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Effective Flow of Incompressible Micropolar Fluid Through a System of Thin Pipes

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Abstract

In this paper, we consider the incompressible micropolar fluid flowing through a multiple pipe system via asymptotic analysis. Introducing the ratio between pipes thickness and its length as a small parameter \(\varepsilon\), we propose an approach leading to a macroscopic model describing the effective flow. In the interior of each pipe (far from the junction), we deduce that the fluid behavior is different depending on the magnitude of viscosity coefficients with respect to \(\varepsilon\). In particular, we prove the solvability of the critical case characterized by the strong coupling between velocity and microrotation. In the vicinity of junction, an interior layer is observed so we correct our asymptotic approximation by solving an appropriate micropolar Leray’s problem. The error estimates are also derived providing the rigorous mathematical justification of the constructed approximation. We believe that the obtained result could be instrumental for understanding the microstructure effects on the fluid flow in pipe networks.

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Notes

  1. In analogy with electrical circuits, we call it the Kirchhoff’s law.

  2. It remained unsolved in [8].

  3. Note that (23) is automatically satisfied.

  4. Same arguments can be applied in the other two cases as well.

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Acknowledgements

The first author of this work has been supported by the project GAČR 13-18652S. The second author of this work has been supported by the Croatian Science Foundation (scientific project 3955: Mathematical modeling and numerical simulations of processes in thin or porous domains). The authors would like to thank the referee for his/her helpful comments and suggestions which helped to greatly improve the paper.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Civil Engineering, Czech Technical University in Prague, Thákurova 7, 166 29, Prague 6, Czech Republic

    Michal Beneš

  2. Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10000, Zagreb, Croatia

    Igor Pažanin

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  1. Michal Beneš
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  2. Igor Pažanin
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Correspondence to Igor Pažanin.

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Beneš, M., Pažanin, I. Effective Flow of Incompressible Micropolar Fluid Through a System of Thin Pipes. Acta Appl Math 143, 29–43 (2016). https://doi.org/10.1007/s10440-015-0026-1

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  • DOI: https://doi.org/10.1007/s10440-015-0026-1

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