318 16 Constitutive Relations
When the field points in the direction of the bisector between thex- andy-axes, viz.
forh=(ex+ey)/
√
2, as in (16.51), the (16.57) implies2 η 45 norm=η(^0 )−η(^2 +)+(
2 η(^1 −)−η(^2 −))
/
√
2.
The corresponding expression for the transverse pressure gradient withhparallel to
the bisector between thex- andz-axes, now forh=(ex+ez)/
√
2, is denoted by
ηtrans 45 ′ and given by
2 ηtrans 45 ′ =3
2
η(^0 )+η(^1 +)−η(^2 +)+√
2
(
η(^1 −)−η(^2 −))
.
Effective viscosities for other directions of the field, e.g. forh=(ex+ey+ez)/
√
3
can be inferred from (16.57)to(16.58).
16.3.4 Senftleben-Beenakker Effect of the Viscosity
The influence of a magnetic field on the transport properties of electrically neutral
gasesisreferredtoasSenftleben-Beenakker effect.Thisphenomenonwasfirstnoticed
around 1930 for the paramagnetic gases O 2 and NO [113]. About thirty years later,
Beenakker and coworkers [114] demonstrated that the influence of a magnetic field
also occurs in diamagnetic gases like N 2. In fact, the effect is typical for all gases
composed of rotating molecules [17], which have a rotational magnetic moment of
the order of the nuclear magnetonμN. The field-induced change of the transport
properties is small, but relative changes can be detected with a high sensitivity.
The Senftleben-Beenakker effect of the viscosity is mainly due to the collisional
coupling between the kinetic part of friction the pressure tensorpμν and the tensor
polarizationaμνT, see Sect.13.6.4. The equations governing the dynamics of these
tenors can be derived from a kinetic equation referred to asWaldmann-Snider equa-
tion[17, 115]. It is a generalized Boltzmann equation for the distribution function
operatorf = f(t,r,p,J), where the the positionrand the linear momentump
are treated as classical variables, the internal angular momentumJis a quantum
mechanical operator, as discussed in Sect.13.6. Furthermore, the collision processes
are treated quantum mechanically. In the presence of the magnetic fieldB=Bh,the
kinetic equation contains a commutator of the distribution operatorfwith the rele-
vant Hamilton operatorHB=−grotμNJ·B, wheregrotthe a gyromagnetic factor
specific for particular molecules. The quantityωB=(grotμNB)/is the frequency
with which the internal angular momentum precesses about the applied field.
Next, for simplicity of notation, pμν ≡ pμνkin
andaμν≡aTμνare used. Fur-
thermore, the ideal pressure of a gas with the number densitynand the temperature
Tis denoted byp 0 =nkBT. The resultingtransport-relaxation equationsare