Tensors for Physics

17.4 Nonlinear Maxwell Model 367

scalar stressπisπ=B/( 2 C)±

√

B^2 /( 4 C^2 )−A/C, provided thatA<B∗/( 4 C),
otherwise one hasπ=0. The caseπ=0, in the absence of a flow corresponds to a
solid state with a yield stress. At the transition temperature, here calledTc, one has
π=πc≡^23 CB, by analogy to the Landau-de Gennes theory for the isotropic-nematic
phase transition. Scaled variables can be introduced in analogy to the treatment in
Sect.16.4.7. In particular,πμνis expressed in units ofπc. For convenience, the
scaled stress tensor is also denoted byπμν. ThenΦμνoccurring in the Maxwell
model equation assumes the form

Φμν=A∗πμν− 3

√

6 πμλπλν + 2 πμνπλκπλκ,

A∗=AA−c^1 , Ac=

2 B^2

9 C

. (17.42)

Furthermore, just as in Sect.16.4.7, the time is expressed in units of a reference
time, here calledτc, the shear rate and the vorticity are made dimensionless by
multiplication withτc =τ 0 A−c^1. A model parameter equivalent to the tumbling
parameter of nematics isλK= 2 /(

√

3 πc). The nonlinear Maxwell model equation
leads to non-stationary periodic and even chaotic solutions of ‘stick-slip’ type, when
thestick-slip parameter, defined by

λ= 2

(√

3 πcπeq

)− 1

+κ/ 3 ,πeq=

1

4

(

3 +

√

9 − 8 A∗

)

, A∗< 1. 125 ,(17.43)

is less than 1. Forλ>1, nonlinear flow behavior with shear thinning and shear
thickening are found, even forκ=0.
Anexampleforthenon-steadyresponseofthesystemtoanappliedsteadyshear,of
stick-slip type, is presented in Fig.17.5for the model parametersπc= 1. 0 ,κ= 0 .0,
η∞= 0. 1 /A∗, and forA∗= 0. 25 , 0. 35 , 0 .42, from top to bottom, at the shear rate
3 .2. Such a behavior is strikingly similar to that one seen in solid friction processes.
Notice that the friction force is proportional to the shear stress. When the plastic

Fig. 17.5Shear stress
versus time for stick-slip
motion

0 20 40 60 80 100

t

0.6

0.8

1

1.2

1.4

1.6

1.8

sh

ea

r stress