Tensors for Physics

326 16 Constitutive Relations

16.3.7 Heat-Flow Birefringence

Also a heat fluxqcan give rise to birefringence. By analogy to the Maxwell effect
(16.67), the constitutive relation for theheat-flow birefringenceis

εμν =− 2 βq∇μqν= 2 βqλ∇μ∇νT. (16.80)

Hereβqis theheat-flow birefringence coefficient. The second equality in (16.80),
involving the second spatial derivative of the temperature fieldTfollows fromqν=
−λ∇νTwhereλis the heat conductivity. The existence of the effect (16.80)was
predicted [62] and calculated [127] for rarefied molecular gases. First measurements
were presented in [128], see also [17].
In gases, the flow birefringence, just like the viscosity, does not depend on the
density, whereas the heat-flow birefringence is inversely proportional to the density.
Theoretical considerations [129] predict the existence of heat-flow birefringence also
in dense fluids. Experiments in a nematic glass [130] are a manifestation of this effect.
In mixtures, a preferential alignment can also be caused by the gradient of a diffu-
sion fluxj. The resulting birefringence is referred to asdiffusio-birefringence[131].

16.3.8 Visco-Elasticity

Elasticity,whichisareversibleprocess,isatypicalpropertyofsolids.Theirreversible
viscous flow behavior is typical for fluids. On a short time scale or for shear rates
varying with high frequencies, however, also fluids show elasticity. TheMaxwell
modelfor the symmetric traceless friction pressure tensor pμν,viz.

τM

∂

∂t

pμν +pμν =− 2 η∇μvν,η=GτM, (16.81)

is a prototype for the description of thevisco-elastic behavior.HereτMis theMaxwell
relaxation time,ηis the shear viscosity andGis the high frequency shear modulus.

Notice that∇μvν =∂∂tuμν. Thus for fast varying processes, whereτM|∂∂t pμν|

|pμν|,(16.81) reduces to∂∂tpμν =− 2 G∂∂tuμν or−pμν = σμν = 2 Guμν,
which corresponds to the constitutive law (16.21) for elasticity.
For a periodic velocity gradient proportional to exp[−iωt], the linear equation
(16.81) implies that the friction pressure has the same dependence on the frequency

ωand the timet. In this case (16.81) can be written as pμν =− 2 η(ω)∇μvν with
the complex frequency dependent viscosity

η(ω)=η( 1 −iωτM)−^1 =η′(ω)+iη′′(ω), (16.82)

η′(ω)=η

1

1 +(ωτM)^2

,η′′(ω)=η

ωτM

1 +(ωτM)^2

.