Paolo Musolino, Vladimir Mityushev
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 . Then for and close to 0 we denote by the longitudinal permeability. We are interested in studying the asymptotic behavior of as tends to 0. We analyze 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 can be written as the sum of a logarithmic term and a power series in . 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).
Submitted June 17, 2015. Published November 20, 2015.
Math Subject Classifications: 76D30, 76D05, 35J05, 35J25, 31B10, 45A05.
Key Words: Longitudinal permeability; asymptotic expansion; rectangular array; singularly perturbed domain; integral equations; logarithmic capacity.
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| Paolo Musolino |
Department of Mathematics
University of Padova, Italy
| Vladimir Mityushev |
Department of Computer Science and Computational Methods
Pedagogical University of Cracow, Poland
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