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

Paolo Musolino, Vladimir Mityushev

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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 KII [∈] the longitudinal permeability. We are interested in studying the asymptotic behavior of KII [∈] as ∈ tends to 0. We analyze KII [∈] 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 KII [∈] 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).

Original languageEnglish
Article numberA290
Pages (from-to)1-20
Number of pages20
JournalElectronic Journal of Differential Equations
Publication statusPublished - 20 Nov 2015


  • Asymptotic expansion
  • Integral equations
  • Logarithmic capacity
  • Longitudinal permeability
  • Rectangular array
  • Singularly perturbed domain


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