16.3 Viscosity and Non-equilibrium Alignment Phenomena 315
The 5 complex viscosity coefficientsη(m)have the propertiesη(m)=(η(−m))∗and
η(^0 ), as well as the real parts ofη(±^1 )andη(±^2 )are positive.
An alternative, but equivalent representation with real viscosity coefficients is
ημν,μ′ν′=η(^0 )Pμν,μ(^0 ) ′ν′ (16.47)+
∑^2
m= 1[
η(m+)(
Pμν,μ(m)′ν′+Pμν,μ(−m)′ν′)
+η(m−)i(
P(μν,μm)′ν′−Pμν,μ(−m)′ν′)]
.
The coefficientsη(m+)=(η(m)+η(−m))/2 andη(m−)=(η(m)−η(−m))/ 2 iare the
real and imaginary parts of the coefficientsη(m). The three non-negative coefficients
η(^0 ),η(^1 +)andη(^2 +)are even functions ofB. The two coefficientsη(^1 −)andη(^2 −)
may have either sign and they are odd functions ofB. The latter two coefficients are
also referred to astransverse viscosity coefficients.
The shear viscosity tensor, as given by (16.46)or(16.47) obeys the symmetry
property
ημνμ′ν′(h)=ημ′ν′μν(−h). (16.48)
The deGroot-Mazur viscositiesηdGMi of [108] are related to the coefficientsη(m)by
η(^0 )=η 1 dGM,η(^1 )=ηdGM 2 +iηdGM 5 ,η(^2 )= 2 ηdGM 2 −η 1 dGMi−ηdGM 4 .(16.49)For an electric fieldEacting on an electrically neutral fluid containing particles with
permanent or induced electric dipole moments the ansatz (16.47) can be used with
the axial vectorhreplaced by a polar unit vector parallel to the electric field, but
the Hall-effect like coefficientsη(^1 −)andη(^2 −)are zero, due to parity conservation.
In fluids containing chiral particles, however, these coefficients can be non-zero. A
pseudo-scalar, characterizing the chirality, occurring as a factor in the Hall-effect
like terms, ensures that the parity is still conserved.
16.3.3 Plane Couette and Plane Poiseuille Flow
To elucidate the meaning of the viscosities introduced in (16.46) and (16.47), a simple
plane Couette flow is considered first. Again, the velocity is in thex-direction and
its gradient in they-direction. Then the velocity gradient tensor∇μvν =γe(μy)e(νx)
with the shear rateγ=∂vx/∂y, is a tensor with the same symmetry as considered in
(14.62) and (14.63). An effective shear viscosityηCouette(h), which depends on the
direction ofh, is defined by
pyx=−ηCouette(h)γ , ηCouette(h)= 2 e(μy)eν(x)ημνμ′ν′(h)eμ(y′)e(νx′). (16.50)