Non-Darcian flow in fibre-reinforced biological tissues

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.