Tensors for Physics

16.3 Viscosity and Non-equilibrium Alignment Phenomena 321

dJλ

dt

+τrot−^1 (Jλ−θωλ)= 0 ,τrot−^1 = 2

m

ρθ

ηrot, (16.65)

with therotational relaxation timeτrot. In the absence of external torques, the angu-
lar momentumJrelaxes toθω, in a time span long compared withτrot. Then the
average rotational velocitywmatches the vorticityωand the antisymmetric part of
the pressure tensor vanishes.

Side Remarks:
(i) Spin Particles

An equation of the type (16.65) which relates, in a stationary situation, the average
internal angular momentum with the vorticity, also applies for particles with spin,
even for electrons. TheBarnett effect, viz. the electron spin polarization and the
ensuing magnetization caused by the rotation of a metal, is an experimental evidence
for this phenomenon [117]. Here the question arises: what is the relevant moment of
inertiaθin this case? Heuristic considerations [21] and a derivation from a general-
ized quantum mechanical Boltzmann equation with a nonlocal collision term [116]
show:θis determined by the mass of the electron times its thermal de Broglie wave
length squared.

(ii) Polymer Coils

Letmbe the mass andri,i= 1 ,...Nbe the position vectors of the monomers of
a polymer chain molecule. Its angular momentum isL=

∑

imr

i× ̇ri. In a liquid,

the polymer molecule forms a coil which is spherical, on average, when the system
is in thermal equilibrium. When the liquid is flowing, the polymer coil is stretched
and it undergoes nonuniform rotations. A remarkable, though approximate, relation
between the average angular velocitywand the shape of the polymer coil was put
forward by Cerf [118]. The shape of the coil is expressed in terms of the radius of
gyration tensorGμν, cf. Sect.5.3.2. More specifically, the long time average〈L〉of
the angular momentum is assumed to be equal tom〈

∑

ir

i×v(ri)〉wherevis the

flow velocity field of the liquid. For a plane Couette flow with the velocity inx- and
its gradient iny-direction, this relation corresponds to

〈Lz〉=−γm〈Gyy〉,

whereγis the shear rate. The resulting average rotational velocitywzis obtained by
dividing〈Lz〉by the effective moment of inertiam(〈Gxx〉+〈Gyy〉), thus

wz=−γ

〈Gyy〉

〈Gxx〉+〈Gyy〉

=ωz

(

1 −

〈Gxx〉−〈Gyy〉

〈Gxx〉+〈Gyy〉

)

. (16.66)

For a weak flow, where the polymer coil retains its effectively spherical shape, one
has〈Gxx〉=〈Gyy〉and consequentlywz=−^12 γ =ωz, i.e. the average angular
velocity matches the vorticity. For larger shear rates, the coil is stretched in the flow