Tensors for Physics

366 17 Tensor Dynamics

17.4.1 Formulation of the Model

Here, the stress tensor rather than the pressure tensor is used. The symmetric traceless
partσμν of the stress tensor is written as

σμν =

√

2 Grefπμν+ 2 η∞Γμν, (17.39)

whereGrefis a reference shear modulus, also called Maxwell modulusGMorG.The
symmetric traceless tensorπμνis the dimensionless stress tensor,η∞is the second
Newtonian viscosity and

Γμν=∇μvν

is the symmetric traceless part of the deformation rate tensor. The generalized non-
linear Maxwell model is formulated for the dimensionless stress tensor [200]

∂

∂t

πμν− 2 εμλκωλπκν− 2 κΓμλπλν+τ 0 −^1 (Φμν−^20 Δπμν)=

√

2 Γμν,

Φμν=

∂

∂πμν

Φ. (17.40)

The relevant relaxation time is calledτ 0 and 0 is a characteristic length. The tensor
Φμνis the derivative of a potential functionΦwith respect toπμν. The standard
Maxwell model with the linear relaxation term corresponds toΦ =^12 Aπμνπμν
andΦμν=Aπμν, with a dimensionless coefficientA>0. In terms of the scalar
invariantsI 2 =πμνπμνandI 3 =

√

6 πμνπνλπλμ= 3

√

6 det(π ), cf. Sect.5.5and
(15.4), the ansatz for the potential, up to the sixth power inπ, is written as

Φ=

1

2

AI 2 −

1

3

BI 3 +

1

4

CI^22 +

1

5

DI 2 I 3 +

1

6

EI^32 +

1

6

FI^23. (17.41)

The dimensionless coefficientsA,B,C,D,E,Fare model parameters. The deriv-

atives ofI 2 andI 3 with respect toπμνare 2πμνand 3

√

6 πμλπλν, respectively.

17.4.2 Special Cases

The most widely studied special case isD=E=F=0 withA=A 0 ( 1 −T 0 /T)
andA 0 ,B,C>0.ThenthepotentialisanalogoustotheLandau-de Gennes potential
and the generalized nonlinear Maxwell model (17.40) is mathematically equivalent
to the dynamic equation for the alignment tensor. For a spatially homogeneous equi-

librium situation, a uniaxial stress tensorπμν=

√

3 / 2 πnμnν is found, where the