12.4 Anisotropic Pair Correlation Function and Static Structure Factor 229
The term involving the productLμνLκσgives contributions to the isotropic part
ofgand to its second and fourth rank irreducible parts. More specifically, the term
γμνLμνγκσLκσgeqis rewritten and computed as
γμνLμνγκσrκrσr−^1 g′eq= 2 γμνγκνrκrμr−^1 g′eq+γμνγκσrνrμrκrσr−^1 (r−^1 g′eq)′.
Due torκrμ= rκrμ +^13 r^2 δμνand with the help of relation (11.59), the second
order distortionδg(^2 )can be decomposed into the partsδg(^2 ), associated with tensors
of ranks= 0 , 2 ,4. The term involving the vorticity contributes toδg( 22 )only. Thus
one has
g=geq+δg(^1 )+δg(^2 )+... , δg(^2 )=δg( 02 )+δg( 22 )+δg( 42 ), (12.123)with
δg( 02 )=τ^2 γμνγμν
(
2
3
rg′eq+2
15
r^3 (r−^1 g′eq)′)
,
δg( 22 )= 2 τ^2 εμκλωλγκνrμrνr−^1 g′eq+τ^2 γμκγκνrνrμ
(
2 r−^1 g′eq+4
7
r(r−^1 g′eq)′)
,
δg( 42 )=τ^2 γμνγκσrνrμrκrσr−^1 (r−^1 g′eq)′. (12.124)
Consequences of corresponding relations for a plane Couette flow are presented
in [55], see also the Exercise12.4. Symmetry considerations for this simple flow
geometry are discussed next.
The shear-induced distortion of the structure of a liquid or of a colloidal disper-
sion can be detected in scattering experiments where the static structure factor, cf.
Sect.12.4.8is analyzed. Direct observations of the distorted pair correlation func-
tion is possible inconfocal microscopyexperiments [56] and, in particular inNon-
Equilibrium Molecular Dynamics(NEMD) computer simulations [57, 58].
12.4.6 Plane Couette Flow Symmetry
A plane Couette flow geometry is considered with the velocityv=v(r)in the
x-direction and its gradient in they-direction, cf. (7.28), thus
vμ=exμe
y
νrν=yexμ, (12.125)