Tensors for Physics

16.3 Viscosity and Non-equilibrium Alignment Phenomena 325

exchanged. This means the relaxation time “matrix” formed by theτ..coefficients is
the reciprocal of the relaxation frequencyν..matrix. Here, reciprocal relaxation time
coefficients have a microscopic interpretation since they are expressed in terms of
collision integrals involving the binary scattering amplitude [17, 62, 64]. To be more
specific, consider a stationary situation in the absence of aBfield. Then the second
equation of (16.59) implies

aμν=−

(√

2 p 0

)− 1

νa−^1 νap pμν

and, with pμν =− 2 η∇μvν,cf.(16.60), one obtains

aμν= 2 η

(√

2 p 0

)− 1

νa−^1 νap∇μvν.

Thus in this case the flow birefringence coefficient is given by [62]

β=−

εaT

√

2

τap,τap=−νap(νpνa)−^1 ( 1 −Apa)−^1. (16.79)

Thisrelationissimilarto(16.76),butwithεaTinsteadofεaandthecouplingcoefficient
τapis expressed in terms of the relaxation frequenciesν... Whereas the Maxwell
effect in colloidal dispersions, molecular liquids and polymeric fluids [123] has been
studied experimentally for over a century, the flow birefringence in molecular gases
was first measured by F. Baas in 1971 [124], see also [17, 125].
The flow birefringence is the manifestation of across effect: a viscous flow causes
an alignment. There is areciprocal effect: a non-equilibrium alignment gives rise to

an extra contribution paνμto the symmetric traceless pressure tensor [126]. The
alignment, in turn, influences the flow properties. This back-coupling underlies the
influence of a magnetic field on the viscosity in molecular gases as discussed above,
and the nonlinear flow behavior in molecular liquids and colloidal dispersions of
non-spherical particles, as treated in Sect.16.3.9.

16.2 Exercise: Acoustic Birefringence
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 notationvμ∼exp[i(k·r−ωt)]andaμν∼exp[i(k·r−ωt)]
to solve the inhomogeneous relaxation equation, then determine the real and the
imaginary part ofaμν.