16.3 Viscosity and Non-equilibrium Alignment Phenomena 329
This type of nonlinearity is to be distinguished from the nonlinear flow effects, e.g.
the turbulence, resulting from the convective termv·∇in the local linear momentum
balance equation (7.52).
Within a phenomenological description, the nonlinear viscous behavior of a plane
Couette flow is characterized by three material coefficients, which depend on the
imposed shear rateγ. The first of these coefficients is thenon-Newtonian viscosity
η(γ )defined via the ratio of theyx-component of the pressure or stress tensor and
the shear rateγ=∂vx/∂y:
σyx=−pyx=η(γ ) γ. (16.89)It is understood thatη(γ )approaches the Newtonian, i.e. shear rate independent,
viscosityηNewin the limit of small shear rates. When nonlinear effects play no role,
it is common practice to use the symbolηinstead ofηNew.
The appropriate ansatz for the friction pressure tensor adapted to the plane Couette
symmetry is
pμν = 2 exμe
y
νΠ++(e
x
μex
ν−ey
μey
ν)Π−+^2 ezμeνzΠ 0. (16.90)Viscosity coefficientsηi, withi=+,−,0 are defined by
Π+=−η+γ, Π−=−η−γ, Π 0 =−η 0 γ. (16.91)The coefficientη+is the non-Newtonian viscosityη(γ ), the coefficientsη−andη 0
characterize the normal pressure differencespxx−pyy= 2 Π−andpzz−^12 (pxx+
pyy)= 2 Π 0. Equivalently, and this is common practice in the rheological literature,
the normal pressure differencespxx−pyyand pyy−pzzare used and referred
to as “first” and “second” normal pressure differences. The corresponding stress
differences, denoted byN 1 andN 2 , are defined by
N 1 =σxx−σyy=pyy−pxx=Ψ 1 γ^2 , N 2 =σyy−σzz=pzz−pyy=Ψ 2 γ^2.
(16.92)
The viscometric functionsΨ 1 andΨ 2 are related to the viscosity coefficientsη−and
η 0 according to
Ψ 1 γ= 2 η−,Ψ 2 γ=− 2 η 0 −η−. (16.93)
In contradistinction toη+, which is positive in order to guarantee positive entropy
production, the coefficientsη−,η 0 , and alsoΨ 1 ,Ψ 2 may have either sign.
The normal pressure differencespxx−pyyandpyy−pzzare zero in the linear flow
regime, however, they are non-zero at higher shear rates. In general, the viscometric
functions depend on the shear rate. When the nonlinearity of the friction pressure
is analytic inγ, i.e. when it can be expressed in terms of a power series inγ, one
haspxy∼γandpxx−pyy∼γ^2 ,pyy−pzz∼γ^2 ,forγ →0. Consequently,