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 $epsilon eq 0$ 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).

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 not equal 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).