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

We consider a Newtonian fluid owing at low Reynolds numbers along a spatially periodic array of cylinders of diameter proportional to a small nonzero parameter ∈. Then for ∈ 6≠ 0 and close to 0 we denote by K_II [∈] the longitudinal permeability. We are interested in studying the asymptotic behavior of K_II [∈] as ∈ tends to 0. We analyze K_II [∈] for ∈ 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 [∈] can be written as the sum of a logarithmic term and a power series in ∈². Then, for small ∈, 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).