Tensors for Physics

416 Appendix: Exercises...

dμdνdλEνEλ=dμ(d·E)^2 −

1

5

[

dμE^2 + 2 Eμ(d·E)

]

was used, cf. (9.5).

16.2 Acoustic Birefringence(p.326)
Sound waves cause an alignment of non-spherical particles in fluids. The ensuing
birefringence is calledacoustic birefringence.Use(16.74)to compute the sound-
induced alignment tensor for the velocity fieldv=v 0 k−^1 kcos(k·r−ωt)wherek
andωare the wave vector and the frequency of the sound wave, v 0 is the amplitude.
Hint: Use the complex notation vμ∼exp[i(k·r−ωt)]and aμν= ̃aμνexp[i(k·
r−ωt)]to solve the inhomogeneous relaxation equation. Discuss the meaning of
the real and imaginary parts ofa ̃μν.

With the recommended ansatz, (16.74) yields

(−iωτa+ 1 )a ̃μν=−i

√

2 τapv 0 k−^1 kμkν.

Consequently, the alignment is determined by

a ̃μν=−i

√

2 τapv 0 kk̂μk̂νA(ω),

where

A(ω)=

1

1 −iωτa

=

1 +iωτa

1 +(ωτa)^2

.

The real part ofA(ω), corresponding to the imaginary part ofa ̃μνdescribes the part
of the alignment which is in phase with the velocity gradient, i.e. it is proportional to
sin(k·r−ωt). The imaginary part ofA(ω)is linked with the part of the alignment
which is in phase with the velocity, i.e. it is proportional to cos(k·r−ωt).

16.3 Diffusion of Perfectly Oriented Ellipsoids(p.337)
In the nematic phase, the fluxjof diffusing particles with number densityρobeys
the equation

jμ=−Dμν∇νρ, Dμν=D‖nμnν+D⊥(δμν−nμnν),

whereD‖andD⊥are the diffusion coefficients for the flux parallel and perpendicular
to the directorn.
Use the volume conserving affine transformation model for uniaxial particles, cf.
Sect.5.7.2,to derive

D‖=Q^4 /^3 D 0 , D⊥=Q−^2 /^3 D 0

for perfectly oriented ellipsoidal particles with axes ratio Q.Here D 0 is the reference
diffusion coefficient of spherical particles.