− 2 κ∇μvκaκν+τa−^1 Φμν=−τa−^1 τap
√
2 ∇νvμ.(16.135)
Apart from the last term on the right hand side of (16.130), the phenomenological
equation corresponds to the equation derived from the Fokker-Planck equation, when
τa−^1 and
√
2 τa−^1 τapare identified withν 2 and−R,asin(16.128).
The symmetric traceless part of the pressure tensor, as it follows from the con-
stitutive relations, is given by pνμ =− 2 ηiso∇νvμ + pνμ
align
, withηiso =
ηNew( 1 −
τap^2
τaτp),ηNew=
ρ
mkBTτp,cf.(16.84) and (16.85), where the friction pressure
associated with the alignment is now
pνμ
align
=
ρ
m
kBT
(√
2
τap
τa
Φμν− 2 κΦμνaκν
)
. (16.136)
A remark on the antisymmetric part of the pressure tensor is in order. Prior to putting
the average angular velocitywμequal to the vorticityωμ, the entropy production
involving pseudo-vectors is proportional topμ(wμ−ωμ), wherepμis the pseudo