16.3 Viscosity and Non-equilibrium Alignment Phenomena 331
tensor∇νvμis symmetric traceless and one has
∇νvμ =ε[
2 ezνezμ−(exνexμ+eνyeμy)]
= 3 εezνezμ. (16.95)By analogy to (16.90), here the symmetry adapted ansatz for the friction pres-
sure tensor is pμν = 2 ezμezνΠ 0. In the linear flow regime, where one has
pμν =− 2 ηNew∇νvμ, with the Newtonian viscosityηNew. In this case, the viscos-
ity coefficientη 0 , defined byΠ 0 =−η 0 ε,cf.(16.91), isη 0 = 3 ηNew. This coefficient
is also referred to asextensional viscosityorTrouton viscosity. The nonlinear case,
an equation governing the componentΠ 0 of the pressure tensor, follows from the
the Maxwell model (16.94). Due to ezμezλ ezλezν =^13 eμzeνz, the resulting equation
forΠ 0 is
τM∂
∂tΠ 0 − 2 κετMΠ 0 +Π 0 =− 3 ηNewε, ηNew=GτM. (16.96)For a stationary situation this leads to
Π 0 =−η 0 ε, η 0 = 3 ηNew( 1 − 2 κετM)−^1 , (16.97)provided that 2κετM<1. This inequality plays no role forκε <0, where the
viscosity is decreasing with an increasing magnitude of the deformation rate. For
κε >0, (16.97) yields an increasing viscosity and the stationary solution breaks
down at the finite deformation rateεlim=( 2 κ)−^1.
(ii) The planar biaxial extensional or compressional flow is considered with the
special geometryvx=εx,vy=−εy, andvz=0. Hereε=∂vx/∂x=−∂vy/∂yis
the extension or compression rate. The symmetry of the flow field is that of the biaxial
squeeze-stretch field as sketched in Fig.7.4. Again, the velocity gradient tensor∇νvμ
is symmetric traceless and one has
∇νvμ=ε(exνexμ−eyνeyμ). (16.98)By analogy to (16.90), here the symmetry adapted ansatz for the friction pressure
tensor ispμν=(exμexν−eμyeyν)Π−+ 2 eμzeνzΠ 0. Viscosity coefficientsη−andη 0 are
defined byΠ−=−η−εandΠ 0 =−η 0 ε. In the linear flow regime, where pμν =
− 2 ηNew∇νvμ applies, with the Newtonian viscosityηNew, one hasη−= 2 ηNew
andη 0 =0. For the planar biaxial flow, the Maxwell model (16.94), and the use of
(exμexλ−eyμeyλ)(exλexν−eyλeyν)=exνexμ+eyνeyμ =−ezνezμ