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), Taber LA (2004)) , a new model for PEM with the variable volume is developed in multidimensional case (the model in the 1D space case is presented in Nonlinear Sci. Numer. Simulat. 132 (2024) 107905 ). Governing equations of the model are constructed using the continuity equations, which reflect the well-known physical laws. The deformation vector is specified using the Terzaghi effective stress tensor. In the two-dimensional space case, the model is studied by analytical methods. The radially-symmetric case is studied in details. It is shown how correct boundary conditions in the case of PEM in the form of a ring and an annulus are constructed. As a result, boundary-value problems with a moving boundary describing the ring (annulus) deformation are constructed. The relevant nonlinear boundary-value problems are analytically solved in the stationary case. In particular, the analytical formulae for unknown deformations and an unknown radius of the annulus are derived. Illustrative plots for parameters, which are typical for the tumour tissue deformation, are presented as well.

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