Tensors for Physics

16.3 Viscosity and Non-equilibrium Alignment Phenomena 323

is made. The term involving the vorticity describes the time change due to a rotation
with an average angular velocity equal to the vorticity. This holds true, when the anti-
symmetric part of the pressure tensor vanishes, see the previous section. The operator
d
dt−^2 ω×is referred to asco-rotational time derivative.Theterm(

δaμν
δt )irrevis the
time change due to irreversible processes which occurs in the entropy production.
The part associated with second rank tensors is now

ρ

m

T

(

δs

δt

)( 2 )

irrev

=−

[

pνμ ∇νvμ+

ρ

m

kBTaμν

(

δaμν

δt

)

irrev

]

. (16.70)

Withaμνand(

√

2 ρmkBT)−^1 pνμchosen as fluxes, and(δaδμνt )irrevand

√

2 ∇νvμ as
forces, constitutive laws for the second rank tensors are

−aμν=τa

(

δaμν

δt

)

irrev

+τap

√

2 ∇νvμ, (16.71)

−

(√

2

ρ

m

kBT

)− 1

pνμ=τpa

(

δaμν

δt

)

irrev

+τp

√

2 ∇νvμ.

Here the quantitiesτ..are relaxation time coefficients where the subscriptsaandp
refer to “alignment” and “pressure”. The non-diagonal coefficients obey the Onsager
symmetry relation
τap=τpa. (16.72)

Positive entropy production is guaranteed by the inequalities

τa> 0 ,τp> 0 ,τaτp>τap^2. (16.73)

Use of the first of the (16.71)in(16.69) yields the inhomogeneous relaxation equation

daμν

dt

− 2 εμλκωλaκν+τa−^1 aμν=−τa−^1 τap

√

2 ∇νvμ. (16.74)

The term involving the vorticity gives rise to effects nonlinear in the shear rate. When
these are disregarded, and for the stationary case, where the time derivative vanishes,
(16.74) leads to flow alignment

aμν=−τap

√

2 ∇νvμ, (16.75)

and consequently. The flow birefringence coefficient is given by

β=−

εa

√

2

τap. (16.76)

Clearly, the coupling coefficientτapis essential for the flow birefringence.