Tensors for Physics

16.3 Viscosity and Non-equilibrium Alignment Phenomena 317

∇μδP=−kμ= 2 ∇νημνμ′ν′∇μ′vν′. (16.55)

For the geometry considered here, this relation is equal to

∇μδP=−kμ=γ′e

y

νημνμ′ν′e

y

μ′e

x

ν′, (16.56)

where∂γ/∂y=γ′is the derivative of the shear rate. The ratio between thelon-
gitudinal pressure gradient exμ∇μδPandγ′defines the effective shear viscosity

ηPois=ηPois(h), is equal to the effective viscosity for the Couette geometry (16.51),
viz.

exμ∇μδP/γ′=exμeyνημνμ′ν′eyμ′eνx′=ηPois=ηCouette(h).

Similarly, the effectivenormal viscosityandtransverse viscositycoefficientsηnorm
andηtranscan be defined via thenormal pressure gradient eyμ∇μδPand thetransverse
pressure gradient−eμz∇μδP. These relations are

e

y

μ∇μδP=γ′e

y

μe

y

νημνμ′ν′e

y

μ′e

x

ν′ =γ

′ηnorm, (16.57)

ηnorm=hxhy

[

η(^0 )

(

3 h^2 y− 1

)

+η(^1 +)

(

2 − 4 h^2 y

)

+η(^2 +)

(

h^2 y− 1

)]

+hy

[

2 η(^1 −)h^2 y+η(^2 −)

(

1 −h^2 y

)]

,

ezμ∇μδP=γ′ezμeyνημνμ′ν′eyμ′exν′ =γ′ηtrans, (16.58)

ηtrans=hxhz

[

3 η(^0 )h^2 y+η(^1 +)

(

1 − 4 h^2 y

)

+η(^2 +)

(

h^2 y− 1

)]

+hz

[

η(^1 −)

(

1 − 2 h^2 y

)

+η(^2 −)

(

h^2 y− 1

)]

.

In contradistinction to the longitudinal viscosity, which is positive, the normal and
transverse effective viscosity coefficients may have either sign. The contributions to
ηnormandηtrans, which involve the coefficientsη(^1 −),η(^2 −), change sign whenhis
replaced by−h.
As above, in connection with the effective shear viscosity, the labels 1, 2 ,3are
used for the field parallel to thex-,y- andz-directions. From (16.57)to(16.58)
follows

ηnorm 1 = 0 ,ηnorm 2 = 2 η(^1 −),η 3 norm= 0 ,

ηtrans 1 = 0 ,ηtrans 2 = 0 ,ηnorm 3 =η(^1 −)−η(^2 −).