Asymptotic behavior of the longitudinal permeability of a periodic array of thin cylinders

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 $\epsilon \neq 0$ 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).