Tensors for Physics

16.3 Viscosity and Non-equilibrium Alignment Phenomena 313

withpλ=ελαβpαβ. The contributions to the entropy production(δδst)dvirrevassociated
with the flow velocityv=v(t,r)follows from the local Gibbs relation linking
the entropy density with the internal energy density and the time change of the
macroscopic kinetic energy which, in turn, is governed by the local conservation law
of the linear momentum, cf. Sect.7.4.3. The resulting expression is

ρ

m

T

(

δs

δt

)dv

irrev

=−(pνμ−Pδμν)∇νvμ,

and consequently, after decomposition into the irreducible parts,

ρ

m

T

(

δs

δt

)

irrev

=−

(

p ̃∇λvλ+ pνμ ∇νvμ +ωλpλ

)

,ωλ=

1

2

ελνμ∇νvμ.
(16.42)
The first and second terms are force-flux pairs involving tensors of ranks=0 and
=2. The discussion of the case=1 associated with the antisymmetric part of
the pressure tensor and the vorticityω, is postponed to Sect.16.3.5.
First, the attention is focussed on the symmetric pressure tensor. This is the case
insimple fluids, where the pressure tensor is symmetric, on account of its symmetric
kinetic and potential constituents, cf. (12.94) and (12.107). It also applies to more
complex molecular fluids when the hydrodynamic processes described by the con-
stitutive relations are slow compared on the time scale over which the antisymmetric
part of the pressure tensor relaxes to zero.
In an isotropic fluid and in the absence of any external fields, the constitutive laws
governing the viscous behavior are, cf. (7.55)

p ̃=−ηV∇λvλ, pνμ=− 2 η∇νvμ.

This ansatz is in accord with the Curie principle and it obeys the condition of positive
entropy production when both the shear viscosityηand the volume viscosityηVare
non-negative.
The general scheme describing the viscous behavior of a fluid with s symmetric
pressure tensor is

pμν =− 2 ημνμ′ν′∇μ′vν′−ζμν(^20 )∇λvλ,

p ̃=−ζμν(^02 )∇μvν−ηV∇λvλ. (16.43)

Here,ημνμ′ν′is the fourth rank shear viscosity tensor,ηVis the volume viscosity and
the symmetric traceless coupling tensorsζ..(..)obey the Onsager relation

ζμν(^20 )=ζμν(^02 ). (16.44)