Tensors for Physics

228 12 Integral Formulae and Distribution Functions

approach ofgto its equilibrium valuegeq=geq(r)when the flow is switched off.
A specific expression forDwas proposed by Kirkwood [53], which is analogous to
what Smoluchowski had used for the diffusion in the presence of an external poten-
tial. For this reason, the kinetic equation is referred to asKirkwood-Smoluchowski
equation. The potential used by Kirkwood is an effective potentialφeffdetermined
by the equilibrium pair correlation functiongeqaccording toφeff =−kBTlngeq.
A generalization and applications are discussed in [54]. The simplerelaxation time
approximation

D(g)=τ−^1 (g−geq),

suffices to analyze the essential features associated with the shear flow induced
distortion ofg.Hereτis a structural relaxation time, sometimes also calledMaxwell
relaxation time. Withg=geq+δgandLλgeq=0, the kinetic equation (12.118)
reduces to

∂

∂t

δg+ωλLλδg+γμνLμνδg+τ−^1 δg=−γμνLμν(geq− 1 ), (12.120)

whereLμνgeq=Lμν(geq− 1 )was used.
Next, a stationary situation is considered, where the time derivative ofgvanishes.
Furthermore,gis written as

g=geq+δg(^1 )+δg(^2 )+... ,

whereg(k)is of the orderkin the shear rate. In first order, the kinetic equation
(12.120) yields

δg(^1 )≡−τγμνLμνgeq=−τγμνrμrνr−^1 g′eq, (12.121)

where the prime denotes the differentiation with respect tor. Comparison with
(12.109) shows that

gμν=−τγμνrg′eq, (12.122)

in this approximation.
The contribution of second order in the shear rate is given by

δg(^2 )=−τ(ωλLλ+γμνLμν)δg(^1 )=τ^2 (ωλLλ+γμνLμν)γκσLκσgeq.

Due to (12.121), the term involving the vorticity yields

2 τ^2 εμκλωλγκνrμrνr−^1 g′eq.