Tensors for Physics

314 16 Constitutive Relations

Positive entropy production requires that

ηsymμνμ′ν′=

1

2

(ημνμ′ν′+ημ′ν′μν)

is positive definite, and thatηV≥0, as stated before. For the isotropic fluid without
external fields, the shear viscosity tensor is proportional to the isotropic fourth rank
tensorΔμν,μ′ν′, thus one has

ημνμ′ν′=ηΔμν,μ′ν′,η> 0 ,

andζμν(^20 )=ζμν(^02 )=0, in this case. Next, the more specific expressions for the
viscosity tensors are discussed for applied magnetic and electric fields.

16.3.2 Influence of Magnetic and Electric Fields

AmagneticfieldinfluencestheviscosityviatheLorentzforce,whenthefluidcontains
mobile electric charges, or via the precession of magnetic moments in electrically
neutral fluids. Examples for the latter substances are ferro-fluids, i.e. colloidal solu-
tions containing particles with permanent or induced magnetic moments [110–112],
as well as gases of rotating molecules [17]. The influence of orienting fields on the
viscous behavior of liquid crystals deserves a separate discussion in Sect.16.4.1.
Application of an electric fieldEon a fluid containing particles with permanent or
induced electric dipole moments also renders the viscosity anisotropic. The resulting
geometric symmetries are alike. The parity of theBandEfields, however, are
different. This implies that terms of odd power inEviolate parity invariance and are
identical to zero, unless one is dealing with chiral substances.
Consider first an isotropic fluid which is subjected to a magnetic fieldB=Bh,
wherehis a constant unit vector. The viscosity coefficients have to be in accord
with this uniaxial symmetry. The obvious ansatz for the coupling tensors occurring
in (16.43)is

ζμν(^20 )=ζμν(^02 )=ζhμhν, (16.45)

with a scalar phenomenological coefficientζ =ζ(B), which is an even function
ofB.
There are multiple ways to construct the fourth rank shear viscosity tensors in
accord with symmetry of the physical situation. First, the fourth rank projection
tensors of Sect.14.2.2are employed as basis tensors. By analogy to the construction
of the rotation tensors, cf. Sect.14.2.3, the viscosity tensor is written as, cf. [42],

ημνμ′ν′=

∑^2

m=− 2

η(m)P(μν,μm)′ν′. (16.46)