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Quantitative estimates for stress concentration of the Stokes flow between adjacent circular cylinders

by H. Ammari and H. Kang and D. Kim and S. Yu

(Report number 2020-14)

Abstract
When two inclusions with high contrast material properties are located close to each other in a homogeneous medium, stress may become arbitrarily large in the narrow region between them. In this paper, we investigate such stress concentration in the two-dimensional Stokes flow when inclusions are the two-dimensional cross sections of circular cylinders of the same radii and the background velocity field is linear. We construct two vector-valued functions which completely capture the singular behavior of the stress and derive an asymptotic representation formula for the stress in terms of these functions as the distance between the two cylinders tends to zero. We then show, using the representation formula, that the stress always blows up by proving that either the pressure or the shear stress component of the stress tensor blows up. The blow-up rate is shown to be one over the square root of the distance between the two cylinders.

Keywords: stress concentration, blow-up, Stokes flow, Stokes system, singular functions, bi-polar coordinates

@Techreport{AKKY20_887,
  author = {H. Ammari and H. Kang and D. Kim and S. Yu},
  title = {Quantitative estimates for stress concentration of the Stokes flow between adjacent circular cylinders},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-14},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-14.pdf },
  year = {2020}
}

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