| Summary: | 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 $\epsilon$. Then for $\epsilon \neq 0$ and
close to $0$ we denote by $K_{II}[\epsilon]$ the longitudinal permeability.
We are interested in studying the asymptotic behavior of $K_{II}[\epsilon]$
as $\epsilon$ tends to $0$. We analyze $K_{II}[\epsilon]$ for $\epsilon$
close to $0$ by an approach based on functional analysis and potential theory,
which is alternative to that of asymptotic analysis. We prove that
$K_{II}[\epsilon]$ can be written as the sum of a logarithmic term and a
power series in $\epsilon^2$. Then, for small $\epsilon$, 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).
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