Tensors for Physics

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