Closed-form evaluation of 2D static lattice sums

In the present paper, employing properties of the complete elliptic integrals of the first and second kind, we deduce closed-form formulae for the lattice sums and other new formulae. Applications to the effective properties of regular and random composites are discussed. We also discuss the Eisenstein summation method and the Rayleigh method used in computations

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 $\epsilon \neq 0$ 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)