A mathematical model for transport in poroelastic materials

Date: Thursday 07 March 2024

Abstract

Fluid and solute transport in poroelastic materials (biological tissue is a typical example) is studied. Mathematical modeling of such transport is a complicated problem because the specimen volume and its form (under special conditions) might change due to swelling or shrinking and the transport processes are nonlinearly linked. The tensorial character of the variables adds also substantial complication in investigation of the fluid and solute transport in poroelastic materials (PEM). Therefore, developing and solving adequate mathematical models is an important and still open problem.

Using modern foundations of the poroelastic theory (see books by Loret B and Simoes FMF (2017), Coussy O. (2010)) the basic equations of the model were constructed. We consider PEM that consists of pores with fluid and solid material (matrix); molecules (e.g. glucose) are dissolved in fluid phase and transported through pores. We assume that fluid is incompressible and there are no internal sources/sinks. The governing equations of the model consist of continuity equations reflecting classical physical laws (the extended Darcy’s law, Newtonian law). The stress-strain relationship was described by the Terzaghi effective stress tensor . The 1D version of this model was studied earlier in [Cherniha R. et al , Symmetry 2020 12, 396. https://doi.org/10.3390/sym12030396; Commun. Nonlinear Sci. Numer. Simulat. 2024 https://doi.org/10.1016/j.cnsns.2024.107905) ]. In this talk, a 2D model is presented. The model consists of six nonlinear PDEs and that is too complicated for deriving substantial analytical results. The Lie symmetry analysis is applied in order to reduce the dimensionality of the model. As a result, the radially-symmetric model is derived. An example describing fluid transport in PEM of the form of a circle is studied in detail.

This is a joint work with Joanna Stachowska-Pietka (IBIB of PAS, Warsaw) and Jacek Waniewski (IBIB of PAS, Warsaw).