We consider a Newtonian fluid flowing at low Reynolds numbers
along a spatially periodic array of cylinders of diameter proportional
to a small nonzero parameter ϵ. Then for ϵ=0 and
close to 0 we denote by KII[ϵ] the longitudinal permeability.
We are interested in studying the asymptotic behavior of KII[ϵ]
as ϵ tends to 0. We analyze KII[ϵ] for ϵ
close to 0 by an approach based on functional analysis and potential theory,
which is alternative to that of asymptotic analysis. We prove that
KII[ϵ] can be written as the sum of a logarithmic term and a
power series in ϵ2. Then, for small ϵ, we provide an
asymptotic expansion of the longitudinal permeability in terms of the sum
of a logarithmic function of the square of the capacity of the cross section
of the cylinders and a term which does not depend of the shape of the unit
inclusion (plus a small remainder)