Tensors for Physics

12.4 Anisotropic Pair Correlation Function and Static Structure Factor 227

It depends on the geometry of applications whether an expansion with respect to
spherical or to Cartesian tensors is preferred. In the case of the distortion of the pair
correlation function caused by a simple shear flow pertaining to the plane Couette
geometry, cf. Fig.7.6, the Cartesian version (12.109) is more appropriate.

12.4.5 Shear-Flow Induced Distortion of the Pair Correlation

The time change of the pair densityn(^2 )(r 1 ,r 2 )involves the flow term

[

vμ(r 1 )

∂

∂r 1 μ

+vμ(r 2 )

∂

∂r 2 μ

]

n(^2 )(r 1 ,r 2 ).

Whenn(^2 ) = n^2 g, with constant number densityn, depends on the difference
variabler=r 1 −r 2 only, this expression reduces to

(

vμ(r 1 )−vμ(r 2 )

) ∂

∂rμ

n^2 g(r).

Consider a linear flow profile, cf. Sect.7.2.2,

vμ(r)=rν(∇νvμ)=rν(ενμλωλ+γμν). (12.116)

Thevorticityωλand thedeformation rateorshear ratetensorγμνare given by

ωλ=

1

2

ελκτ∇κvτ,γμν=∇νvμ. (12.117)

Here∇νvν=0, i.e. a divergence-free flow is assumed.
The flow term in the kinetic equation forgcan be split into two contributions
involvingωλandγμν, which induce a local rotation and deformation, respectively,
in pair-space. This equation reads

∂

∂t

g+ωλLλg+γμνLμνg+D(g)= 0 (12.118)

with the differential operators

Lλ=ελκτrκ

∂

∂rτ

, Lμν= rμ∂∂rτ. (12.119)

The vector operatorLλ,cf.(7.80), is the generator of the infinitesimal rotation. The
symmetric traceless second rank tensor operatorLμνis associated with an infin-
itesimal volume conserving deformation. The damping termD(g)guarantees the