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Non-Darcian flow in fibre-reinforced biological tissues

  • Active Behavior in Soft Matter and Mechanobiology
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Abstract

Under suitable conditions, the motion of a fluid in a porous medium can be studied by assuming the validity of Darcy’s law. Since many biological tissues can be thought of as porous media, Darcy’s law is invoked in several biomechanical contexts, like the transport of the chemical species needed for the metabolism of tissue cells. Although Darcy’s law supplies physically sound results in many circumstances, there may be cases in which the dynamic behaviour of a biological fluid deviates from the Darcian one. The scope of this work is to analyse some possible consequences of such deviations, with emphasis on the fluid velocity and pressure, which, in turn, influence the health and correct functioning of the tissue cells. In particular, our study addresses the flow of an interstitial fluid through a fibre-reinforced tissue, in which the fibres are oriented statistically. We take articular cartilage as a representative tissue of this type, and study the deviation from Darcy’s law known as “Forchheimer’s correction”. Moreover, we introduce two models of tissue permeability, which lead to discrepant results when the fluid velocity is described by Darcy’s law. We show, however, that the discrepancies in the description of the flow can be reduced if Forchheimer’s correction is applied.

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Acknowledgements

This work has been supported in part by the Politecnico di Torino (Italy) [AG] and [MC], in part by the Fondazione Cassa di Risparmio di Torino (Italy), through the “La Ricerca dei Talenti” (“HR Excellence in Research”) programme [AG] and in part by the Natural Science and Engineering Research Council of Canada, through the NSERC Discovery Programme [SF]. Finally, we would like to acknowledge the contribution of Mr. Alberto Stracuzzi (former student of the Politecnico di Torino) to the material contained in Sect. 5.1 [45].

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Authors and Affiliations

  1. Department of Mathematical Sciences (DISMA) “G.L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Turin, Italy

    Alfio Grillo & Melania Carfagna

  2. Department of Mechanical and Manufacturing Engineering, The University of Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4, Canada

    Salvatore Federico

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  1. Alfio Grillo
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  2. Melania Carfagna
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Correspondence to Alfio Grillo.

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In memory of our master Prof. Gaetano Giaquinta (1945–2016).

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Grillo, A., Carfagna, M. & Federico, S. Non-Darcian flow in fibre-reinforced biological tissues. Meccanica 52, 3299–3320 (2017). https://doi.org/10.1007/s11012-017-0679-0

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