332 16 Constitutive Relations
and
(eμxeλx−eyμeyλ)ezλezν =−1
3
(exμexν−eμyeνy),lead to two coupled equations forΠ−andΠ 0 :
τM∂
∂tΠ−+
4
3
κετMΠ 0 +Π−=− 2 ηNewε, (16.99)τM∂
∂tΠ 0 +κετMΠ−+Π 0 = 0.Here the stationary solution is given byΠ−=−η−ε,Π 0 =−η 0 ε, with
η−= 2 ηNew(
1 −
4
3
(κετM)^2)− 1
,η 0 =−κετMη−, (16.100)provided that 4(κετM)^2 <3.
16.4 Viscosity and Alignment in Nematics
16.4.1 Well Aligned Nematic Liquid Crystals and Ferro Fluids
A viscous flow with a moderate shear rate does not affect the magnitude of the order
parameter of a nematic liquid crystal, cf. Sect.15.2.1. The direction of the director
n, however, is influenced by the flow geometry and by external fields. First, the case
is considered, where a magnetic field is applied, which is strong enough such that it
practically fixes the orientation of the director. On the other hand, it should not be so
strong, that it alters the order parameter. The tensornn determines the anisotropy
of the fluid. It is understood thatnis parallel or anti-parallel to the direction of the
applied magnetic fieldB=Bh, the sign ofnhas no meaning for nematics. The
symmetry considerations used for the viscosity coefficients of well aligned nematic
liquid crystals also apply to ferro-fluids in the presence a of strong magnetic field.
Ferro-fluidsare colloidal dispersions containing practically spherical particles with
permanent or induced magnetic dipoles [110].
Point of departure for the set up of the constitutive relations governing the friction
pressure tensor are the expression (16.42) for the entropy production, and the second
rank tensornn, or equivalentlyhh specifying the anisotropy of the unperturbed
state. The external field exerts a torque on the system. As a consequence, the antisym-
metric part of the pressure tensor is not zero. Compared with the ansatz (16.43), an
additional constitutive relation is needed and coupling terms between the symmetric
traceless and antisymmetric parts of the pressure tensor occur: